Jan

01/08/24
Michael Celentano  UC Berkeley
Debiasing in the inconsistency regime
AbstractIn this talk, I will discuss semiparametric estimation when nuisance parameters cannot be estimated consistently, focusing in particular on the estimation of average treatment effects, conditional correlations, and linear effects under highdimensional GLM specifications. In this challenging regime, even standard doublyrobust estimators can be inconsistent. I describe novel approaches which enjoy consistency guarantees for lowdimensional target parameters even though standard approaches fail. For some target parameters, these guarantees can also be used for inference. Finally, I will provide my perspective on the broader implications of this work for designing methods which are less sensitive to biases from highdimensional prediction models.

01/09/24
Junichiro Matsuda  Kyoto University
Algebraic connectedness and bipartiteness of quantum graphs
AbstractQuantum graphs are a noncommutative analogue of classical graphs related to operator algebras, quantum information, quantum groups, etc. In this talk, I will give a brief introduction to quantum graphs and talk about spectral characterizations of properties of quantum graphs. We introduce connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms, and these properties have algebraic characterizations in the same way as classical cases. We also see the equivalence between bipartiteness and twocolorability of quantum graphs defined by two notions of graph homomorphisms: one respects adjacency matrices, and the other respects edge spaces.
This talk is based on arXiv:2310.09500.

01/09/24
Sam Spiro
Sidorenko Hypergraphs and Random Tur\'an Numbers
AbstractLet $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an $F$free subgraph of the random $r$uniform hypergraph $G_{n,p}^r$. Following recent work of Conlon, Lee, and Sidorenko, we prove nontrivial lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $F$ is not Sidorenko. This connection between Sidorenko's conjecture and random Tur\'an problems gives new lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $F$ is not Sidorenko, and further allows us to bound how ``far'' from Sidorenko an $r$graph $F$ is whenever upper bounds for $\mathrm{ex}(G_{n,p}^r,F)$ are known. This is joint work with Jiaxi Nie.

01/09/24
Runjie Hu  Stony Brook University
Galois symmetry on manifolds
AbstractHow to understand the Galois group of Qbar over Q? We want to analyze its action on the nonsingular complex varieties defined over finite extensions of Q. This action preserves the underlying etale homotopy type but permutes the manifold structures over it. In 1970, Sullivan proposed that there is an abelianized Galois symmetry on higher dimensional simplyconnected TOP manifolds by the Adams conjecture and it is compatible with the Galois symmetries on varieties. It is still an ongoing project to describe this mysterious Galois symmetry in a more geometric way by branched coverings. Indeed, this agrees with Grothendieck's discussion of dessin d'enfants on Riemann surfaces in the 1980's. I will report our ongoing works on a generalization to higher dimensions.

01/10/24
Prof. Gian Maria Dall’Ara  Istituto Nazionale di Alta Matematica "Francesco Severi"
An uncertainty principle for the dbar operator
AbstractI will present a rather elementary inequality and discuss its application to dbar equations with weights on the whole complex Euclidean space and to subelliptic estimates for the dbarNeumann problem.
The latter is joint work with Samuele Mongodi (Univ. MilanoBicocca, Italy).

01/10/24
Prof. Brendon Rhoades  UC San Diego
Increasing subsequences, standard bases, and shadow play
AbstractAn {\em increasing subsequence} of a permutation $w \in S_n$ is a sequence of numbers $1 \leq i_1 < \cdots < i_k \leq n$ such that $w(i_1) < \cdots < w(i_k)$. Increasing subsequences have appeared in various guises in combinatorics, probability, and representation theory. We present an algebraic interpretation in terms of a quotient ring inspired by a problem in cryptography. A link between standard monomial bases and Viennot's `shadow line' construction for the Schensted correspondence will play a key role.

01/10/24
Prof. Weixia Zhu  University of Vienna
Deformation of CR structures and Spectral Stability of the Kohn Laplacian
AbstractThe interplay between deformation of complex structures and stability of spectrum for the complex Laplacian on compact complex manifolds was studied extensively by Kodaira and Spencer in the 1950s.
In this talk, we will discuss analogous problems for complex manifolds with boundaries and for compact CR manifolds. This talk is based on joint work with Howard Jacobowitz and Siqi Fu.

01/11/24
Pieter Spaas  University of Copenhagen
Local HilbertSchmidt stability
AbstractWe will introduce a local notion of HilbertSchmidt stability (HSstability), partially motivated by the recent introduction of local permutation stability by Bradford. We will discuss some basic properties, and then establish a local character criterion for local HSstability of amenable groups, by analogy with the character criterion for HSstability of Hadwin and Shulman. We will then discuss further examples of (flexible versions of) local HSstability. Finally, we show that infinite sofic (resp. hyperlinear) property (T) groups are never locally permutation (resp. HS) stable, answering a question by Lubotzky. This is based on joint work with Francesco FournierFacio and Maria Gerasimova.

01/11/24
Dmitriy Drusvyatskiy  Department of Mathematics, University of Washington
Optimization for largescale learning: beyond smoothness and convexity
AbstractEstimation and learning algorithms are dramatically improving our capacity to extract information from massive datasets, with impressive consequences for technology and society at large. Although these algorithms have had widespread empirical success, we have yet to find a coherent mathematical foundation that can explain why these algorithms succeed on such a wide array of problems. The challenge is that the two assumptions that underpin classical optimization theorysmoothness and convexityrarely hold in contemporary settings. Nonetheless, simple optimization algorithms often do succeed, and over the last few years, I have studied when and why this happens. In this talk, I will survey some recent work in this area covering optimization theory, algorithms, and applications in signal processing and machine learning. In the process, we will encounter a surprisingly rich array of mathematical tools spanning nonsmooth analysis, semialgebraic geometry, and high dimensional probability and statistics.

01/12/24
Jessica Fintzen  Bonn
Representations of padic groups and Hecke algebras
AbstractRepresentations of padic groups and Hecke algebras Abstract: An explicit understanding of the category of all (smooth, complex) representations of padic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of representations of padic groups decomposes into subcategories, called Bernstein blocks, which are indexed by equivalence classes of so called supercuspidal representations of Levi subgroups. In this talk, I will give an overview of what we know about an explicit construction of supercuspidal representations and about the structure of the Bernstein blocks. In particular, I will discuss a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara in which we show that general Bernstein blocks are equivalent to much better understood depthzero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras and allows to reduce a lot of problems about the (category of) representations of padic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

01/16/24
Dr. Jorge GarzaVargas  California Institute of Technology
Finite free amalgamated convolutions: Towards a unified theory for proving root bounds
AbstractBetween 2013 and 2015, Marcus, Spielman and Srivastava wrote a sequence of papers where they famously solved the KadisonSinger problem, proved the existence of infinitely many Ramanujan graphs of any fixed degree, and derived strong restricted invertibility results. With the goal of putting their results (and the results of other follow up work) under the same umbrella, we introduce amalgamated finite free probability, which is a framework that draws connections between real stable polynomials and free probability. This is joint work with Nikhil Srivastava. 
01/16/24
Prof. Jonathan Novak  UC San Diego
From Graph Theory to YangMills Theory via Math 202B
AbstractThere are many interesting matrices associated to graphs. We all know about the adjacency matrix and the Laplacian, the basic matrices of spectral graph theory. The distance matrix is another interesting one  it was famously shown by Graham and Pollack that distance determinants of trees depend only on the number of vertices. The characteristic polynomial of distance matrices of trees was further studied by Graham and Lovasz, who found many interesting properties. Recently, graph theorists have begun to consider "exponential distance matrices" of graphs, obtained by taking the entrywise exponential of the usual distance matrix, and have proved some basic theorems on their eigenvalues for simple families of graphs. Taking a less myopic view of the mathematical landscape quickly reveals that exponential distance matrices appeared some thirty years ago in quantum physics, when Zagier explicitly evaluated the determinant of the exponential distance matrix of the CoxeterCayley graph of the symmetric group as the main step in proving the existence of a Hilbert space representation of deformed commutation relations interpolating between bosons and fermions. I will describe parallel results for the HurwitzCayley graph of the symmetric group and explain their relation to gaugestring/dualities in YangMills theory. As in Zagier's study, the main tools come from discrete harmonic analysis, aka the character theory of finite groups, and some basic aspects of symmetric function theory also play an important role. From a pedagogical perspective, the moral of the story is that it's good to imbibe some algebra with your combinatorics, and plain old matrices just don't cut it.

01/16/24
Prof. Ruobing Zhang  Princeton University
Metric geometric aspects of Einstein manifolds
AbstractThis lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics.
My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress in the field. If time permits, I will propose several important open questions

01/17/24
Linghao Zhang  UCSD
Polynomial Optimization Over Unions of Sets

01/17/24
Prof. Adrian Gonzalez Casanova  UC Berkeley
(Markov) Duality
AbstractHeuristically, two stochastic processes are dual if one can find a function to study one process by using the other. Implicitly, this technique can be traced back to the work of Blaise Pascal. Explicitly, it has been studied in different contexts, including interacting particle systems, and it is a crucial concept in population genetics.
Additionally, we will explore the duality between theoretical and applied mathematics. Specifically, we will examine instances in which theoretical probability is employed to study biological problems and situations where biological questions inspire interesting mathematical models. This discussion will encompass examples from population genetics, experimental evolution, and public health. 
01/17/24
Prof. Or Hershkovits  Hebrew University of Jerusalem
Mean curvature flow in spaces with positive cosmological constant
AbstractIn this talk, I will describe an approach of using Lorentzian mean curvature flow (MCF) to probe "expanding universes" (such as, presumably, ours) with matter that is assumed to be attracted to matter (formally, this assumption is called the "strong energy condition")Assuming 2dimensional symmetry, I will explain how the mean curvature flow can be used to show that such universes become asymptotic, in some sense, to the maximally symmetric such universe  de Sitter space. This proves a special case of the de Sitter no hair conjecture of Hawking and Gibbons.Unfortunately, the early universe did not support such twodimensional symmetry, rendering the above mentioned result physically insignificant. As a first step for removing the above symmetry assumption, I will illustrate a condition, natural in the above context, such that any local graphical mean curvature flow (without symmetry) in de Sitter space satisfying that condition converges to a certain "universal flow".Effort will be made to make the talk accessible to the wide mathematical audience. In particular, no "physics reasoning" will be involved. This is based on a joint work with Creminelli, Senatore and Vasy, and on a joint work with Senatore. 
01/18/24
Francis Wagner  Ohio State University
TBA

01/18/24
Eugenia Rosu  Leiden
A higher degree Weierstrass function
AbstractThe Weierstrass pfunction plays a great role in the classic theory of complex elliptic curves. A related function, the Weierstrass zetafunction, is used by Guerzhoy to construct preimages under the xioperator of newforms of weight 2, corresponding to elliptic curves. In this talk, I will discuss a generalization of the Weierstrass zetafunction and an application to harmonic Maass forms. More precisely, I will describe a construction of a preimage of the xioperator of a newform of weight k for k>2. This is based on joint work with C. AlfesNeumann, J. Funke and M. Mertens.

01/18/24
Dr. Robert Webber  Caltech
Randomized matrix decompositions for faster scientific computing
AbstractTraditional numerical methods based on expensive matrix factorizations struggle with the scale of modern scientific applications. For example, kernelbased algorithms take a data set of size $N$, form the kernel matrix of size $N x N$, and then perform an eigendecomposition or inversion at a cost of $O(N^3)$ operations. For data sets of size $N \geq 10^5$, kernel learning is too expensive, straining the limits of personal workstations and even dedicated computing clusters. Randomized iterative methods have emerged as a faster alternative to the classical approaches. These methods combine randomized exploration with information about which matrix structures are important, leading to significant speed gains.
In this talk, I will review recent developments concerning two randomized algorithms. The first is "randomized block Krylov iteration", which uses an array of random Gaussian test vectors to probe a large data matrix in order to provide a randomized principal component analysis. Remarkably, this approach works well even when the matrix of interest is not lowrank. The second algorithm is "randomly pivoted Cholesky decomposition", which iteratively samples columns from a positive semidefinite matrix using a novelty metric and reconstructs the matrix from the randomly selected columns. Ultimately, both algorithms furnish a randomized approximation of an N x N matrix with a reduced rank $k << N$, which enables fast inversion or singular value decomposition at a cost of $O(N k^2)$ operations. The speedup factor from $N^3$ to $N k^2$ operations can be 3 million. The newest algorithms achieve this speedup factor while guaranteeing performance across a broad range of input matrices.

01/19/24
Dr. Iacopo Brivio  Harvard University
Anti Iitaka conjecture in positive characteristic
AbstractGiven a fibration of complex projective manifolds $f:X\rightarrow Y$ with general fiber $F$, if the stable base locus of $K_X $ is vertical then a theorem of Chang establishes the inequality $\kappa(K_X)\leq \kappa(K_Y) +\kappa(K_F)$. In this talk I am going to discuss a generalization of this result to fibrations in positive characteristic satisfying certain tameness conditions. This is based on a joint project with Marta Benozzo and ChiKang Chang.

01/19/24
Dr. Christopher Eur  Harvard
Geometry of independence
AbstractMatroids combinatorially abstract the ubiquitous notion of "independence" in various contexts such as linear algebra and graph theory. Recently, an algebrogeometric perspective known as "combinatorial Hodge theory" led by June Huh produced several breakthroughs in matroid theory. We first give an introduction to matroid theory in this light. Then, we introduce a new geometric model for matroids that unifies, recovers, and extends various results from previous geometric models of matroids. We conclude with a glimpse of new questions that further probe the boundary between combinatorics and algebraic geometry. Joint works with Andrew Berget, Alex Fink, June Huh, Matt Larson, Hunter Spink, and Dennis Tseng.

01/22/24
Subha Maity  University of Michigan, Department of Statistics
Investigations of algorithmic biases caused by underrepresentation of minority groups
AbstractThe problem of algorithmic bias, where machine learning algorithms reflect biases that are prevalent in their training datasets, is widely recognized as a major concern. In this talk, I will discuss two of my projects related to algorithmic biases that are caused by underrepresentation of minority groups. In the first project, we demonstrate that when learning representations from standard contrastive learning methods, the representations of minority groups merge with the representations of certain similar majority groups. We refer to this phenomenon as representation harm and demonstrate that it leads to allocation harms in downstream classification tasks. In the second project, we investigate whether enforcing group fairness is aligned with improving model performance. In light of the longheld belief that enforcing fairness comes at the cost of reduced model performance, we present an alternative perspective on the problem. In cases where the machine bias is due to the underrepresentation of minority groups, we show that enforcing fairness is often in line with improving model performance on a balanced test dataset. Furthermore, we derive necessary and sufficient conditions for such an alignment.

01/22/24
Harold Blum  Utah
Moduli of Fano varieties and Kstability
AbstractAlgebraic geometry is the study of shapes defined by polynomial equations called algebraic varieties. One natural approach to study them is to construct a moduli space, which is a space parameterizing such shapes of a given type (e.g. algebraic curves). After surveying this topic, I will focus on the problem of constructing moduli spaces parametrizing Fano varieties, which are a class of positively curved complex manifolds that form one of the three main building blocks of varieties in algebraic geometry. While algebraic geometers once considered this problem intractable due to various pathologies that occur, it has recently been solved using Kstability, which is an algebraic definition introduced by differential geometers to characterize when a Fano variety admits a KahlerEinstein metric.

01/23/24
Dr. Junhwi Lim  Vanderbilt University
An index for quantum cellular automata on fusion spin chains
AbstractThe index for 1D quantum cellular automata (QCA) was introduced to measure the flow of the information by Gross, Nesme, Vogts, and Werner. Interpreting the index as the ratio of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators on 2D topologically ordered spin systems. We introduce our generalization of index and show that it is a complete invariant for the group of QCA modulo finite depth circuits for the fusion spin chains built from the fusion category Fib. This talk is based on a joint work with Corey Jones.

01/23/24
Martin Balko  Charles University Prague
Ordered Ramsey numbers: some recent progress
AbstractWe survey some of the newest results about ordered Ramsey numbers of graphs, that is, about a variant of Ramsey numbers for graphs with linearly ordered vertex sets. In particular, we will focus on one of the wellknown problems in the area about estimating on offdiagonal ordered Ramsey numbers of ordered matchings versus a triangle. This is a joint work with Marian Poljak.

01/23/24
Andrew Lawrie  MIT
Continuous bubbling for the harmonic map heat flow
AbstractI will discuss joint work with Jacek Jendrej and Wilhelm Schlag about the two dimensional harmonic map heat flow for maps taking values in the sphere. It has been known since the 80s90’s that solutions can exhibit bubbling along a wellchosen sequence of times — the solution decouples into a superposition of wellseparated harmonic maps and a body map accounting for the rest of the energy. We prove that every sequence of times contains a subsequence along which such bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multibubbles in continuous time. The proof is partly motivated by the classical theory of dynamical systems and uses the notion of “minimal collision energy” developed in joint work with Jendrej on the soliton resolution conjecture for nonlinear waves.

01/24/24
Prof. Alexander Strang  UC Berkeley
Solution Continuation Methods for Bayesian Estimation and Sampling
AbstractBayesian estimation and uncertainty quantification depend on prior assumptions. These assumptions are often chosen to promote specific features in the recovered estimate like sparsity. The form of the chosen prior determines the shape of the posterior distribution, thus the behavior of the estimator, and the complexity of the associated optimization and sampling problems. Here, we consider a family of Gaussian hierarchical models with generalized gamma hyperpriors designed to promote sparsity in linear inverse problems. By varying the hyperparameters we can move continuously between priors that act as smoothed ℓp penalties with flexible p, smoothing, and scale. We introduce methods for efficiently tracking MAP solutions along paths through hyperparameter space. Path following allows a user to explore the space of possible estimators under varying assumptions and to test the robustness and sensitivity of solutions to changes in the prior assumptions. By tracing paths from a convex region to a nonconvex region, the user can find local minimizers in nonconvex, strongly sparsitypromoting regimes that are consistent with a convex relaxation drawn from the same family of posteriors. We show experimentally that these solutions are less errorprone than direct optimization of the nonconvex problem. The same relaxation approach allows sampling from highly nonconvex multimodal posteriors in high dimension via a variational Bayesian formulation. We demonstrate predictorcorrector methods for estimator and sample continuation.

01/24/24
Prof. Ery AriasCastro
An introduction to Classical Scaling, with some new developments
AbstractClassical Scaling is perhaps the main method for Multidimensional Scaling (MDS), which is an area of Statistics (although initiated in Psychometrics) where the central task is the embedding of a weighted graph as a configuration of points in a Euclidean space in such a way as to match, as much as possible, the edges weights with the Euclidean distance between the corresponding points. The presentation will introduce this old method (dating back to the 1930s) and then go over more recent advances (last 20 years) in terms of computation, perturbation bounds, and more.

01/25/24
Prof. Jane Wang  University of Maine
The topology of the moduli space of dilation surfaces
AbstractTranslation surfaces are geometric objects that can be defined as a collection of polygons with sides identified in parallel opposite pairs by translation. If we generalize slightly and allow for polygons with sides identified by both translation and dilation, we get a new family of objects called dilation surfaces. While translation surfaces are wellstudied, much less is known about dynamics on dilation surfaces and their moduli spaces. In this talk, we will survey recent progress in understanding the topology of moduli spaces of dilation surfaces. We will do this by understanding the action of the mapping class group on the moduli space of dilation surfaces. This talk represents joint work with Paul Apisa and Matt Bainbridge.

01/25/24
Bo Li  Department of Mathematics and qBio Ph.D. program, UCSD
Variational Implicit Solvation and Fast Algorithms for Molecular Binding and Unbinding
AbstractLigandreceptor binding and unbinding are fundamental molecular processes, whereas water fluctuations impact strongly their thermodynamics and kinetics. We develop a variational implicitsolvent model (VISM) and a fast binary levelset method to calculate the potential of mean force and the moleculewater interfacial structures for dry and wet states. Monte Carlo simulations with our model and method provide initial configurations for efficient molecular dynamics simulations. Moreover, combined with the string method and stochastic simulations of ligand molecules, our hybrid approach enables the prediction of the transition paths and rates for the drywet transitions and the mean firstpassage times for the ligandpocket binding and unbinding. Without any explicit description of individual water molecules, our predictions are in a very good, qualitative and semiquantitative, agreement with existing explicitwater molecular dynamics simulations.
This talk reviews a series of works done in collaboration with L.T. Cheng, S. Zhou, Z. Zhang, S. Liu, H.B. Cheng, J. Dzubiella, C. Ricci, and J. A. McCammon.

01/25/24
John Yin  Wisconsin
A Chebotarev Density Theorem over Local Fields
AbstractI will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.
[pretalk at 1:20PM] 
01/25/24
Uri Bader  Weizmann Institute of Science
Cohomology of Arithmetic Groups, Higher Property T and Spectral Gap
AbstractGroups of matrices with integer entries, aka arithmetic groups, are prominent objects of mathematics.From a geometric point of view, they appear as the fundamental groups of locally symmetric space. Topological invariants of such spaces could be seen as group invariants and vice versa.
In my talk I will relate this useful link between topology and arithmetics with the theory of unitary representations. More precisely, I will focus on the cohomology of arithmetic groups with unitary coefficients, presenting a recent joint work with Roman Sauer which completely clarifies the theory in small degrees.
By the end of the talk I will discuss the relation of the above with the phenomenon of spectral gap and state various related conjectures.
I will make an effort to present the subject to a general audience.

01/25/24
Anthony Sanchez  UCSD
Translation surfaces and renormalization dynamics
AbstractA translation surface is a collection of polygons with edge identifications given by translations. In spite of the simplicity of the definition, the space of translation surfaces has connections to different areas of math such as the moduli space of hyperbolic surfaces. A guiding principle centers on turning questions on a fixed translation surface into a dynamical one on the space of all translation surfaces. We consider an instance of this philosophy related to the slope gap distribution of holonomy vectors of a translation surface. We use this as a jumping off point to consider expanding translates in different spaces such as nonarithmetic hyperbolic manifolds. Aspects of this talk represent different works with L. Kumandari and J. Wang, and with K. Ohm.

01/25/24
Dr. Sen Na  UC Berkeley
Practicality meets Optimality: RealTime Statistical Inference under Complex Constraints
AbstractConstrained estimation problems are prevalent in statistics, machine learning, and engineering. These problems encompass constrained generalized linear models, constrained deep neural networks, physicsinspired machine learning, algorithmic fairness, and optimal control. However, existing estimation methods under hard constraints rely on either projection or regularization, which may theoretically exhibit optimal efficiency but are impractical or unreasonably fail in reality. This talk aims to bridge the significant gap between practice and theory for constrained estimation problems.
I will begin by introducing the critical methodology used to bridge the gap, called Stochastic Sequential Quadratic Programming. We will see that SQP methods serve as the workhorse for modern scientific machine learning problems and can resolve the failure modes of prevalent regularizationbased methods. I will demonstrate how to make SQP adaptive and scalable using various modern techniques, such as stochastic line search, trust region, and dimensionality reduction. Additionally, I will show how to further enhance SQP to handle inequality constraints online.
Following the methodology, I will present some selective theories, emphasizing the consistency and efficiency of the SQP methods. Specifically, I will show that online SQP iterates asymptotically exhibit normal behavior with a mean of zero and optimal covariance in the Hajek and Le Cam sense. Significantly, the covariance does not deteriorate even when we apply modern techniques driven by practical concerns. The talk concludes with experiments on both synthetic and real datasets.

01/29/24
Paul Orland  UC San Diego
Lookahead SAT Solvers for Cube and Conquer

01/30/24
Dr. Sutanu Roy  National Institute of Science Education and Research (NISER), India
Anyonic quantum permutation groups
AbstractIn this talk, I shall present the anyonic version of the permutation groups and show that it represents the anyonic symmetry of finite sets. I shall also discuss an application of anyonic quantum permutation groups in computing anyonic symmetries of circulant graphs. The content of this talk is based on joint work with Anshu, Suvrajit Bhattacharjee, and Atibur Rahaman [ https://doi.org/10.1007/s11005023017361 ]. 
01/30/24
Prof. Zach Hamaker  University of Florida
Duality for polynomials
AbstractIn recent years, many mathematicians have contributed to a combinatorial theory for the polynomial ring ${\mathbb C}[x_1, x_2, \ldots]$ similar to symmetric function theory. Beginning with Schubert polynomials and later key polynomials, numerous bases have been introduced whose monomials have combinatorial interpretations. In the theory of harmonic polynomials, there is a natural inner product for the polynomial ring with monomials as an orthogonal basis. Duality with respect to this inner product is characterized by a Cauchy type identity. We show how to interpret this duality combinatorially. As a byproduct, we recover Postnikov and Stanley’s dual Schubert polynomials and introduce a novel family of dual key polynomials whose further properties remain uninvestigated.

01/30/24
Prof. Angxiu Ni  Yau Mathematical Sciences Center, Tsinghua University, China
Backpropagation and adjoint differentiation of chaos
AbstractComputing the derivative of longtimeaveraged observables with respect to system parameters is a central problem for many numerical applications. Conventionally, there are three straightforward formulas for this derivative: the pathwise perturbation formula (including the backpropagation method used by the machine learning community), the divergence formula, and the kernel differentiation formula. We shall explain why none works for the general case, which is typically chaotic (also known as the gradient explosion phenomenon), highdimensional, and smallnoise.
We present the fast response formula, which is a 'MonteCarlo' type formula for the parameterderivative of hyperbolic chaos. It is the average of some function of umany vectors over an orbit, where u is the unstable dimension, and those vectors can be computed recursively. The fast response overcomes all three difficulties under hyperbolicity assumptions. Then we discuss how to further incorporate the kernel differentiation trick to overcome nonhyperbolicity. 
01/30/24
Morgan Oppie  UCLA
Applications of higher real Ktheory to enumeration of stably trivial vector bundles
AbstractThe zeroeth complex topological Ktheory of a space encodes complex vector bundles up to stabilization. Since complex topological Ktheory is highly computable, this is a great place to start when asking bundletheoretic questions. However, in general, many nonequivalent bundles represent the same Ktheory class. Bridging the gap between Ktheory and actual bundles is challenging even for the simplest CW complexes.
For example, given random r and n, the number of rank r bundles on complex projective rspace that are trivial in Ktheory is unknown. In this talk, we will compute the pprimary portion of the number of rank r bundles on $\mathbb CP^n$ in infinitely many cases. We will give lower bounds for this number in more cases.
Building on work of Hu, we use Weisstheoretic techniques in tandem with a little chromatic homotopy theory to translate bundle enumeration to a computation of the higher real Ktheory of particular simple spectra. The result will involve actual numbers! This is joint work with Hood Chatham and Yang Hu.
Feb

02/01/24
Daren Wang  University of Notre Dame
Nonparametric Estimation via VarianceReduced Sketching
AbstractNonparametric models are of great interest in various scientific and engineering disciplines. Classical kernel methods, while numerically robust and statistically sound in lowdimensional settings, become inadequate in higherdimensional settings due to the curse of dimensionality.
In this talk, we will introduce a new framework called VarianceReduced Sketching (VRS), specifically designed to estimate density functions and nonparametric regression functions in higher dimensions with a reduced curse of dimensionality. Our framework conceptualizes multivariable functions as infinitesize matrices, facilitating a new matrixbased biasvariance tradeoff in various nonparametric contexts.
We will demonstrate the robust numerical performance of VRS through a series of simulated experiments and realworld data applications. Notably, VRS shows remarkable improvement over existing neural network estimators and classical kernel methods in numerous density estimation and nonparametric regression models. Additionally, we will discuss theoretical guarantees for VRS to support its ability to deliver nonparametric estimation with a reduced curse of dimensionality. 
02/01/24
Jinho Jeoung  Seoul National University
$\operatorname{PGL}_2(\mathbb{Q}_p)$orbit closures on a $p$adic homogenenous space of infinite volume
AbstractWe proved closed/dense dichotomy of $\operatorname{PGL}_2(\mathbb{
Q}_p)$orbit closures in the renormalized frame bundle of a $p$adic homogeneous space of infinite volume. Our result is a generalization of Ratner’s theorem and the result of McMullen, Mohammadi, and Oh in 2017 into nonArchimedean local fields. Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_p$. Our homogeneous space is a quotient space of $\operatorname{\mathbb{K}}$ by a certain class of Schottky subgroups. Using the main tools of McMullen, Mohammadi, and Oh, we introduced the necessary properties of Schottky subgroups and used the BruhatTits tree $\operatorname{PGL}_2$. In this talk, we introduce the highlybranched Schottky subgroups and steps for the proof of the main theorem.
This is a joint work with Seonhee Lim.

02/02/24
Patricia Muñoz Ewald  UT Austin
What dirt and ChatGPT have in common
AbstractOne day, mathematicians started thinking really hard about moving piles of dirt around, and the Wasserstein distance was born. It measures the difference between two probability distributions, in a way that is different (and sometimes better) than entropy and the L^p metrics. In this talk, I will introduce the field known as optimal transport, and talk about some applications, mainly to machine learning.

02/05/24
Chan Park  Department of Statistics, Wharton School of Business, University of Pennsylvania
Single Proxy (Synthetic) Control
AbstractA negative control outcome (NCO) is an outcome that is associated with unobserved confounders of the effect of a treatment on an outcome in view, and is a priori known not to be causally impacted by the treatment. In the first half of the talk, we discuss the single proxy control (SPC) framework, a formal NCO method to detect and correct for residual confounding bias. We establish nonparametric identification of the average causal effect for the treated (ATT) by treating the NCO as an errorprone proxy of the treatmentfree potential outcome, a key assumption of the SPC framework. We characterize the efficient influence function for the ATT under a semiparametric model in which nuisance functions are a priori unrestricted. Moreover, we develop a consistent, asymptotically linear, and locally semiparametric efficient estimator of the ATT using modern machine learning theory. Shifting to the second half of the talk, we introduce the single proxy synthetic control (SPSC) framework, an extension of the SPC framework designed for a synthetic control setting, where a single unit is treated and pre and posttreatment time series data are available on the treated unit and a heterogeneous pool of untreated control units. Similar to SPC, the SPSC framework views the outcomes of untreated control units as proxies of the treatmentfree potential outcome of the treated unit, a perspective we formally leverage to construct a valid synthetic control. Under this framework, we establish alternative identification and estimation methodology for synthetic controls and, in turn, for the ATT. Additionally, we adapt a conformal inference approach to perform inference on the treatment effect, obviating the need for a large number of posttreatment data. We illustrate the SPC and SPSC approaches with realworld applications from the Zika virus outbreak in Brazil and the 1907 financial crisis.

02/06/24
Yuhua Zhu  UCSD
A PDE based Bellman equation for Continuoustime Reinforcement Learning
AbstractIn this paper, we address the problem of continuoustime reinforcement learning in scenarios where the dynamics follow a stochastic differential equation. When the underlying dynamics remain unknown and we have access only to discretetime information, how can we effectively perform policy evaluation? We first demonstrate that the commonly used Bellman equation is a firstorder approximation to the true value function. We then introduce a higher order PDEbased Bellman equation called PhiBE. We show that the solution to the ith order PhiBE is an ith order approximation to the true value function. Additionally, even the firstorder PhiBE outperforms the Bellman equation in approximating the true value function when the system dynamics change slowly. We develop a numerical algorithm based on Galerkin method to solve PhiBE when we possess only discretetime trajectory data. Numerical experiments are provided to validate the theoretical guarantees we propose.

02/06/24
Prof. Michael Molloy  University of Toronto
kregular subgraphs near the kcore threshold of a random graph
AbstractWe prove that $G_{n,p=c/n}$ whp has a $k$regular subgraph if $c$ is at least $e^{\Theta(k)}$ above the threshold for the appearance of a subgraph with minimum degree at least $k$; i.e. an nonempty $k$core. In particular, this pins down the threshold for the appearance of a $k$regular subgraph to a window of size $e^{\Theta(k)}$.
This is a joint work with Dieter Mitsche and Pawel Pralat; see arXiv:1804.04173 
02/07/24
Prof. Daniel Grier  UCSD
Quantum computing and the search for provable advantage over classical devices
AbstractIn the first half of the talk, I will give a brief introduction to quantum computing from the perspective of a computer scientist/mathematician. While it may seem obvious that quantum computers should be better than classical computers, this can be surprisingly hard to rigorously prove, especially using the types of quantum computers that are available today. In the second half of the talk, I will describe this quest for provable quantum advantage and some of the research directions I find most interesting.

02/07/24
Dr. Dmitriy (Tim) Kunisky  Yale
The computational cost of detecting hidden structures: from random to deterministic
AbstractI will present a line of work on the computational complexity of several algorithmic tasks on random inputs, including hypothesis testing, sampling, and "certification" for optimization problems (where an algorithm must output a bound on a problem's optimum rather than just a highquality solution). Surprisingly, these diverse tasks admit a unified analysis involving the same two main ingredients. The first is the study of algorithms that output lowdegree polynomial functions of their inputs. Such algorithms are believed to be optimal for many statistical tasks and can be understood with the theory of orthogonal polynomials, leading to strong evidence for the hardness of certain hypothesis testing problems. The second is a strategy of "planting" unusual structures in problem instances, which gives reductions from hypothesis testing to tasks like sampling and certification. I will focus on examples of the latter motivated by statistical physics: (1) sampling from Ising models, and (2) certifying bounds on the Hamiltonian of the SherringtonKirkpatrick spin glass model.
Next, by examining the sumofsquares hierarchy of semidefinite programs, I will demonstrate how reasoning with planted solutions can show computational hardness of certification problems not only in random settings under strong distributional assumptions, but also for more generic problem instances. As an extreme example, I will show how some of the above ideas may be completely derandomized and applied in a deterministic setting. Using as a testbed the longstanding open problem in number theory and Ramsey theory of bounding the clique number of the Paley graph, I will give an analysis of semidefinite programming that suggests both new theoretical approaches to proving stronger bounds on the clique number and refined notions of pseudorandomness capturing deterministic versions of phenomena from random matrix theory. 
02/08/24
Prof. Darren Creutz  U.S. Naval Academy
Word complexity cutoffs for mixing properties of subshifts
AbstractIn the setting of zeroentropy transformations, the class of subshiftsclosed shiftinvariant subsets $X$ of $\mathcal{A}^{\mathbb{Z}}$ for a finite alphabet $\mathcal{A}$possesses a quantitative measure of complexity: the number of distinct `words' of a given length $p(q) = \{ w \in \mathcal{A}^{q} : \exists x \in X \text{ s.t. w is a substring of x}\}$.
I will discuss my work, some joint with R. Pavlov, pinning down the relationship between this quantitative notion of complexity with the qualitative dynamical complexity properties of probabilitypreserving systems known as strong and weak mixing.
Specifically, I will present results that strong mixing can occur with word complexity arbitrarily close to linear but cannot occur when $\liminf p(q)/q < \infty$ and that weak mixing can occur when $\limsup p(q)/q = 1.5$ but cannot occur when $\limsup p(q)/q < 1/5$.
The condition that $\limsup p(q)/q < 1.5$ is a (much) stronger version of zero entropy. A corollary of our work is that the celebrated Sarnak conjecture holds for all such systems.

02/08/24
Dr. KarlTheodor Sturm  University of Bonn
Wasserstein Diffusion on Multidimensional Spaces
AbstractGiven any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space $\mathcal{P}(M)$ of probability measures on $M$ that is
 reversible w.r.t. the entropic measure $\mathbb{P}^\beta$ on $\mathcal{P}(M)$, heuristically given as
$$d\mathbb{P}^\beta(\mu) =\frac{1}{Z} e^{\beta \, \text{Ent}(\mu  m)}\ d\mathbb{P}^0(\mu);$$
 associated with a regular Dirichlet form with carré du champ derived from the Wasserstein gradient in the sense of Otto calculus
$$\mathcal{E}_W(f)=\liminf_{\
tilde f\to f}\ \frac12\int_{\mathcal{P}(M)} \big\\nabla_W \tilde f\big\^2(\mu)\ d\mathbb{P}^\beta(\mu);$$  nondegenerate, at least in the case of the $n$sphere and the $n$torus.

02/08/24
Dietmar Bisch  Vanderbilt University
New hyperfinite subfactors with small Jones index
AbstractSince Vaughan Jones introduced the theory of subfactors in 1983, it has been an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Not much is known about this set.
My student Julio Caceres and I could recently show that certain indices between 4 and 5 are realized by new hyperfinite subfactors with TemperleyLiebJones standard invariant. This leads to a conjecture regarding Jones' problem. Our construction involves commuting squares, a graph planar algebra embedding theorem, and a few tricks that allow us to avoid solving large systems of linear equations to compute invariants of our subfactors. If there is time, I will mention a few connections to quantum Fourier analysis and quantum information theory. 
02/08/24
Mckenzie West  University of WisconsinEau Claire
A Robust Implementation of an Algorithm to Solve the $S$Unit Equation
AbstractThe $S$unit equation has vast applications in number theory. We will discuss an implementation of an algorithm to solve the $S$unit equation in the mathematical software Sage. The mathematical foundation for this implementation and some applications will be outlined, including an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded discriminant in which 2 is totally ramified. We will conclude with a discussion on current and future work toward improving the existing Sage functionality.
[pretalk at 1:20PM, in person only] 
02/08/24
Prof. Dominik Wodarz  Biology, UC San Diego
Mathematical models of tissue homeostasis and stemcell driven cancer growth
AbstractI will discuss mathematical models of stem cell dynamics in tissues at homeostasis, focusing on the ability of negative feedback loops within cell lineages to contribute to homeostatic control. These dynamics will be examined both in nonspatial and spatially explicit computational models, highlighting how spatial interactions can change dynamics and conclusions. The talk will further discuss the evolution of cells towards escape from homeostatic control, which gives rise to cancerous growth of cells. In the cancer cell growth dynamics, the models will be used to examine factors that determine the fraction of cancer stem cells in tumors, which in turn can determine the degree to which tumors respond to chemotherapies. Higher stem cell fractions correlate with increased resistance to therapy. This theory will be applied to data from bladder cancer, with the aim to better understand the heterogeneity that is observed in the responses among different patients to treatments.

02/08/24
Amit Ophir  UCSD
Stable lattices in representations over $p$adic field
AbstractRepresentations of groups over $p$adic fields arise naturally in Number Theory. Stable lattices serve as integral models for such representations. I will provide an example of these representations. I will discuss the connection between the set of lattices and a combinatorial object called the BruhatTits building. If time permits, I will discuss open problems.

02/09/24
Vitor Borges
Waves in a Pond and Strong Nuclear Forces
AbstractHave you ever thrown a rock into a still pond, stared at the concentric waves for a while and realized that the farther the waves travel, the smaller their crests seem to get? In this talk, we'll discuss a class of equations that, like the water waves in a pond, disperse. We'll discuss common techniques to study existence and qualitative properties of solutions to nonlinear dispersive PDEs using equations that model the strong nuclear force: the DiracKleinGordon system.

02/12/24
Nicholas Sieger  UC San Diego
The Satisfiability Threshold in Random kSAT
AbstractConsider a uniformly random kSAT instance with a fixed ratio of the number of clauses to the number of variables. As the clausevariable ratio increases, a curious phenomenon appears. With high probability the random instance is satisfiable below a certain threshold and unsatisfiable above the same threshold. This talk will give an overview of the problem of determining the precise threshold for random kSAT, including various algorithmic approaches, the connections to statistical physics, Friedman's sharp threshold theorem, and the determination of the kSAT threshold for large k by Ding, Sun, and Sly in 2014.

02/13/24
Adriana Fernandez I Quero  The University of Iowa
Rigidity results for group von Neumann algebras with diffuse center
AbstractWe introduce the first examples of groups G with an infinite center which in a natural sense are completely recognizable from their von Neumann algebras, L(G). Specifically, assume that G=A x W, where A is an infinite abelian group and W is an ICC wreathlike product group with property (T) and trivial abelianization. Then whenever H is an arbitrary group such that L(G) is isomorphic to L(H), via an arbitrary isomorphism preserving the canonical traces, it must be the case that H= B x H_0 where B is infinite abelian and H_0 is isomorphic to W. Moreover, we completely describe the isomorphism between L(G) and L(H). This yields new applications to the classification of group C*algebras, including examples of nonamenable groups which are recoverable from their reduced C*algebras but not from their von Neumann algebras. This is joint work with Ionuţ Chifan and Hui Tan. 
02/13/24
Dr. Marcelo Sales  UC Irvine
On Pisier type problems
AbstractA subset $A\subseteq\mathbf{Z}$ of integers is free if for every two distinct subsets $B,B'\subseteq A$ we have $$\sum_{b\in B}b\neq\sum_{b'\in B'}b'.$$ Pisier asked if for every subset $A\subseteq\mathbf{Z}$ of integers the following two statement are equivalent:
(i) $A$ is a union of finitely many free sets.
(ii) There exists $\varepsilon>0$ such that every finite subset $B\subseteq A$ contains a free subset $C\subseteq B$ with $\vert C\vert\geq \varepsilon \vert B\vert$.
In a more general framework, the Pisier question can be seen as the problem of determining if statements (i) and (ii) are equivalent for subsets of a given structure with prescribed property. We study the problem for several structures including $B_h$sets, arithmetic progressions, independent sets in hypergraphs and configurations in the Euclidean space.
This is joint work with Jaroslav Nešetřil, Christian Reiher and Vojtěch Rödl. 
02/14/24
Prof. Yian Ma  UCSD
MCMC, variational inference, and reverse diffusion Monte Carlo
AbstractI will introduce some recent progress towards understanding the scalability of Markov chain Monte Carlo (MCMC) methods and their comparative advantage with respect to variational inference. I will factcheck the folklore that "variational inference is fast but biased, MCMC is unbiased but slow". I will then discuss a combination of the two via reverse diffusion, which holds promise of solving some of the multimodal problems. This talk will be motivated by the need for Bayesian computation in reinforcement learning problems as well as the differential privacy requirements that we face.

02/14/24
Prof. Yuhau Zhu  UC San Diego
A PDEbased Bellman Equation for ContinuousTime Reinforcement Learning
AbstractIn this talk, we address the problem of continuoustime reinforcement learning in scenarios where the dynamics follow a stochastic differential equation. When the underlying dynamics remain unknown and we have access only to discretetime information, how can we effectively conduct policy evaluation? We first demonstrate that the commonly used Bellman equation is a firstorder approximation to the true value function. We then introduce higher order PDEbased Bellman equation called PhiBE. We show that the solution to the ith order PhiBE is an ith order approximation to the true value function. Additionally, even the firstorder PhiBE outperforms the Bellman equation in approximating the true value function when the system dynamics change slowly. We develop a numerical algorithm based on Galerkin method to solve PhiBE when we possess only discretetime trajectory data. Numerical experiments are provided to validate the theoretical guarantees we propose.

02/14/24
Prof. Fay Dowker  Imperial College London
Combinatorial Geometry: a tale of two signatures
AbstractCan a purely combinatorial object be approximated by a continuum geometry? I will describe evidence that the answer is "yes'' if that object is a transitive directed acyclic graph, otherwise known as a discrete order, otherwise known as a causal set. In which case, the approximating continuum geometry must be pseudoRiemannian with a "Lorentzian'' signature of $(, +, +, \ldots, +)$. I will, along the way, explain the crucial difference between Riemannian and Lorentzian geometry: in the former case the geometry is local and in the latter the geometry is, if not actually nonlocal then teetering on the edge of being nonlocal. If there is time I will describe a model of random orders called Transitive Percolation, which is the Lorentzian analogue of the ErdősRenyi random graph and is an interesting toy model for a physical dynamics of discrete spacetime.

02/15/24
Prof. Ben Hayes  University of Virginia
Growth dichotomy for unimodular random rooted trees
AbstractWe show that the growth of a unimodular random rooted tree (T,o) of degree bounded by d always exists, assuming its upper growth passes the critical threshold of the square root of d1. This complements Timar's work who showed the possible nonexistence of growth below this threshold. The proof goes as follows. By BenjaminiLyonsSchramm, we can realize (T,o) as the cluster of the root for some invariant percolation on the dregular tree. Then we show that for such a percolation, the limiting exponent with which the lazy random walk returns to the cluster of its starting point always exists. We develop a new method to get this, that we call the 23method, as the usual pointwise ergodic theorems do not seem to work here. We then define and prove the CohenGrigorchuk cogrowth formula to the invariant percolation setting. This establishes and expresses the growth of the cluster from the limiting exponent, assuming we are above the critical threshold.

02/15/24
Lili Zheng  Rice University
Uncertainty Quantification for Interpretable Machine Learning
AbstractInterpretable machine learning has been widely deployed for scientific discoveries and decisionmaking, while its reliability hinges on the critical role of uncertainty quantification (UQ). In this talk, I will discuss UQ in two challenging scenarios motivated by scientific and societal applications: selective inference for largescale graph learning and UQ for modelagnostic machine learning interpretations. Specifically, the first part concerns graphical model inference when only irregular, patchwise observations are available, a common setting in neuroscience, healthcare, genomics, and econometrics. To filter out lowconfidence edges due to the irregular measurements, I will present a novel inference method that quantifies the uneven edgewise uncertainty levels over the graph as well as an FDR control procedure; this is achieved by carefully disentangling the dependencies across the graph and consequently yields more reliable graph selection. In the second part, I will discuss the computational and statistical challenges associated with UQ for feature importance of any machine learning model. I will take inspiration from recent advances in conformal inference and utilize an ensemble framework to address these challenges. This leads to an almost computationally free, assumptionlight, and statistically powerful inference approach for occlusionbased feature importance. For both parts of the talk, I will highlight the potential applications of my research in science and society as well as how it contributes to more reliable and trustworthy data science.
Bio: Lili Zheng is a current postdoctoral researcher in the Department of Electrical and Computer Engineering at Rice University, mentored by Prof. Genevera I. Allen. Prior to this, she obtained her Ph.D. degree from the Department of Statistics at the University of WisconsinMadison, mentored by Prof. Garvesh Raskutti. Her research interests include graph learning, interpretable machine learning, uncertainty quantification, tensor data analysis, ensemble methods, and time series. Her website can be found at https://lilizhengstat.github.io 
02/15/24
Yuchen Zhou  University of Pennsylvania
Towards More Reliable Tensor Learning – heteroskedastic tensor clustering and uncertainty quantification for lowrank tensors
AbstractTensor data, which exhibits more sophisticated structures than matrix data and brings unique statistical and computational challenges, has attracted a flurry of interest in modern statistics and data science. While tensor estimation has been extensively studied in recent literature, most existing methods rely heavily on idealistic assumptions (e.g., i.i.d. noise), which are often violated in real applications. In addition, uncertainty quantification for lowrank tensors, also known as statistical inference in this context, remains vastly underexplored.
In this talk, I will present our recent progress on tensor learning. The first part of the talk is concerned with heteroskedastic tensor clustering, which seeks to extract underlying cluster structures from tensor observations in the presence of heteroskedastic noise. A novel tensor clustering algorithm will be introduced to achieve exact clustering under an (almost) necessary signaltonoise ratio condition for polynomialtime algorithms. The second part of the talk focuses on uncertainty quantification for tensor learning. Under a classical tensor PCA model, I will present a twoiteration alternating minimization procedure, and demonstrate that inference of principal components can be efficiently accomplished. These two developments represent the prolific interplay between statistics and computation in tensor learning.

02/15/24
Shahed Sharif  CSU San Marcos
Number theory and quantum computing: Algorithmists, Assemble!
AbstractQuantum computing made its name by solving a problem in number theory; namely, determining if factoring could be accomplished efficiently. Since then, there has been immense progress in development of quantum algorithms related to number theory. I'll give a perhaps idiosyncratic overview of the computational tools quantum computers bring to the table, with the goal of inspiring the audience to find new problems that quantum computers can solve.
[pretalk at 1:20PM] 
02/15/24
Nigel Goldenfeld  Physics Department, UC San Diego
Topological scaling laws and the mathematics of evolution
AbstractFor the last 3.8 billion years, the largescale structure of evolution has followed a pattern of speciation that can be described by branching trees. Recent work, especially on bacterial sequences, has established that despite their apparent complexity, these socalled phylogenetic or evolutionary trees exhibit two unexplained broad structural features which are consistent across evolutionary time. The first is that phylogenetic trees exhibit scaleinvariant topology, which quantifies the fact that their branching lies in between the two extreme cases of balanced binary trees and maximally unbalanced ones. The second is that the backbones of phylogenetic trees exhibit bursts of diversification on all timescales. I present a coarsegrained statistical mechanics model of ecological niche construction coupled to a simple model of speciation, and use renormalization group arguments to show that the statistical scaling properties of the resultant phylogenetic trees recapitulate both the scaleinvariant topology and the bursty pattern of diversification in time. These results show in principle how dynamical scaling laws of phylogenetic trees on long timescales may emerge from generic aspects of the interplay between ecological and evolutionary processes, leading to scale interference.
Finally, I will argue that these sorts of simplistic, minimal arguments might have a place in understanding other largescale aspects of evolutionary biology. In particular I will mention two important biological questions, which, I will argue, require significant mathematical advances in order to answer them. At present, we do not have even a qualitative understanding let alone a quantitative one: (1) the spontaneous emergence of the openended growth of complexity; (2) the response of evolving systems to perturbations and the implications for their control.
Even though biology is intimidatingly complex, "everything has an exception", and there are a huge number of undetermined parameters, this work shows that mathematical reasoning may lead to useful new insights into the existence and universal characteristics of living systems.
Work performed in collaboration with Chi Xue and Zhiru Liu and supported by NASA through cooperative agreement NNA13AA91A through the NASA Astrobiology Institute for Universal Biology.

02/16/24
Johnny Jingze Li  UCSD
Mathematical Theory for Emergent Effects
AbstractEmergent effects are commonly understood as novel properties, patterns, or behaviors of systems that are not present in their components, sometimes expressed as “the whole is more than the sum of its parts”. I will discuss a framework that gives a measure of emergent effect as the “loss of exactness” computed from local structures, through category theory, homological algebra and quiver representations, and show that the derived functor that encodes emergent effects is related to information loss. I will also discuss potential connections to neural networks. I can also talk about other maths that could be used for quantifying emergence if you are not that into homological algebra.

02/16/24
Prof. SongYing Li  UC Irvine
Supnorm Estimates for $\bar{d}$ and Corona Problems
AbstractIn this talk, we will present the development of Corona problem in sereval complex variables and discuss its relation to the solution of the supnorm estimates for the CauchyRiemann equations. It includes the Berndtsson conjecture and its application to Corona problem. As well as the application of the H\"ormander weighted $L^2$estimates for $\overline{\partial}$ to the corona problem.

02/16/24
Prof. Min Ru  University of Houston
Recent developments in the theory of holomorphic curves
AbstractIn this talk, I will discuss some recent developments and techniques in the study of the theory of holomorphic curves (Nevanlinna theory). In particular I will discuss the recent techniques of the socalled G.C.D. method as the applications of my recent work with Paul Vojta.

02/20/24
Prof. Bill Helton  UC San Diego
Perfect quantum strategies for XOR games
AbstractThe talk will describe some of the structure associated with 'perfect quantum strategies’ for a class of cooperative games. In such problems one has a (noncommutative) algebra A which encodes quantum mechanical laws and a list of matrix equations. A solution to these amounts to a perfect quantum strategy; 1 x 1 matrix solutions give a perfect classical strategy. The focus will be on 3XOR games. There is now a way to determine if a perfect quantum strategy exists and construct it if it does. The core of the construction is a variant on the classical 3XOR SAT problem. The talk will describe current understanding of this variant, in particular the sharp transition between solvability and unsolvability as numbers of constraints vs unknowns vary. The work is joint with Adam Bene Watts, Zehong Zhao, Jared Huges and Daniel Kane. 
02/20/24
Jiyang Gao  Harvard University
Quantum Bruhat Graphs and Tilted Richardson Varieties
AbstractThe quantum Bruhat graph, initially introduced by Brenti, Fomin, and Postnikov, is a weighted directed graph defined on finite Weyl groups. It serves as a valuable tool for exploring the quantum cohomology ring of the flag variety. In this presentation, we present a combinatorial formula for the minimal weights between any pair of permutations within the quantum Bruhat graph. Furthermore, for an ordered pair of permutations $u$ and $v$, we introduce the tilted Richardson variety $T_{u,v}$ demonstrating its equivalence to the twopointed curve neighborhood of opposite Schubert varieties $X_u$ and $X^v$ in the minimal degree $d$. We establish a Deodharlike decomposition for tilted Richardson varieties, leveraging it to prove several results. This is joint work with Shiliang Gao and Yibo Gao.

02/21/24
Prof. Jacques Verstraete  UCSD
Ramsey Theory and Pseudorandom Graphs
AbstractThe use of randomness in Ramsey Theory has been a key tool in nonconstructive lower bounds for Ramsey Numbers.
In this talk, I will describe a new direction in Ramsey Theory, which employs pseudorandom graphs instead of random graphs, and leads to breakthroughs on longstanding open problems in the area.

02/22/24
Bradley Zykoski  University of Michigan
Strongly Obtuse Rational Lattice Triangles
AbstractThe dynamics of a billiard ball on a triangular table can be studied by considering geodesic trajectories on an associated singular flat metric structure called a translation surface when the angles of the triangle are commensurable with pi. In the case of the isosceles right triangle, this surface is a torus, whose geodesic trajectories in any direction are either all periodic or all uniquely ergodic. Triangles satisfying such a dichotomy are called lattice triangles, and our work contributes to the ongoing classification of such triangles. We make use of a numbertheoretic criterion of Mirzakhani and Wright to classify such triangles with a large obtuse angle. This work is joint with Anne Larsen and Chaya Norton.

02/22/24
Prof. Jason Schweinsberg  Math, UCSD
Using Coalescent Theory to Estimate the Growth Rate of a Tumor
AbstractConsider a birth and death process in which each individual gives birth at rate $\lambda$ and dies at rate $\mu$, so that the population size grows at rate $r = \lambda  \mu$. Lambert (2018) and Harris, Johnston, and Roberts (2020) came up with methods for describing the exact genealogy of a sample of size $n$ taken from this population after time $T$. We use the construction of Lambert, which is based on the coalescent point process, to obtain asymptotic results for the site frequency spectrum associated with the sample. This allows us to derive point and interval estimates for the growth rate $r$, which are valid when $T$ and $n$ are large. We apply this method to the problem of estimating the growth rate of clones in blood cancer. This is joint work with Kit Curtius, Brian Johnson, and Yubo Shuai.

02/22/24
Marco Carfagnini  UCSD
Small Fluctuations, Spectral Theory, and Random Geometry
AbstractThe goal of this talk is to discuss new developments of random geometry. We will focus on small fluctuations (small balls) for degenerate diffusions and their connection to subRiemannian geometry. In particular, such diffusions can be used to describe spectral properties of their (hypoelliptic) generators, where the lack of ellipticity makes the analytic approach more challenging. Moreover, we will discuss random loops on Riemann surfaces which can be described in terms of SLEk loop measures. These are measures on the space of simple loops, and we will provide their asymptotics on small balls. Lastly, we will focus on the geometry of random Laplace eigenfunctions on the sphere and their application to physics and statistics.

02/22/24
Ivan Corwin  Columbia University
Scaling limit of a model of random transpositions
AbstractEach site x in Z is initially occupied by a particle of color x. Across each bond (x,x+1) particles swap places at rate 1 or q<1 depending on whether they are in reverse order (e.g. color 2 then 1) or order (color 1 then 2). This process describes a bijection of Z>Z which starts maximally in reverse order and randomly drifts towards being ordered. Another name for this model is the "colored asymmetric simple exclusion process". I will explain how to use the YangBaxter equation along with techniques involving Gibbs measures to extract the spacetime scaling limit of this process, as well as a discrete time analog known as the "stochastic six vertex model". The limit is described by objects in the KardarParisiZhang universality class, namely the Airy sheet, directed landscape and KPZ fixed point. This is joint work with Amol Aggarwal and Milind Hegde.

02/23/24
Hugo Jenkins  UCSD
The Group Algebra
AbstractI’ll prove and discuss the isomorphism $CG \cong \prod M_{d_i}(C)$, its abstract and concrete forms, and the relationship between the centers of the two algebras.

02/23/24
Dr. Valery Lunts  Indiana University
Vector field on the plane
AbstractGiven a vector field on a complex plane C^2 with polynomial coefficients one would like to know if all the integral curves of this vector field are algebraic. The problem seems to be very difficult. I will discuss an approach to this problem using the reduction to characteristic p, which produces a conjectural answer. Also I will explain the relation with the famous GrothendieckKatz pcurvature conjecture. Probably the main beauty of the subject is its completely elementary nature, which makes it accessible to a first year student.

02/27/24
Linghao Zhang  UCSD
Polynomial Optimization over Unions of Sets
AbstractThis paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of MomentSOS relaxations to solve it globally. Under some assumptions, we prove the asymptotic or finite convergence of the unified hierarchy. Special properties for the univariate case are discussed. The numerical experiments demonstrate that solving this unified hierarchy takes less computational time than optimizing the objective over each individual constraining subset separately. The application for computing (p,q)norms of matrices is also presented.

02/27/24
Dr. Brent Nelson  Michigan State University
Von Neumann dimension for faithful normal strictly semifinite weights
AbstractThe notion of von Neumann dimension for a tracial von Neumann algebra $(M,\tau)$ has been used extensively throughout the theory, particularly in defining numerical invariants from Jones' index of a subfactor to Connes and Shlyakhtenko's $\ell^2$Betti numbers of von Neumann algebras. The latter relies on work of Lück showing that Murray and von Neumann's original definition could be extended to purely algebraic $M$modules, and more recently Petersen further extended von Neumann dimension to pairs $(M,\tau)$ where $\tau$ is a faithful normal semifinite tracial weight. In this talk, I will introduce a yet further extension of this theory to pairs $(M,\varphi)$ where $\varphi$ is a faithful normal strictly semifinite weight. Here strict semifiniteness means the restriction of $\varphi$ to the centralizer subalgebra $M^\varphi \subset M$ is still semifinite (note this condition is automatic for faithful normal states), and by work of Takesaki this is equivalent to the existence of a $\varphi$invariant faithful normal conditional expectation $\mathcal{E}_\varphi\colon M \to M^\varphi$. Consequently, one can consider the Jones basic construction for the inclusion $M^\varphi \subset M$, and this is the key ingredient in our definition of von Neumann dimension for the pair $(M,\varphi)$. I will discuss properties of this dimension and how it can be used to recover the index for certain inclusions of factors. This is based on joint work with Aldo Garcia Guinto and Matthew Lorentz.

02/27/24
Prof. Dmitri Zaitsev  Trinity College Dublin
Global regularity in the dbarNeumann problem and finite type conditions
AbstractThe celebrated result of Catlin on global regularity of the $\bar\partial$Neumann operator for pseudoconvex domains of finite type links local algebraic and analytic geometric invariants through potential theory with estimates for $\bar\partial$equation. Yet despite their importance, there seems to be a major lack of understanding of Catlin's techniques, resulting in a notable absence of an alternative proof, exposition or simplification.
The goal of my talk will be to present an alternative proof based on a new notion of a ''tower multitype''. The finiteness of the tower multitype is an intrinsic geometric condition that is more general than the finiteness of the regular type, which in turn is more general than the finite type. Under that condition, we obtain a generalized stratification of the boundary into countably many level sets of the tower multitype, each covered locally by strongly pseudoconvex submanifolds of the boundary. The existence of such stratification implies Catlin's celebrated potentialtheoretic ''Property (P)'', which, in turn, is known to imply global regularity via compactness estimate. Notable applications of global regularity include Condition R by Bell and Ligocka and its applications to boundary smoothness of proper holomorphic maps generalizing a celebrated theorem by Fefferman.

02/27/24
Prof. Victor Reiner  University of Minnesota
Descents, peaks and configuration spaces
AbstractLouis Solomon observed in the 1970s that, within the group algebra of the symmetric group, there is an interesting subalgebra spanned linearly by sums of permutations with the same sets of descents. Later work of several authors showed that this contains a further subalgebra spanned by sums of permutations with the same numbers of descents, and that this has a connection with the topology of configuration spaces of n labeled distinct points in odd dimensional Euclidean spaces.
We review some of this story, as well as an analogous story that replaces descent sets with peak sets of permutations. We then report on the connection to topology, which is new. Joint work with Marcelo Aguiar and Sarah Brauner.

02/27/24
Prof. John D'Angelo  UIUC
Some old work of Kohn and some of my contributions
AbstractI will discuss in detail some work of Joe Kohn involving subelliptic estimates. I hope to provide an understandable account of some of the technical matters from his 1979 Acta paper, and I will discuss some of my own work on points of finite type. Although there is no new theorem to present, I will provide several new approaches to these ideas. To prevent the talk from being too technical, I will also include several elementary interludes that can be understood by graduate students.

02/27/24
Yutao Liu & Guoqi Yan  University of Washington & University of Notre Dame
The generalized Tate diagram of the equivariant slice spectral sequence
Abstracthe generalized Tate diagram developed by Greenlees and May is a fundamental tool in equivariant homotopy theory. In this talk, we will discuss an integration of the generalized Tate diagram with the equivariant slice filtration of Hill—Hopkins—Ravenel, resulting in a generalized Tate diagram for equivariant spectral sequences. This new diagram provides valuable insights into various equivariant spectral sequences and allows us to extract information about isomorphism regions between these equivariant filtrations.
As an application, we will begin by proving a stratification theorem for the negative cone of the slice spectral sequence. Building upon the work of Meier—Shi—Zeng, we will then utilize this stratification to establish shearing isomorphisms, explore transchromatic phenomena, and analyze vanishing lines within the negative cone of the slice spectral sequences associated with periodic Hill—Hopkins—Ravenel and Lubin—Tate theories. This is joint work of Yutao Liu, XiaoLin Danny Shi and Guoqi Yan. 
02/28/24
Prof. Tamás Terlaky  Lehigh University
Novel Quantum Interior Point Methods with Iterative Refinement for Linear and Semidefinite Optimization
AbstractQuantum Interior Point Methods (QIPMs) build on classic polynomial time IPMs. With the emergence of quantum computing we apply Quantum Linear System Algorithms (QLSAs) to Newton systems within IPMs to gain quantum speedup in solving Linear Optimization (LO) and Semidefinite Optimization (SDO) problems. Due to their inexact nature, QLSAs mandate the development of inexact variants of IPMs which, due to the inexact nature f their computations, by default are inexact infeasible methods. We propose “quantum inspired‘’ InexactFeasible IPMs (IFIPM) for LO and SDO problems, using novel Newton systems to generate inexact but feasible steps. We show that IFQIPMs enjoys the todate best iteration complexity. Further, we explore how QLSAs can be used efficiently in iterative refinement schemes to find optimal solutions without excessive calls to QLSAs. Finally, we experiment with the proposed IFIPM’s efficiency using IBMs QISKIT environment.
Speaker Bio:
Dr. Terlaky has published four books, edited over ten books and journal special issues and published over 200 research papers. Topics include theoretical and algorithmic foundations of mathematical optimization; nuclear reactor core reloading, oil refinery, VLSI design, radiation therapy treatment, and inmate assignment optimization; quantum computing.
Dr. Terlaky is EditorinChief of the Journal of Optimization Theory and Applications. He has served as associate editor of ten journals and has served as conference chair, conference organizer, and distinguished invited speaker at conferences all over the world. He was general Chair of the INFORMS 2015 Annual Meeting, a former Chair of INFORMS’ Optimization Society, Chair of the ICCOPT Steering Committee of the Mathematical Optimization Society, Chair of the SIAM AG Optimization, and Vice President of INFORMS. He received the MITACS Mentorship Award; Award of Merit of the Canadian Operational Society, Egerváry Award of the Hungarian Operations Research Society, H.G. Wagner Prize of INFOMRS, Outstanding Innovation in Service Science Engineering Award of IISE. He is Fellow of INFORMS, SIAM, IFORS, The Fields Institute, and elected Fellow of the Canadian Academy of Engineering. He will be a Plenary Speaker at ISMP’2024 in Montreal. 
02/28/24
Prof. Natalia Komarova  UCSD
Mathematical methods in cancer dynamics
AbstractEcoEvolutionary dynamics are at the core of carcinogenesis. Mathematical methods can be used to study ecological and evolutionary processes, and to shed light into cancer origins, progression, and mechanisms of treatment. I will present two broad approaches to cancer modeling that we have developed. One is concerned with nearequilibrium dynamics of stem cells, with the goal of figuring out how tissue cell turnover is orchestrated, and how control networks prevent “selfish” cell growth. The other direction is studying evolutionary dynamics in random environments. The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Applications to biomedical problems will be discussed.

02/29/24
Carsten Petersen  Paderborn University
Quantum ergodicity on the BruhatTits building for PGL(3) in the BenjaminiSchramm limit
AbstractOriginally, quantum ergodicity concerned equidistribution properties of Laplacian eigenfunctions with large eigenvalue on manifolds for which the geodesic flow is ergodic. More recently, several authors have investigated quantum ergodicity for sequences of spaces which "converge" to their common universal cover and when one restricts to eigenfunctions with eigenvalues in a fixed range. Previous authors have considered this type of quantum ergodicity in the settings of regular graphs, rank one symmetric spaces, and some higher rank symmetric spaces. We prove analogous results in the case when the underlying common universal cover is the BruhatTits building associated to PGL(3, F) where F is a nonarchimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a nonarchimedean analogue of the symmetric space setting.

02/29/24
Aranya Lahiri  UC San Diego
Dagger groups and $p$adic distribution algebras
AbstractLocally analytic representations were introduced by Peter Schneider and Jeremy Teitelbaum as a tool to understand $p$adic Langlands program. From the very beginning the theory of $p$valued groups played an instrumental role in the study of locally analytic representations. In a previous work we attached a rigid analytic group to a $\textit{$p$saturated group}$ (a class of $p$valued groups that contains uniform pro$p$ groups and pro$p$ Iwahori subgroups as examples). In this talk I will outline how to elevate the rigid group to a $\textit{dagger group}$, a group object in the category of dagger spaces as introduced by Elmar GrosseKlönne. I will further introduce the space of $\textit{overconvergent functions}$ and its strong dual the $\textit{overconvergent distribution algebra}$ on such a group. Finally I will show that in analogy to the locally analytic distribution algebra of compact $p$adic groups, in the case of uniform pro$p$ groups the overconvergent distribution algebra is a Fr´echetStein algebra, i.e., it is equipped with a desirable algebraic structure. This is joint work with Claus Sorensen and Matthias Strauch.

02/29/24
Prof. Tamás Terlaky  Lehigh University
The Quantum Computing Revolution and Optimization: Trends, and Perspectives
AbstractThe Quantum Computing (QC) revolution is spreading fast and has the potential of disrupting all industries. It is widely expected that QC can revolutionize the way we perform and think about computation and optimization, and QC will be the backbone of thrilling new technologies and products. Governments and private investors are already investing billions of dollars annually to accelerate developments in QC technologies and to explore a myriad of potential applications. The focus of this presentation will be on the impact of Quantum Computing on optimization sciences, the potential of making optimized decisions faster and better, let it be engineering design, systems performance, supply chain, or finance. Specifically, in the mathematical optimization area, Quantum Computing has the potential to speed up problem solving tremendously and solve very largescale problems that are not solvable to date. Just to mention, almost all results and claims about “Quantum Supremacy” are about solving optimization problems. Projected trends of QC hardware development with challenges ahead are discussed. Computing, algorithmic, and software stack developments, along with actual and potential applications of QC Optimization, and related areas will be discussed.
Speaker Bio:
Dr. Terlaky has published four books, edited over ten books and journal special issues and published over 200 research papers. Topics include theoretical and algorithmic foundations of mathematical optimization; nuclear reactor core reloading, oil refinery, VLSI design, radiation therapy treatment, and inmate assignment optimization; quantum computing.
Dr. Terlaky is EditorinChief of the Journal of Optimization Theory and Applications. He has served as associate editor of ten journals and has served as conference chair, conference organizer, and distinguished invited speaker at conferences all over the world. He was general Chair of the INFORMS 2015 Annual Meeting, a former Chair of INFORMS’ Optimization Society, Chair of the ICCOPT Steering Committee of the Mathematical Optimization Society, Chair of the SIAM AG Optimization, and Vice President of INFORMS. He received the MITACS Mentorship Award; Award of Merit of the Canadian Operational Society, Egerváry Award of the Hungarian Operations Research Society, H.G. Wagner Prize of INFOMRS, Outstanding Innovation in Service Science Engineering Award of IISE. He is Fellow of INFORMS, SIAM, IFORS, The Fields Institute, and elected Fellow of the Canadian Academy of Engineering. He will be a Plenary Speaker at ISMP’2024 in Montreal.
Mar

03/04/24
William Jack Wesley  UC San Diego
Symmetry Breaking in SAT Solving
AbstractSymmetry breaking is a useful technique that prevents a solver from looking for solutions in isomorphic parts of the search space, which often leads to massive speedups. In this talk we will give an overview of the theory behind symmetry in SAT and show its applications in some concrete problems.

03/05/24
Zirui Zhang  UC Irvine
Personalized Predictions of Glioblastoma Infiltration: Mathematical Models, PhysicsInformed Neural Networks and Multimodal Scans
AbstractPredicting the infiltration of Glioblastoma (GBM) from medical MRI scans is crucial for understanding tumor growth dynamics and designing personalized radiotherapy treatment plans.Mathematical models of GBM growth can complement the data in the prediction of spatial distributions of tumor cells. However, this requires estimating patientspecific parameters of the model from clinical data, which is a challenging inverse problem due to limited temporal data and the limited time between imaging and diagnosis. This work proposes a method that uses PhysicsInformed Neural Networks (PINNs) to estimate patientspecific parameters of a reactiondiffusion PDE model of GBM growth from a single 3D structural MRI snapshot. PINNs embed both the data and the PDE into a loss function, thus integrating theory and data. Key innovations include the identification and estimation of characteristic nondimensional parameters, a pretraining step that utilizes the nondimensional parameters and a finetuning step to determine the patient specific parameters. Additionally, the diffuse domain method is employed to handle the complex brain geometry within the PINN framework. Our method is validated both on synthetic and patient datasets, and shows promise for realtime parametric inference in the clinical setting for personalized GBM treatment.

03/05/24
Dr. Mehrdad Kalantar  University of Houston
Operator space complexification revisited
AbstractThe complexification of a real space can be described as an induced representation (in the sense of Frobenius). In this language, in particular, the analytical aspects of the concept and its generalizations (e.g. quaternification of real spaces), have very canonical descriptions, which allow vast generalizations of some of the key results, such as Ruan’s uniqueness theorem for “reasonable” operator space complexification.This is joint work with David Blecher. 
03/05/24
Prof. Tom Bohman  Carnegie Mellon University
Notes on 2point concentration in the random graph
AbstractWe say that an integervalued random variable $X$ defined on $G_{n,p}$ is concentrated on 2 values if there is a function $f(n)$ such that the probability that $X$ equals $f(n)$ or $ f(n)+1$ tends to 1 as $n$ goes to infinity. 2point concentration has been a central issue in the study of random graphs from the beginning. In this talk we survey some recent progress in our understanding of this phenomenon, with an emphasis on the independence number and domination number of the random graph.
Joint work with Jakob Hofstad, Lutz Warnke and Emily Zhu. 
03/06/24
Prof. Rayan Saab  UC San Diego
Stochastic algorithms for quantizing neural networks
AbstractNeural networks are highly nonlinear functions often parametrized by a staggering number of weights. Miniaturizing these networks and implementing them in hardware is a direction of research that is fueled by a practical need, and at the same time connects to interesting mathematical problems. For example, by quantizing, or replacing the weights of a neural network with quantized (e.g., binary) counterparts, massive savings in cost, computation time, memory, and power consumption can be attained. Of course, one wishes to attain these savings while preserving the action of the function on domains of interest.
We discuss connections to problems in discrepancy theory, present datadriven and computationally efficient stochastic methods for quantizing the weights of already trained neural networks and we prove that our methods have favorable error guarantees under a variety of assumptions.

03/06/24
Prof. Tingting Tang  San Diego State University
On computing the nonlinearity interval and MAPs of SDPs
AbstractIn this talk, I will talk about the parametric analysis of semidefinite optimization problems w.r.t. the perturbation of the objective function along a fixed direction and on a compact set. For the perturbation along a fixed direction, it is proven that the continuity of the optimal set mapping could fail on a nonlinearity interval and the set of points where this failure occurs is finite. A numerical method is developed to numerically compute the nonlinearity interval and generalize to perturbations on a compact set. For multivariable perturbations, a maximal analytic perturbation set (MAPs) is defined on which the analyticity of the optimal mapping holds. Numerical examples are given to demonstrate the performance.

03/07/24
Michael Zshornack  UC Santa Barbara
Twist flows and the arithmetic of surface group representations
AbstractMargulis's work on lattices and a number of questions on the existence of surface subgroups motivate the need for understanding arithmetic properties of spaces of surface group representations. In recent work with Jacques Audibert, we outline one possible approach towards understanding such properties for the Hitchin component, a particularly nice space of representations. We utilize the underlying geometry of this space to reduce questions about its arithmetic to questions about the arithmetic of certain algebraic groups, which in turn, allows us to characterize the rational points on these components. In this talk, I'll give an overview of the geometric methods behind the proof of our result and indicate some natural questions about the nature of the resulting surface group actions that follow.

03/07/24
Rene Schoof  Universita' di Roma Tor Vergata, Italy
Greenberg’s $\lambda=0$ conjecture
AbstractRecent and not so recent computations by Mercuri and Paoluzi have verified Greenberg’s $\lambda=0$ conjecture in Iwasawa theory in many cases. We discuss the conjecture and the computations.

03/07/24
Aranya Lahiri  UCSD
Why look at padic groups?
AbstractDo I really do number theory? Sometimes I have no idea how I belong to the number theory group, and not say functional analysis group? Even though the only books I pretend to read are: padic Lie groups, nonarchimidean functional analysis and Lecture notes on formal and rigid geometry? But then I realize I really don't know any functional analysis for that matter. In this talk, in very broad and crude strokes I will try to convince myself that I do number theory. Come burst my bubble.

03/07/24
Prof. Gunther Uhlmann  University of Washington
Journey to the Center of the Earth
AbstractWe will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It also has several applications in optics and medical imaging among others.
The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will survey some of the known results about this problem.No previous knowledge of differential geometry will be assumed.

03/08/24
Runqiu Xu  UCSD
How does the discrete Fourier transform on symmetric groups walk you through Hurwitz Cayley graphs?
AbstractIn this talk, I will give a quick review of representation theory and graph theory. I will explain the symmetric group algebra and its Fourier transform with an explanation of the corresponding characters. I will hint how it could be used to count the number of a specific type of walks on the Cayley graph of permutations.

03/12/24
Jeb Runnoe  UCSD
SecondDerivative SQP Methods for LargeScale Nonconvex Optimization
AbstractThe class of sequential quadratic programming (SQP) methods solve a nonlinearly constrained optimization problem by solving a sequence of related quadratic programming (QP) subproblems. Each subproblem involves the minimization of a quadratic model of the Lagrangian function subject to the linearized constraints. In contrast to the quasiNewton approach, which maintains a positivedefinite approximation of the Hessian of the Lagrangian, secondderivative SQP methods use the exact Hessian of the Lagrangian. In this context, we will discuss a dynamic convexification strategy with two main features. First, the method makes minimal matrix modifications while ensuring that the iterates of the QP subproblem are bounded. Second, the solution of the convexified QP is a descent direction for a merit function that is used to force convergence from any starting point. This talk will focus on the dynamic convexification of a class of primaldual SQP methods. Extensive numerical results will be presented.

03/12/24
Maciej Dolega  Polish Academy of Sciences
Weighted bHurwitz numbers from Walgebras
AbstractWeighted Hurwitz numbers were introduced by Harnad and GuayPaquet as objects covering a wide class of Hurwitz numbers of various types. A particularly strong property of Hurwitz numbers is that they are governed by the celebrated topological recursion (TR) of ChekhovEynardOrantin: a universal algorithm that allows computation of them recursively with respect to their topology. The program of understanding how TR can be used to compute different types of Hurwitz numbers was carried out over the last two decades by considering each case separately, and finally, the general case of rationallyweighted Hurwitz numbers was recently proved by BychkovDuninBarkowski
KazarianShadrin. We will discuss a more general case of weighted $b$Hurwitz numbers that arise naturally in the context of symmetric functions theory and matrix models. We show that their generating function satisfies the socalled $W$constraints  certain explicit differential equations arising from representations of $W$algebras. We will focus on a transition from an algebraic/geometric background to a combinatorial one, which turned out to be crucial in our work. Our result gives a new explanation of the remarkable enumerative properties of Hurwitz numbers following from TR, and extends it to the $b$deformed case. This is joint work with Nitin Chidambaram and Kento Osuga.

03/12/24
Alireza Golsefidy  UCSD
Closure of orbits of the pure mapping class group on the character variety
AbstractFor every surface S, the pure mapping class group G_S acts on the (SL_2)character variety Ch_S of a fundamental group P of S. The character variety Ch_S is a scheme over the ring of integers. Classically this action on the real points Ch_S(R) of the character variety has been studied in the context of the Teichmuller theory and SL(2,R)representations of P.
In a seminal work, Goldman studied this action on a subset of Ch_S(R) which comes from SU(2)representations of P. In this case, Goldman showed that if S is of genus g>1 and zero punctures, then the action of G_S is ergodic. Previte and Xia studied this question from topological point of view, and when g>0, proved that the orbit closure is as large as algebraically possible.
Bourgain, Gamburd, and Sarnak studied this action on the F_ppoints Ch_S(F_p) of the character variety with boundary trace equal to 2 where S is a puncture torus. They conjectured that in this case, this action has only two orbits, where one of the orbits has only one point. Recently, this conjecture was proved for large enough primes by Chen. When S is an npunctured sphere, the finite orbits of this action on Ch_S(C) are connected to the algebraic solutions of Painleve differential equations.
I will report on my joint work with Natallie Tamam in this area.

03/12/24
Elham Izadi  UCSD
Hyperkahler manifolds and Lagrangian fibrations
AbstractHyperkahler manifolds are one of the main classes of manifolds appearing in Berger’s classification of holonomy groups of Riemannian manifolds. It is known that for any nonconstant map f from a hyperkahler manifold of dimension 2n, the generic fibers of f are either finite or abelian varieties of dimension n. The latter are Lagrangian fibrations. I will discuss some open problems and some results concerning Lagrangian fibrations on hyperkahler manifolds.

03/13/24
Prof. Lutz Warnke  UC San Diego
Random Structures and Statistical Inference: a teaser
AbstractIn the upcoming Spring Quarter 2024 I will teach MATH262A 'Random Structures and Statistical Inference:', which is a topics course at the intersection of combinatorial statistics, algorithms and probabilistic combinatorics. The goal of this informal lecture is to give a glimpse into the kind of questions we intend to cover in this course. To this end we shall review the 'hidden clique' problem, which is a simple prototypical example with a surprisingly rich and interesting structure behind.

03/14/24
David Gao  UCSD
Sofic actions on sets and applications to generalized wreath products
AbstractInspired by the work of Hayes and Sale showing wreath products of two sofic groups are sofic, we define a notion of soficity for actions of countable discrete groups on countable discrete sets. We shall prove that, if the action $\alpha$ of G on X is sofic, G is sofic, and H is sofic, then the generalized wreath product H $\wr_\alpha$ G is sofic. We shall demonstrate several examples of sofic actions, including actions of sofic groups with locally finite stabilizers, all actions of amenable groups, and all actions of LERF groups. This talk is based on joint work with Srivatsav Kunnawalkam Elayavalli and Gregory Patchell.

03/14/24
Prof. Caroline Moosmueller  UNC
Manifold learning in Wasserstein space
AbstractThis talk discusses computationally feasible algorithms to uncover lowdimensional structures in the Wasserstein space. This line of research is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, and that finding lowdimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. One of our algorithms, LOT Wassmap, leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speedup computations, and in particular, avoids computing a pairwise distance matrix. Experiments demonstrate that LOT Wassmap attains correct embeddings, and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.
This talk is based on joint work with Alex Cloninger, Keaton Hamm, Varun Khurana, Matthew Thorpe, and Bernhard Schmitzer.

03/14/24
Prof. Natalia Komarova  Math, UCSD
Mathematical Methods in Evolution and Medicine
AbstractEvolutionary dynamics permeates life and lifelike systems. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to make sense of many phenomena in the life sciences. How likely is a single mutant to take over a population of individuals? What is the speed of evolution, if things have to get worse before they can get better (aka, fitness valley crossing)? Can cooperation, hierarchical relationships between individuals, spatial interactions, or randomness influence the speed or direction of evolution? Applications to biomedicine will be discussed.

03/15/24
Morgan Makhina  UCSD
The Merits and Drawbacks of the National Parks System
AbstractIn this talk I'll present a brief history of the National Parks System and discuss some of its merits and drawbacks. I will also compare and contrast it with the National Forest Service and Bureau of Land Management. Since notions of "merit" are inherently subjective, to balance the discussion audience members are encouraged to contribute their views on this subject. Perspectives from those who have had experience with public land management systems in other countries are especially welcome. To close, we'll talk about some of the ways you can get out there and enjoy your public lands!

03/15/24
Dr. Olivier Martin  IMPA
Isotrivial Lagrangian fibrations of compact hyperKähler manifolds
AbstractCompact hyperKähler manifolds and their Lagrangian fibrations are higherdimensional generalizations of K3 surfaces and their elliptic fibrations. I will present a recent exploration of the geometry of isotrivial Lagrangian fibrations conducted with Y. Kim and R. Laza. We show that the smooth fiber of such a fibration is isogenous to the power of an elliptic curve and present a trichotomy arising from the Kodaira dimension of the minimal Galois cover of the base which trivializes monodromy. We are motivated in part by the search for new deformation types of hyperKähler manifolds and the boundedness problem.

03/18/24
Abhik Pal  University of California San Diego
Sheaf Cohomology of the Supergrassmannian and the Representation Theory of $\mathfrak{gl}(mn)$

03/19/24
Sylvester Zhang  University of Minnesota
Schubert calculus and the bosonfermion correspondence
AbstractOriginally appearing in string theory, the BosonFermion correspondence has found connection to symmetric functions, through its application by the Kyoto school for deriving soliton solutions of the KP equations. In this framework, the space of Young diagrams is conceived as the Fermionic Fock space, while the ring of symmetric functions serves as the Bosonic Fock space. Then the (second part of) BF correspondence asserts that the map sending a partition to its Schur function forms an isomorphism as Hmodules, with H being the Heisenberg algebra. In this talk, we give a generalization of this correspondence into the context of Schubert calculus, wherein the space of infinite permutations plays the role of the Fermionic space, and the ring of backstable symmetric functions represents the Bosonic space.

03/21/24
Philip Easo  Caltech
The critical percolation probability is local
AbstractAround 2008, Schramm conjectured that the critical percolation probability $p_c$ of a transitive graph is entirely determined by the local geometry of the graph, subject to the global constraint that $p_c<1$. Previous works had verified the conjecture in various special cases, including nonamenable graphs of high girth (Benjamini, Nachmias and Peres 2012); Cayley graphs of abelian groups (Martineau and Tassion 2013); nonunimodular graphs (Hutchcroft 2017 and 2018); graphs of uniform exponential growth (Hutchcroft 2018); and graphs of (automatically uniform) polynomial growth (Contreras, Martineau and Tassion 2022). In this talk I will describe joint work with Hutchcroft (https://arxiv.org/abs/2310.
10983 ) in which we resolve this conjecture.
Apr

04/01/24
Prof. Padmini Rangamani  UC San Diego
The mathematics of cellular mechanotransduction
AbstractCellular function often integrates biochemical and mechanical cues in what is known as mechanotransduction. Mechanotransduction is closely tied to cell shape during development, disease, and wound healing. In this talk, I will showcase how mathematical models have helped shed light on some fundamental problems in this area of research including how cell shape can alter biochemical signaling and how cell mechanics can alter cell shape. Throughout, I will highlight the challenges and opportunities for integrating mathematical models with experimental measurements.

04/01/24
Prof. Lisa Fauci  Tulane University
Insights from biofluidmechanics: A tale of tails
AbstractThe motion of undulating or rotating elastic `tailsâ€™ in a fluid environment is a common element in many biological and engineered systems. At the microscale, we will consider models of the journey of extremely long and flexible insect flagella through narrow and tortuous female reproductive tracts, and the penetration of mucosal tissue by helical flagella of bacteria. At the macroscale, we will probe the neuromechanics and fluid dynamics of the lamprey, the most primitive vertebrate and, hence, a model organism. Using a closedloop model that couples neural signaling, muscle mechanics, fluid dynamics and sensory feedback, we examine the hypothesis that amplified proprioceptive feedback could restore effective locomotion in lampreys with spinal injuries.

04/02/24
Dr. Gavril Farkas  Humboldt University
The birational geometry of $M_g$ via tropical geometry and nonabelian BrillNoether theory
AbstractI will discuss how novel ideas from nonabelian BrillNoether theory coupled with tropical geometry can be used to prove that the moduli space of genus 16 is uniruled. This is the highest genus for which the moduli space is known not to be of general type. For the much studied question of determining the Kodaira dimension of $M_g$, this case has long been understood to be crucial in order to make further progress. This is joint work with Verra

04/02/24
Prof. Brian Hall  University of Notre Dame
Heat flow on polynomials with connections to random matrices and random polynomials
AbstractIt is an old result of Polya and Benz the backward heat flow preserves the set of polynomials with all real roots. Recent results have shown a surprising connection between the evolution of real roots under the backward heat flow and the notion of “free convolution” in free probability. Free convolution, in turn, is the operation that allows one to compute the eigenvalue distribution for sums of independent Hermitian matrices in terms of the individual eigenvalue distributions.
The story gets even more interesting when one considers polynomials with complex roots. Recent work of mine with Ho indicates that under the heat flow, the complex roots of highdegree polynomials should evolve in straight lines with constant speed. This behavior also connects to random matrix theory and free probability. I will present some conjectures as well as recent rigorous results with Ho, Jalowy, and Kabluchko.

04/02/24
Andre Jurgen Massing  Norwegian University of Science and Technology
Cut finite element methods for complex multiphysics problems
AbstractMany advanced computational problems in engineering and biologyrequire the numerical solution of multidomain, multidimension, multiphysics and multimaterial problems with interfaces. When the interface geometry is highly complex or evolving in time, the generation of conforming meshes may become prohibitively expensive, thereby severely limiting the scope of conventional discretization methods.
In this talk we focus on recent, socalled cut finite element methods (CutFEM) as one possible remedy. The main idea is to design a discretization method which allows for the embedding of purely surfacebased geometry representations into structured and easytogenerate background meshes.In the first part of the talk, we explain how the CutFEM framework leads to accurate and optimal convergent discretization schemes for a variety of PDEs posed on complex geometries. Furthermore, we demonstrate their effectiveness when discretizing PDEs on evolving domains, including NavierStokes equations and fluidstructure interaction problems with large deformations. In the second part of the talk, we show that the CutFEM framework can also be used to discretize surfacebound PDEs as well as mixeddimensional problems where PDEs are posed on domains of different topological dimensionality.
As a particular example, we consider the socalled ExtracellularMembranIntracel
lular (EMI) model which couples an elliptic partial differential equation on the extra/intracellular domains with a system of nonlinear ordinary differential equations (ODEs) over the cell membranes to model of electrical activity of explicitly resolved brain cells. 
04/04/24
Sam Freedman  Brown University
Periodic points of Veech surfaces
AbstractWe will consider the dynamics of automorphisms acting on highlysymmetric flat surfaces called Veech surfaces. Specifically, we'll examine the points of the surface that are periodic, i.e., have a finite orbit under the whole automorphism group. While this set is known to be finite for primitive Veech surfaces, for applications it is desirable to determine the periodic points exactly. In this talk we will classify periodic points for the case of minimal Prym eigenforms, certain primitive Veech surfaces in genera 2, 3, and 4.

04/04/24
Dr. Lev S. Tsimring  Synthetic Biology Institute, UC San Diego
Statistical theory of asymmetric damage segregation in clonal cell populations
AbstractAsymmetric damage segregation (ADS) is ubiquitous among unicellular organisms: After a mother cell divides, its two daughter cells receive sometimes slightly, sometimes strongly different fractions of damaged proteins accumulated in the mother cell. Previous studies demonstrated that ADS provides a selective advantage over symmetrically dividing cells by rejuvenating and perpetuating the population as a whole. In this work we focus on the statistical properties of damage in individual lineages and the overall damage distributions in growing populations for a variety of ADS models with different rules governing damage accumulation, segregation, and the lifetime dependence on damage. We show that for a large class of deterministic ADS rules the trajectories of damage along the lineages are chaotic, and the distributions of damage in cells born at a given time asymptotically becomes fractal. By exploiting the analogy of linear ADS models with the Iterated Function Systems known in chaos theory, we derive the Frobenius–Perron equation for the stationary damage density distribution and analytically compute the damage distribution moments and fractal dimensions. We also investigate nonlinear and stochastic variants of ADS models and show the robustness of the salient features of the damage distributions.

04/04/24
Christopher Gartland  UCSD
Metric Embeddings
AbstractWe will survey the theory of embeddings between metric spaces. Most attention will be paid to biLipschitz embeddings between particular metric spaces of interest such as Banach spaces, Wasserstein spaces, and finitely generated groups.

04/04/24
Mikael de la Salle  CNRS, Lyon and IAS
Fourier Analysis with arithmetic groups
AbstractI will explain how ideas of classical harmonic analysis about convergence of Fourier series, Hilbert transform and other Fourier multipliers can be extended and applied to the setting of semisimple Lie groups and their lattices, obtaining interesting applications to operator algebras and representation theory.

04/09/24
Dr. Tattwamasi Amrutam  Ben Gurion University, Negev
On amenable subalgebras of the group von Neumann algebra
AbstractIn a joint work with Yair Hartman and Hanna Oppelmayer, we study the subvon Neumann Algebras of the group von Neumann algebra $L\Gamma$. We will first show that $L\Gamma$ admits a maximal invariant amenable subalgebra. We will also introduce the notion of invariant probability measures on the space of subvon Neumann algebras (IRAs), which is analogous to the concept of Invariant Random Subgroups. We shall show that amenable IRAs are supported on the maximal amenable invariant subalgebra. 
04/09/24
Ewan Cassidy  Durham University
SchurWeyl duality for symmetric groups
AbstractSchurWeyl duality involves the commuting actions of the general linear group and the symmetric group on a tensor space, relating the irreducible representations of these two groups. The idea can be generalised to other groups using the partition algebra and its subalgebras. I will discuss one such generalisation, `SchurWeylJones duality', as well as a refinement of this used to obtain a combinatorial formula for irreducible characters of the symmetric group. Time permitting, I will discuss an application of this formula towards obtaining new bounds on the expected irreducible character of a wrandom permutation, that is, a random permutation obtained via a word map $w : S_n \times \cdots \times S_n \rightarrow S_n$.

04/11/24
Tariq Osman  Brandeis University
TBA

04/11/24
Moritz Voss  UCLA
Equilibrium in functional stochastic games with meanfield interaction
AbstractWe study a general class of finiteplayer stochastic games with meanfield interaction where the linearquadratic cost functional includes linear operators acting on controls in L^2. We propose a new approach for deriving the Nash equilibrium of these games in terms of operator resolvents, by reducing the associated first order conditions to a system of stochastic Fredholm equations which can be solved. Moreover, by deriving stability results for the system of Fredholm equations, we obtain the convergence of the finiteplayer Nash equilibrium to the meanfield equilibrium in the infinite player limit. Our general framework includes examples of stochastic Volterra linearquadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.
This is joint work with Eduardo Abi Jaber (Ecole Polytechnique) and Eyal Neuman (Imperial College London). The paper is available at https://ssrn.com/abstract=4470883 .

04/11/24
Prof. Weitao Chen  Department of Mathematics, UC Riverside
A Mechanochemical Coupled Model to Understand Budding Behavior in Aging Yeast
AbstractCell polarization, in which a uniform distribution of substances becomes asymmetric due to internal or external stimuli, is a fundamental process underlying cell mobility and cell division. Budding yeast provides a good system to study how biochemical signals and mechanical properties coordinate with each other to achieve stable cell polarization and give rise to certain morphological change in a single cell. Recent experimental data suggests yeast budding develops into two trajectories with different bud shapes as mother cells become old. We first developed a 2D model to simulate biochemical signals on a shapechanging cell and investigated strategies for robust yeast mating. Then we extended and coupled this biochemical signaling model with a 3D subcellular element model to take into account cell mechanics, which was applied to investigate how the interaction between biochemical signals and mechanical properties affects the cell polarization and budding initiation. This 3D mechanochemical model was also applied to predict mechanisms underlying different bud shape formation due to cellular aging.

04/11/24
Claus Sorensen  UC San Diego
Projective smooth representations mod $p$
AbstractThis talk will be colloquial and geared towards people from other fields. I will talk about smooth mod $p$ representations of $p$adic Lie groups. In stark contrast to the complex case, these categories typically do not have any (nonzero) projective objects. For reductive groups this is a byproduct of a stronger result on the derived functors of smooth induction. The talk is based on joint work with Peter Schneider.

04/11/24
Ilijas Farah  York University
CORONA RIGIDITY
AbstractIn the early years of the XX century, Weyl initiated study of compact perturbations of pseudodifferential operators. The Weylvon Neumann theorem asserts that two selfadjoint operators on a complex Hilbert space are unitarily equivalent modulo compact perturbations if and only if their essential spectra coincide. Berg and Sikonia (independently) extended this result to normal operators. New impetus to the subject was given in 1970s by Brown, Douglas, and Fillmore, who replaced single operators with (separable) C*algebras and realized that compact perturbations can be considered as extensions by the ideal of compact operators. After passing to the quotient (the Calkin algebra, Q) and identifying an extension with a *homomorhism into Q, analytic methods had been supplemented with methods from algebraic topology, homological algebra, and (most recently) logic. Around the same time, Shelah proved one of his many influential results, by showing that the assertion `all automorphisms of $\ell_\infty/c_0$ are trivial' is relatively consistent with ZFC. Surprisingly, these two directions of research are intimately connected.
This talk will be about rigidity of quotient structures and it is partially based on the preprint Farah, I., Ghasemi, S., Vaccaro, A., and Vignati, A. (2022). Corona rigidity. arXiv preprint arXiv:2201.11618
https://arxiv.org/abs/2201.
11618 and some more recent results. 
04/13/24
Southern California Geometric Analysis Seminar  April 1314, 2024
AbstractThe 29th SCGAS will be held at the Department of Mathematics of University of California at San Diego on Saturday, April 13, 2024 and Sunday, April 14, 2024. The lectures will be held in Natural Science Building Auditorium (04/13) and Center Hall 105 (04/14) due to the campus event of Triton Day. For directions on how to get to Natural Science Building see map; For the Center Hall, here is a map.
And here are directions from the BWDel Mar to the UCSD campus.
Registration starts at 10am Saturday morning. The first talk will be at 11:00am and the last talk will finish at 12:30pm on Sunday, to allow for travel.
Graduate students, recent Ph.D.s and underrepresented minorities are especially encouraged to join our annual seminar. Partial financial support is available.
The Seminar is supported by the NSF and by the School of Physical Sciences at UC San Diego.
Invited Speakers: Guido De Philippis (CIMS), Bruce Kleiner (CIMS), Yi Lai (Stanford), Bill Minicozzi (MIT), Song Sun (Berkeley/Zhejiang Univ.), Guofang Wei (UCSB), Xin Zhou (Cornell)
Registration: Participants are asked to register online: the electronic registration form is now available.

04/16/24
Dr. Ian Charlesworth & Dr. David Jekel  Cardiff University/Fields Institute for Research in Mathematical Sciences
Algebraic soficity and graph products
AbstractWe show that a graph product of tracial von Neumann algebras is strongly $1$bounded if the first $\ell^2$Betti number vanishes for an associated dense $*$subalgebra. Graph products of tracial von Neumann algebras were studied by Caspers and Fima, and generalize Green's graph product of groups. Given groups $G_v$ for each vertex of a graph $\Gamma$, the graph product is the free product modulo the relations that $G_v$ and $G_w$ commute when $v \sim w$; for von Neumann algebras, graph products are described by a certain moment relation. In our paper, the crux of the argument is a generalization to tracial von Neumann algebras of the statement that soficity of groups is preserved by graph products. We replace soficity for groups with a more general notion of algebraic soficity for a $*$algebra $A$, which means the existence of certain approximations for the generators of $A$ by matrices with algebraic integer entries and approximately constant diagonal. We show algebraic soficity is preserved under graph products through a random permutation construction, inspired by previous work of Charlesworth and Collins as well as AuC{\'e}bronDahlqvist
GabrielMale. In particular, this gives a new probabilistic proof of CiobanuHoltRees's result that soficity of groups is preserved by graph products. This is based on joint work with Rolando de Santiago, Ben Hayes, Srivatsav Kunnawalkam Elayavalli, Brent Nelson.

04/18/24
Prof. Keaton Hamm  University of Texas at Arlington
Tensor decompositions by mode subsampling
AbstractWe will overview variants of CUR decompositions for tensors. These are lowrank tensor approximations in which the constituent tensors or factor matrices are subtensors of the original data tensors. We will discuss variants of Tucker decompositions and those based on tproducts in this framework. Characterizations of exact decompositions are given, and approximation bounds are shown for data tensors contaminated with Gaussian noise via perturbation arguments. Experiments are shown for image compression and time permitting we will discuss applications to robust PCA.

04/18/24
Professor Ruth J. Williams  UCSD
Stochastic Analysis of Chemical Reaction Networks with Applications to Epigenetic Cell Memory
AbstractEpigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multicellular organisms. Simulation studies have shown how stochastic dynamics and timescale differences between establishment and erasure processes in chromatin modifications (such as histone modifications and DNA methylation) can have a critical effect on epigenetic cell memory.
In this talk, we describe a mathematical framework to rigorously validate and extend beyond these computational findings. Viewing our stochastic model of a chromatin modification circuit as a singularly perturbed, finite state, continuous time Markov chain, we extend beyond existing theory in order to characterize the leading coefficients in the series expansions of stationary distributions and mean first passage times. In particular, we characterize the limiting stationary distribution in terms of a reduced Markov chain, provide an algorithm to determine the orders of the poles of mean first passage times, and describe a comparison theorem which can be used to explore how changing erasure rates affects system behavior. These theoretical tools not only allow us to set a rigorous mathematical basis for the computational findings of prior work, highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains especially those associated with chemical reaction networks.
Based on joint work with Simone Bruno, Felipe Campos, Yi Fu and Domitilla Del Vecchio.

04/18/24
Aranya Lahiri  UC San Diego
Distribution algebras of padic groups
AbstractMy goal will be to motivate why looking at distribution algebras associated to padic lie groups is natural in the context of number theory. More specifically I will try to briefly outline their importance in the padic Langlands program. And then I will give a simple example of an overconvergent distribution algebra of certain kinds of padic groups with an eye towards illuminating techniques used in my work Dagger groups and padic distribution algebras (joint w/ Matthias Strauch and Claus Sorensen).

04/18/24
David Jekel  Fields Institute for Research in Mathematical Sciences
Infinitedimensional, noncommutative probability spaces and their symmetries
AbstractThere is a deep analogy between, on the one hand, matrices and their trace, and on the other hand, random variables and their expectation. This idea motivates "quantum" or noncommutative probability theory. Tracial von Neumann algebras are infinitedimensional analogs of matrix algebras and the normalized trace, and there are several ways to construct von Neumann algebras that represent suitable "limits" of matrix algebras, either through inductive limits, random matrix models, or ultraproducts. I will give an introduction to this topic and discuss the ultraproduct of matrix algebras and its automorphisms or symmetries. This study incorporates ideas from model theory as well as probability and optimal transport theory.

04/19/24
Hugo Jenkins  UCSD
No Prerequisites CayleyBacharach
AbstractThe CayleyBacharach theorem says that if two plane cubics intersect in exactly 9 points, then any third cubic passing through eight of these must pass through the ninth. We'll give a weird, elementary but cute proof which shows something a tiny bit stronger. The prerequisites will be not nil but nilpotent, limited to Bezout's theorem which I'll state carefully in the form I need. This proof came from Math 262A, which apparently got it from Terence Tao's blog.

04/22/24
Runqiu Xu  UCSD
A Comparison of U(N) and SU(N) Weingarten functions
AbstractU(N) Weingarten function, known in computing the U(N) link integral, is an essential ingredient in physics. Although fewer people pay attention to SU(N), the SU(N) Weingarten function is important in the lattice gauge theory and it differs from U(N) . In this talk, I will present the derivation of the SU(N) Weingarten function using character theory and emphasize some details about how it differs from the perspective of polynomial representation of $GL_N$. We will also explore the nice combinatorial interpretation of the 1/N expansion of the Weingarten function using HurwitzCayley graph which serves as the Feynman diagram

04/23/24
Haoyu Zhang  UCSD
An interacting particle consensus method for constrained global optimization
AbstractThis talk presents a particlebased optimization method designed for addressing minimization problems with equality constraints, particularly in cases where the loss function exhibits nondifferentiability or nonconvexity. A rigorous meanfield limit of the particle system is derived, and the convergence of the meanfield limit to the constrained minimizer is established.

04/23/24
Jesse Peterson  Vanderbilt University
Biexact von Neumann algebras
AbstractThe notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. In joint work with Changying Ding, we extended this notion from the group theory setting to the setting of von Neumann algebras, thereby giving a unified setting for proving solidity type results. We will discuss biexactness and solidity and give examples of solid von Neumann algebras that are not biexact.

04/23/24
Bryan Hu  UCSD
Advancement to Candidacy

04/23/24
Colleen Robichaux  UCLA
Exploring Kohnert’s rule for Grothendieck polynomials
AbstractA 2015 conjecture of Ross and Yong proposes a KKohnert rule for Grothendieck polynomials. In this talk we discuss the utility of Kohnert rules then prove a special case of the RossYong conjecture. We then show the conjecture fails in general.

04/24/24
Prof. Yuhua Zhu  UCSD
An interacting particle method for global optimization
AbstractThis talk presents a particlebased optimization method designed for addressing global optimization problems, particularly in cases where the loss function exhibits nondifferentiability or nonconvexity. Numerically, we show that it outperforms gradientbased method in finding global optimizer. Theoretically, A rigorous meanfield limit of the particle system is derived, and the convergence of the meanfield limit to the global minimizer is established. In addition, we will talk about its application to the constrained optimization problems and federated learning.

04/25/24
Prof. Konstantinos Panagiotou  LMU Munich
Limit Laws for Critical Dispersion on Complete Graphs
AbstractWe consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time.
In this work we study the case where $G$ is the complete graph on $n$ vertices and the number of particles is $M=n/2+\alpha n^{1/2} + o(n^{1/2})$, $\alpha\in \mathbb{R}$.This choice of $M$ corresponds to the critical window of the process, with respect to the dispersion time.We show that the dispersion time, if rescaled by $n^{1/2}$, converges in $p$th mean, as $n\rightarrow \infty$ and for any $p \in \mathbb{R}$, to a continuous and almost surely positive random variable $T_\alpha$.
We find that $T_\alpha$ is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that $\mathbb{E}[T_0] = \pi^{3/2}/\sqrt{7}$, and furthermore we formulate explicit asymptotics when~$\alpha$ gets large that quantify the transition into and out of the critical window. We also study the random variable counting the \emph{total number of jumps} that are performed by the particles until the dispersion time is reached and prove that, if rescaled by $n\ln n$, it converges to $2/7$ in probability.
Based on joint work with Umberto De Ambroggio, Tamás Makai, and Annika Steibel; see arXiv:2403.05372

04/25/24
Jon Aycock  UC San Diego
Congruences Between Automorphic Forms
AbstractWe will introduce an analytic notion of automorphic forms. These automorphic forms encode arithmetic data by way of their Fourier theory, and we will explore two different families of automorphic forms which have interesting congruences between their Fourier coefficients.

04/25/24
Prof. Tsachik Gelander  Northwestern University
Things we can learn by looking at random manifolds
AbstractIn mathematics in general, it is fruitful to allow randomness. Indeed, it is often easier to deal with random rather than deterministic objects. It seems miraculous, however, when we are able to say more about deterministic objects by treating them as random ones.
This idea applies in particular to the theory of discrete subgroups of Lie groups and locally symmetric manifolds.
The theory of invariant random subgroups (IRS), which has been developed quite rapidly during the last decade, has been very fruitful to the study of lattices and their asymptotic invariants. However, restricting to invariant measures limits the scope of problems that one can approach (in particular since the groups involved are highly nonamenable). It was recently realised that the notion of stationary random subgroups (SRS) is still very effective and opens paths to deal with questions which were thought to be unreachable.
In the talk I will describe various old and new results concerning arithmetic groups and general locally symmetric manifolds of finite as well as infinite volume that can be proved using `randomness', e.g.:
1. KazhdanMargulis minimal covolume theorem.
2. Most hyperbolic manifolds are nonarithmetic (a joint work with A. Levit).
3. Higher rank manifolds of large volume have a large injectivity radius (joint with Abert, Bergeron, Biringer, Nikolov, Raimbault and Samet).
4. Higher rank manifolds of infinite volume have infinite injectivity radius  conjectured by Margulis (joint with M. Fraczyk).

04/26/24
Srivatsa Srinivas  UCSD
Can You Shove an Arithmetic Progression into a Geometric Progression?
AbstractWhat is the longest arithmetic progression that is a subset of a geometric progression? This problem is not as benign as it looks. But I bet we could do something or the other...

04/29/24
Professor Tsachik Gelander  Northwestern University
Spectral gap for irreducible subgroups and a strong version of Margulis normal subgroup theorem.
AbstractLet \(\Gamma\) be a discrete group. A subgroup \(N\) is called confined if there is a finite set \(F\) in \(\Gamma\) which intersects every conjugate of \(N\) outside the trivial element. For example, a nontrivial normal subgroup is confined.
A discrete subgroup of a semisimple Lie group is confined if the corresponding locally symmetric orbifold has bounded injectivity radius. We proved a generalization of the celebrated NST: Let \(\Gamma\) be an irreducible lattice in a higher rank semisimple Lie group G. Let \(N<\Gamma\) be a confined subgroup. Then \(N\) is of finite index.
The case where \(G\) has Kazhdan's property (T) was established in my joint work with Mikolaj Frakzyc. As in the original NST, without property (T) the problem is considerably harder. The main part is to prove a spectral gap for \(L_2(G/N)\).
This is a joint work with Uri Bader and Arie Levit.

04/30/24
Dr. Pawel Kasprzak  University of Warsaw, Poland
Quantum Mycielski Graphs
AbstractQuantum graphs and their characteristics are intriguing generalizations of notions and tools known from discrete mathematics into the quantum world. Their nontrivial relations with quantum information theory provide a bridge between this branch of mathematics and quantum mechanics. In classical graph theory, there are several characteristics that one can associate with given graphs, e.g., chromatic or clique numbers. The famous problem, solved by Mycielski, was to construct a graph that contains a given graph as a subgraph and can have an arbitrarily large chromatic number, but no larger clique is produced. We propose an analog of the Mycielski transformation and its generalizations in the quantum setting and study how they affect the (quantum) characteristics of quantum graphs. Moreover we study relations between quantum automorphism groups of a quantum graph and its Mycielskian. Based on joint work with A. Bochniak (arXiv:2306.09994) and work in progress with A. Bochniak, P.M. Sołtan, and I. Chełstowski.
May

05/02/24
Albert Artiles Calix  University of Washington
Statistics of Minimal Denominators and Short Holonomy Vectors of Translation Surfaces
AbstractThis talk will explore the connection between Diophantine approximation and the theory of homogeneous dynamics. The first part of the talk will be used to define and study the minimal denominator function (MDF). We compute the limiting distribution of the MDF as one of its parameters tends to zero. We do this by relating the function to the space of unimodular lattices on the plane.
The second part of the talk will be devoted to describing equivariant processes. This will give a general framework to generalize the main theorem in two directions:
1. Higher dimensional Diophantine approximation
2. Statistics of short saddle connections of Veech surfaces
If time allows, we will compute formulas for the statistics of short holonomy vectors of translation surfaces.

05/02/24
Zichao Wang  UCSD
Nonlinear spiked covariance matrices and signal propagation in neural network models
AbstractIn this talk, we will discuss recent work on the extreme eigenvalues of the sample covariance matrix with a spiked population covariance. Extending previous random matrix theory, we will characterize the spiked eigenvalues outside the bulk distribution and their corresponding eigenvectors for a nonlinear version of the spiked covariance model. Our result shows the universality of the spiked covariance model with the same quantitative spectral properties as a linear spiked covariance model. In the proof, we will present a deterministic equivalent for the Stieltjes transform for any spectral argument separated from the support of the limit spectral measure. Then, we will apply this new result to deep neural network models. We will describe how spiked eigenstructure in the input data propagates through the hidden layers of a neural network with random weights. As a second application, we can study a simple regime where the weight matrix has a rankone signal component over gradient descent training and characterize the alignment of the target function. This is a joint work with Denny Wu and Zhou Fan.

05/02/24
Wei Yin  UC San Diego
Higher CoatesSinnott Conjectures for CMFields
AbstractThe classical CoatesSinnott Conjecture and its refinements predict the dee relationship between the special values of Lfunctions and the structure of the étale cohomology groups attached to number fields. In this talk, we aim to delve deeper along this direction to propose what we call the “Higher CoatesSinnott Conjectures" which reveal more information about these two types of important arithmetic objects. We introduce the conjectures we formulate and our work towards them. This is joint work with C. Popescu.

05/02/24
Srivatsav Kunnawalkam Elayavalli  UCSD
Sequential commutation
AbstractI will discuss a new conceptual framework called sequential commutation that has applications to von Neumann algebra theory. These focus on joint works by the speaker and others including Patchell, Gao and Tan.

05/02/24
Prof. Weng Kee Wong  UCLA
Using Animal Instincts to Find Optimal Designs for Early Phase Clinical Trials
AbstractNatureinspired metaheuristics is widely used in computer science and engineering but seems greatly underused in pharmaceutical research, clinical science research, and somewhat in statistical science as well. This class of algorithms is appealing because they are essentially assumptions free, fast and have been shown that they are capable of tackling all sorts of high dimensional complex optimization problems. We first review optimal design theory, some exemplary natureinspired metaheuristic algorithms and show how they can be applied to (i) find efficient designs for estimating the Biologically Optimal Dose (BOD), (ii) extend Simon’s 2 stage designs for a Phase II trial with a single alternative hypothesis to one with multiple alternative hypotheses to capture the uncertainty of the efficacy of the drug more accurately, and if time permits (iii) find a Doptimal designs for estimating parameters in 10 interacting factors. We also indicate how metaheuristics can be applied to develop more realistic and flexible adaptive designs for early phase clinical trials.

05/07/24
Prof. Keivan MallahiKarai  Constructor University
A Central limit theorem for random walks on horospherical products of Gromov hyperbolic spaces
AbstractLet \(G\) be a countable group acting by isometries on a metric space \((M, d)\), and let \(\mu\) denote a probability measure on \(G\). The \(\mu\)random walk on \(M\) is the random process defined by
\[Z_n=X_n \dots X_1 o,\]
where \(o \in M\) is a fixed base point, and \(X_i\) are independent \(\mu\)distributed random variables.Studying statistical properties of the displacement sequence \(D_n:= d(Z_n, o)\) has been a topic of current research.
In this talk, which is based on a joint work with Amin Bahmanian, Behrang Forghani, and Ilya Gekhtman, I will discuss a central limit theorem for \(D_n\) in the case that \(M\) is the horospherical product of Gromov hyperbolic spaces.

05/07/24
Patrick Hiatt  UCLA
On the Singular Abelian Rank of Ultraproduct II$_1$ Factors
AbstractI will present some recent joint work with Sorin Popa where we show that, under the continuum hypotheses, any ultraproduct II$_1$ factor contains more than continuum many mutually disjoint singular MASAs. In other words, the singular abelian rank of any ultraproduct II$_1$ factor $M$, $\text{r}(M)$, is larger than $\mathfrak{c}$. Moreover, if the strong continuum hypothesis $2^\mathfrak{c}=\aleph_2$ is assumed, then $\text{r}(M) = 2^\mathfrak{c}$. More generally, these results hold true for any II$_1$ factor $M$ with unitary group of cardinality $\mathfrak{c}$ that satisfies the bicommutant condition $(A_0'\cap M)'\cap M=M$, for all $A_0\subset M$ separable abelian.

05/07/24
Prof. Carl Mautner  UC Riverside
Symmetric groups, Schur algebras and Hilbert schemes
AbstractIn his 1901 thesis, Issai Schur discovered a connection between the representation theory of the symmetric group and general linear group. One way to understand this connection is through a finite dimensional algebra called the Schur algebra. I will outline this picture and then describe a new algebra, defined in joint work with Tom Braden, which enhances the Schur algebra and provides a new window into the representation theory of symmetric groups. Finally, I will explain how we came to discover this algebra by studying the geometry of Hilbert schemes of points in the plane and how this fits into my larger program to uncover representation theory in the geometry of symplectic singularities and their resolutions.

05/07/24
Víctor Rivero  Center of Research in Mathematics, Guanajuato, Mexico
An excursion from selfsimilar Markov processes to Markov additive processes
AbstractIn stochastic modeling we often need to deal with one of two apparently unrelated objects. One is selfsimilar processes and the other is additive functionals. Selfsimilar Markov processes are the class of Markovian models that arise as scaling limits of stochastic processes, that are obtained after renormalization of time and space. Additive functionals arise commonly when one considers, for instance, rewards associated to a Markovian model.
On the one hand, the socalled Lamperti transform ensures that any $R^d$valued selfsimilar Markov process admits a polar decomposition, and the argument and the radius of the process are related to a Markov additive process via an explicit time change. On the other hand, any additive functional A of a Markov process X is such that the pair (A, X) is a Markov additive process. A Markov additive process (MAP) is a stochastic process with two components: one that is additive, and real valued, the ordinator, and a general one, the modulator, that rules the behavior of the ordinator. The ordinator has independent and stationary increments, given the modulator. This general structure emulates the structure of processes with independent and stationary increments, Levy processes, as for instance Brownian motion, Cauchy and stable processes, Gamma processes, etc.
In general, it is too ambitious to try to determine explicitly the whole law of a selfsimilar Markov process or of an additive functional. But we can aim at understanding properties of the extremes of these processes and to be ready for the best and worst scenarios. In the fluctuation theory of Markov additive processes we aim at developing tools for studying the extremes of the additive part, ordinator, of the process. This has been done in a systematic way during the last four decades under the assumption that the modulator is a constant process, and hence the ordinator is a real valued Levy process. Also, in the 198090 period, some foundations were laid to develop a fluctuation theory for MAPs in a general setting.
In this talk we aim at giving a brief overview of the fluctuation theory of Markov additive processes, to describe some recent results and to provide some applications to the theory of selfsimilar Markov processes. These applications are mainly related to stable processes, a class of processes that arises often in mathematical physics, potential and harmonic analysis, and in other areas of mathematics. We aim at making this overview accessible to graduate and advanced undergraduate students, with some knowledge of Markov chains and Levy processes, and to point out at some open research questions.

05/08/24
Scotty Tilton  UC San Diego
Distinguishing diffeomorphisms with Equivariant BauerFuruta Invariants
AbstractSimply connected smooth 4manifolds are complicated; understanding anything about them is good progress toward the larger goal of classification. There have been some discoveries in the past few years that distinguish exotic diffeomorphisms (which are topologically isotopic, but not smoothly so) using the families BauerFuruta invariant. The goal of this talk is to provide the context of this area, the background for the families BauerFuruta invariant, and some ideas for my future research directions.

05/09/24
Dr. John PecaMedlin  University of Arizona, Tucson
Random permutations using GEPP
AbstractGaussian elimination with partial pivoting (GEPP) remains the most used dense linear solver. For a n x n matrix A, GEPP results in the factorization PA = LU where L and U are lower and upper triangular matrices and P is a permutation matrix. If A is a random matrix, then the associated permutation from the P factor is random. When is this a uniform permutation? How many cycles are in its disjoint cycle decomposition (which equivalently answers how many GEPP pivot movements are needed on A)? What is the length of the longest increasing subsequence of this permutation? We will provide some statistical answers to these questions for select random matrix ensembles and transformations. For particular butterfly permutations, we will present full distributional descriptions for these particular statistics. Moreover, we introduce a random butterfly matrix ensemble that induces the Haar measure on a full 2Sylow subgroup of the symmetric group on a set of size 2ⁿ.

05/09/24
Nandagopal Ramachandran  UC San Diego
Euler factors in Drinfeld modules
AbstractIn this talk, I'll first give a quick introduction to the theory of Drinfeld modules and talk about an equivariant $L$function associated to Drinfeld modules as defined by FerraraHigginsGreenPopescu in their work on the ETNC. As is usual, these $L$functions are defined as an infinite product of Euler factors, and the main focus of this talk is a result relating these Euler factors to a certain quotient of Fitting ideals of some algebraically relevant modules. This is joint work with Cristian Popescu.

05/09/24
Prof. Feng Fu  Dartmouth College
Evolutionary Dynamics of Human Behavior
AbstractHuman behavior impacts the world around us. From disease control to climate change, understanding human behavior through the lens of evolutionary dynamics provides useful insights and implications for making the world a better place. This multidisciplinary, datadriven modeling approach combines various introspective processes with interpersonal interactions by accounting for interdependent biological and social network processes across different yet interconnected network layers. In this talk, we will present recent work on modeling complex, multifaceted human behavior across diverse domains in critical issues of societal importance, ranging from sociocognitive biases to pandemic compliance. The talk will also discuss the importance of bottomup behavior and attitude changes, as well as largescale human cooperation, in addressing urgent challenges facing our common humanity.

05/13/24
Prof. Xindong TANG  Hong Kong Baptist University
Biquadratic Games and MomentSOS relaxations
AbstractWe consider the mixedstrategy zerosum game such that each player’s objective function is quadratic in its own variables. By considering each player’s value function and duality, the biquadratic games are reformulated as linear programs over the cone of copositive (COP) and completely positive (CP) matrices. We apply moment and SOS relaxations for the conic constraints of CP and COP matrices, respectively, and obtain a hierarchy of semidefinite relaxations. Under certain conditions, the finite convergence for this hierarchy is guaranteed, and the tightness can be checked via flat truncation. We present numerical experiments to show the effectiveness of our approach.

05/14/24
PerOlof Persson  UC Berkeley
A FaceUpwinded Spectral Element Method and Machine Learning for Optimal Meshes
AbstractThe talk will cover two new developments in our work on efficient highorder methods for conservation laws: (1) A simple stabilization technique for spectral element methods, which uses continuous solution spaces and is provably convergent for linear problems at arbitrary orders of accuracy. The main motivation for the new scheme is its lower cost, which comes from having fewer degrees of freedom, no Riemann solvers, and a linebased sparsity pattern. However, it also has other attractive properties such as an improved CFL condition and allowing for other solvers including static condensation. (2) A deep reinforcement learning approach for generation of meshes with optimal connectivities. Starting from a Delaunay mesh, we formulate the mesh optimization process as a "game" where the moves are standard topological element operations, and the goal is to maximize the number of regular nodes. The agent is trained in a selfplay framework using the proximal policy optimization (PPO) algorithm running on GPUs. Our approach works for 2D triangular and quadrilateral meshes with minimal modification, and it routinely produces closetoperfect meshes.

05/14/24
Jack Xin  UCI
Computing Entropy Production Rates and Chemotaxis Dynamics in High Dimensions by Stochastic Interacting Particle Methods
AbstractWe study stochastic interacting particle methods with and without field coupling for high dimensional concentration and singularity formation phenomena. In case of entropy production of reversetime diffusion processes, the method computes concentrated invariant measures meshfree up to dimension 16 at a linear complexity rate based on solving a principal eigenvalue problem of nonselfadjoint advectiondiffusion operators. In case of fully parabolic chemotaxis nonlinear dynamics in 3D, the method captures critical mass for finite time singularity formation and blowup time at low costs through a smoother field without relying on selfsimilarity.

05/14/24
Aldo Garciaguinto  Michigan State University
Schreier's Formula for some Free Probability Invariants
AbstractLet $G\stackrel{\alpha}{\
curvearrowright}(M,\tau)$ be a tracepreserving action of a finite group $G$ on a tracial von Neumann algebra. Suppose that $A \subset M$ is a finitely generated unital $*$subalgebra which is globally invariant under $\alpha$. We give a formula relating the von Neumann dimension of the space of derivations on $A$ valued on its coarse bimodule to the von Neumann dimension of the space of derivations on $A \rtimes^{\text{alg}}_\alpha G$ valued on its coarse bimodule, which is reminiscent of Schreier's formula for finite index subgroups of free groups. This formula induces a formula for $\dim \text{Der}_c(A,\tau)$ (defined by Shlyakhtenko) and under the assumption that $G$ is abelian we obtain the formula for $\Delta$ (defined by Connes and Shlyakhtenko). These quantities and the free entropy dimension quantities agree on a large class of examples, and so by combining these results with known inequalities, one can expand the family of examples for which the quantities agree. 
05/14/24
Prof. Jane Gao  University of Waterloo
Evolution of random representable matroids
AbstractInspired by the classical random graph process introduced by Erdos and Renyi in 1960, we discuss two analogous processes for random representable matroids, one introduced by Kelly and Oxley in 1982 and the other one introduced by Cooper, Frieze and Pegden in 2019. In the talk we address the evolution of the rank, circuits, connectivity, and the critical number (corresponding to the logarithm of the chromatic number of graphs) of the first random matroid, and then we focus on the minors in both matroid models.

05/15/24
Prof. Suhan Zhong  Texas A&M University
Twostage stochastic optimization
AbstractThis talk discusses the challenging problem of finding global optimal solutions for twostage stochastic programs with continuous decision variables and nonconvex recourse functions. We introduce a twophase approach, which does not only generate global lower bounds for the nonconvex stochastic program but also simplifies the computation of the expected value of the recourse function by using moments of random vectors. This makes our overall algorithm particularly suitable for the case where the random vector follows a continuous distribution or when dealing with many scenarios. Numerical experiments are given to demonstrate the effectiveness of our proposed approach.

05/15/24
Cameron Cinel  UCSD
Linearly Sofic Lie Algebras

05/16/24
Qingyuan Chen  UCSD
Shannon Orbit Equivalences Preserve KolmogorovSinai Entropy
AbstractWe will consider the behavior of the KolmogorovSinai entropies of amenable group actions under a Shannon orbit equivalence. Although dynamical entropy is in general not invariant under orbit equivalences, recent works have shown that various notions of restricted orbit equivalences will preserve entropy. We focus on the case where the orbit equivalence is Shannon, and both groups are finitely generated amenable. In this talk, we will present a proof for our main result.

05/16/24
Garrett Tresch  Texas A&M University
Stochastic Embeddings of Graphs into Trees
AbstractAs the shortest path metric on a weighted tree can be embedded isometrically into a finite $\ell_1$ space, a Lipschitz embedding of a given graph into $\ell_1$ can be obtained by constructing a low distortion embedding into a tree. Conversely, while there are various topological properties of graphs that guarantee controlled distortion Lipschitz embeddings into $\ell_1$ ($k$outerplanar, seriesparallel, low Euler characteristic), it is still often the case that such a graph embeds quite poorly into a tree.
By introducing the notion of a stochastic embedding into a family of trees one can find more general concrete embeddings into $\ell_1$ then those limited by a single tree. In fact, it is known that every graph with n vertices embeds stochastically into trees with distortion O(log(n)). Nevertheless, this upper bound is sharp for graphs such as expanders, grids and, by a recent joint work with Schlumprecht, a large class of "fractallike" seriesparallel graphs called slash powers.
In this talk we introduce an equivalent characterization of stochastic distortion called expected distortion and after proving a mild extension of a result of Gupta regarding poor tree embeddings of a cycle, inductively lower bound the expected distortion of generalized Laakso graphs found in most nontrivial slash power families.

05/16/24
Bryan Hu  UC San Diego
Critical values of the adjoint Lfunction of U(2,1) in the quaternionic case
AbstractWe will discuss questions surrounding automorphic Lfunctions, particularly Deligne’s conjecture about critical values of motivic Lfunctions. In particular, we study the adjoint Lfunction of U(2,1).
Hundley showed that a certain integral, involving an Eisenstein series on the exceptional group G_2, computes this Lfunction at unramified places. We discuss the computation of this integral at the archimedean place for quaternionic modular forms, and how this relates to Deligne's
conjecture. 
05/16/24
Patrick Girardet  UCSD
Automorphisms of Hilbert Schemes of Points of Abelian Surfaces
AbstractGiven an automorphism of a variety $X$, there is an induced ''natural'' automorphism on $X^{[n]}$, the Hilbert scheme of $n$ points of $X$. While unnatural automorphisms of $X^{[n]}$ are known to exist for certain varieties $X$ and integers $n$, all previously known examples could be shown to be unnatural because they do not preserve multiplicities. Belmans, Oberdieck, and Rennemo thus asked if an automorphism of a Hilbert scheme of points of a surface is natural if and only if it preserves the diagonal of nonreduced subschemes.
We give an answer in the negative for all $n\ge 2$ by constructing explicit counterexamples on certain abelian surfaces $X$. These surfaces are not generic, and hence we prove a partial converse statement that all automorphisms of the Hilbert scheme of two points on a very general abelian surface are natural for certain polarization types (including the principally polarized case).

05/16/24
Davide Parise  UCSD
Monotonicity formulae in analysis and geometry
AbstractWhen trying to solve partial differential equations, a common practice is to enlarge the space of possible solutions to the class of nondifferentiable functions, where it is easier to find “weak” solutions (i.e. potentially very irregular). As we are usually interested in “strong” solutions (i.e very regular), one is then confronted with the following problem: how do we upgrade the regularity? A fundamental tool in these situations is a monotonicity formula, an object that allows to study the infinitesimal behavior of solutions of PDEs by reducing it to a classification problem. More concretely, a monotonicity formula is an identity implying that a certain quantity related to the problem at hand is monotone, or conserved. I will try to convey the gist of this idea that has found applications in many areas at the intersection of geometry and analysis, e.g. harmonic maps, minimal surfaces, free boundary problems, YangMills connections to name just a few. I will try to maintain the level of analysis needed at a minimum, you only need to remember that the first derivative of a smooth function at an interior minimum is zero. I will explain the rest.

05/16/24
Paul K. Newton  University of Southern California
Control of evolutionary mean field games and tumor cell population models
AbstractMean field games are played by populations of competing agents who derive their update rules by comparing their own state variable with that of the mean field. After a brief introduction to several areas where they have been used recently, we will focus on models of competing tumor cell populations based on the replicator dynamics mean field evolutionary game with prisoner’s dilemma payoff matrix. We use optimal and adaptive control theory on both deterministic and stochastic versions of these models to design multidrug chemotherapy schedules that suppress the competitive release of resistant cell populations (to avoid chemoresistance) by maximizing the Shannon diversity of the competing subpopulations. The models can be extended to networks where spatial connectivity can influence optimal chemotherapy scheduling.

05/20/24
Ji Zeng
Variation of nothreeinline problem
AbstractThe famous nothreeinline problem by Dudeney more than a century ago asks whether one can select 2n points from the grid $[n]^2$ such that no three are collinear. We present two results related to this problem. First, we give a nontrivial upper bound for the maximum size of a set in $[n]^4$ such that no four are coplanar. Second, we characterize the behavior of the maximum size of a subset such that no three are collinear in a random set of $\mathbb{F}_q^2$, that is, the plane over the finite field of order q. We discuss their proofs and related open problems.

05/21/24
Dr. Pawel Kasprzak  University of Warsaw, Poland
Quantum Mycielski Graphs and their Quantum Groups
AbstractWe propose a quantum analog of the Mycielski transformation of graphs and iwe study relations between quantum automorphism groups of a quantum graph and its Mycielskian. Based on joint work with A. Bochniak (arXiv:2306.09994) and work in progress with A. Bochniak, P.M. Sołtan, and I. Chełstowski.

05/21/24
Anuj Kumar  UC Berkeley
Sharp nonuniqueness of the transport equation with Sobolev vector fields
AbstractWe construct nonunique solutions of the transport equation in the class $L^\infty$ in time and $L^r$ in space, for divergence free Sobolev vector fields from $W^{1,p}$. We achieve this by introducing two novel ideas: (1) in the construction, we interweave scaled copies of the vector field itself, and (2) asynchronous translation of cubes, which makes the construction heterogeneous in space. These new ideas allow us to prove nonuniqueness in the range of exponents going beyond what is available using the method of convex integration, and sharply match with the range of uniqueness of solutions from Bruè, Colombo, De Lellis ’21.

05/21/24
Prof. Michael Molloy  University of Toronto
An improved bound for the List Colouring Conjecture
AbstractThe List Colouring Conjecture posits that the list edge chromatic number of any graph is equal to the edge chromatic number, and thus is at most D+1 where D is the maximum degree. This means that if each edge is assigned a list of $D+1$ colours then it is always possible to obtain a proper edge colouring by choosing one colour from each list.
In the 1990's, Khan proved that one can always obtain a proper edge colouring from lists of size $D+o(D)$, then Molloy and Reed obtained $D+D^{1/2}\mathrm{poly}(\log D)$. We improve that bound to $D+D^{2/5}\mathrm{poly}(\log D)$

05/22/24
David Cavender
Mean Field Games for Quadrotor Control

05/23/24
Prof. Joshua Bowman  Pepperdine University
Horocycle flow on $\mathcal{H}(2)$ and the gap distribution for slopes of saddle connections
AbstractSaddle connections on a translation surface generalize both diagonals in a polygon and primitive vectors in a 2dimensional lattice. Their slopes thus contain geometric and algebraic information about the surface. Slopes of saddle connections can be studied using the action of a horocycle subgroup of $\mathrm{SL}_2(\mathbb{R})$ on the moduli space of all translation surfaces. In particular, gaps between slopes are directly related to the return time function of a Poincaré section for the horocycle flow.
In this talk, we will describe a Poincaré section for horocycle flow in the smallest nontrivial stratum $\mathcal{H}(2)$ and see how to compute the return time function. Then we will examine some consequences for gap distributions. This is joint work with Anthony Sanchez.

05/23/24
Michele Caselli  Scuola Normale Superiore, Pisa
Geometric features of fractional minimal surfaces and their generalization to higher codimension
AbstractIn this talk, I will explain why fractional (or nonlocal) minimal surfaces are ideal objects to which minmax methods can be applied on Riemannian manifolds. After a short introduction about these objects and how they approximate minimal surfaces, I will present a vision for the future on how to generalize this setting to higher codimension.

05/23/24
Christian Klevdal  UC San Diego
Local systems on Shimura varieties
AbstractA large area of modern number theory (the Langlands program) studies a deep correspondence between the representation theory of Galois groups, algebraic varieties and certain analytic objects (automorphic forms). Many spectacular theorems have come from this area, for example the key insight in Wiles' proof of Fermat's last theorem was a connection between elliptic curves, modular forms and Galois representations.
The goal of this talk is to explain how geometric constructions, particularly related to Shimura varieties, arise naturally in the Langlands program. I will then talk about joint work with Stefan Patrikis, stating that Galois representations arising from certain Shimura varieties satisfy the properties predicted by the correspondence introduced above. 
05/23/24
Dr. Pearson Miller  Flatiron Institute, Simons Foundation
Hierarchical Control of Biological SelfOrganization
AbstractClassic mechanisms of spatial pattern formation in developmental biology are characterized by high degrees of multistability and sensitivity to initial conditions. These traits are commonly seen as undermining the capacity of these processes to exhibit robust morphogenesis. However, a growing body of experimental evidence suggests developing organisms can accomplish robust pattern selection in reactiondiffusion processes with relatively simple spatiotemporal forcings. To better understand this phenomenon, we perform a series of systematic investigations into the optimal controllability of a minimal patternforming system. Using machinelearninginspired techniques, we generate simple optimal control protocols to drive an underactuated system to a desired steady state. We numerically demonstrate the effectiveness of control in two universal scenarios of pattern formation: within a weakly nonlinear regime associated with a supercritical Turing instability and for localized states associated with homoclinic snaking.

05/24/24
Zhichao Wang  UC San Diego, Department of Mathematics
Spectral Properties of Neural Network Modelsectra

05/24/24
Dehao Dai
Advancement to Candidacy

05/24/24
Nandagopal Ramachandran  University of California San Diego
Some Fitting ideal computations in Iwasawa theory over Q and the theory of Drinfeld module

05/28/24
Poornima B  UC San Diego
Extension of modules over the Robba ring

05/28/24
Dr. Jacob Campbell  The University of Virginia
Even hypergeometric polynomials and finite free probability
AbstractIn 2015, Marcus, Spielman, and Srivastava realized that expected characteristic polynomials of sums and products of randomly rotated matrices behave like finite versions of Voiculescu's free convolution operations. In 2022, I obtained a similar result for commutators of such random matrices; one feature of this result is the special role of even polynomials, in parallel with the situation in free probability.
It turns out that a certain family of special polynomials, called hypergeometric polynomials, arises naturally in relation to convolution of even polynomials and finite free commutators. I will explain how these polynomials can be used to approach questions of realrootedness and asymptotics for finite free commutators. Based on arXiv:2209.00523 and ongoing joint work with Rafael Morales and Daniel Perales. 
05/28/24
Sebastian Pardo Guerra  UCSD
Extending undirected graph techniques to directed graphs via Category Theory
AbstractIt is well known that any directed graph induces an undirected graph by forgetting the direction of the edges and keeping the underling structure. In fact, this assignment can be extended to consider graph morphisms, thus obtaining a functor from the category of simple directed graphs and directed graph morphism, to the category of undirected graphs and undirected graph morphisms. This particular functor is known as a “forgetful” functor, since it forgets the notion of direction.
In this talk, I will present a bijective functor that relates the category of simple directed graphs with a particular category of undirected graphs, whose objects we call “prime graphs”. Intuitively, prime graphs are undirected bipartite graphs endowed with a label that evokes a notion of direction. As an application, we use two undirected graph techniques to study directed graphs: spectral clustering and network alignment.

05/29/24
Evangelos A. Nikitopoulos  UCSD
On differentiating maps induced by functional calculus and applications to free stochastic calculus

05/29/24
Zeyu Liu  UC San Diego
Prismatic crystals over $\mathcal{O}_K$

05/29/24
Varun Khurana  University of California, San Diego
Learning With MeasureValued Data
AbstractThis talk discusses computationally feasible machine learning methods, based on optimal transport and neural network theory, applied to measurevalued data. We first analyze linearized optimal transport (LOT), which essentially embeds measurevalued data into an $L^2$ space, where outofthebox machine learning techniques are available. We analyze the situations when LOT provides an isometric embedding with respect to the Wasserstein2 distance and provide necessary bounds when we can achieve a prespecified linear separation level in the LOT embedding space. Second, we produce a computationally feasible algorithm to recover lowdimensional structures in measurevalued data by using the LOT embedding along with dimensionality reduction techniques. Using computational methods for solving optimal transport problems such as the Sinkhorn algorithm or linear programming, we provide approximation guarantees in terms of the sampling rates. Third, we study structured approximations of measures in Wasserstein space by a scaled Voronoi partition of $\mathbb{R}^d$ generated from a full rank lattice. We show that these structured approximations match rates of optimal quantizers and empirical measure approximation in most instances. We then extend these results for noncompactly supported measures that decay fast enough. Finally, we study methods for comparing probability measures by analyzing a neural network twosample test. In particular, we perform timeanalysis on a related neural tangent kernel (NTK) twosample test and extend the analysis to the neural network twosample test with a smalltime training regime. We also show the amount of time needed before the twosample test detects a deviation $\epsilon > 0$ in the case the probability measures considered are different versus when they are the same.

05/29/24
Zichen He  UC San Diego
Noisy Holographic Quantum Error Correcting Codes
AbstractWe introduce repetition noise into quantum errorcorrecting codes with a tensor network structure. Our approach employs a quantum channel, which is a superposition of exact encodings and repetition encoding with a small probability. The boundary states of our models capture key features of conformal field theory states, particularly the power law of the twopoint function and logarithmic entanglement, which are precisely obeyed in many cases. The noisy holographic quantum errorcorrecting codes on trees and tilings of twodimensional hyperbolic space preserve the bulk/boundary duality in AdS/CFT, and their boundary states exhibit the features of conformal field theory accordingly.

05/30/24
Carlos Ospina  University of Utah
Some Real Rel trajectories in $\mathcal{H}(1,1)$ that are not recurrent
AbstractIn this talk we will define the Rel foliation for a stratum of translation surfaces with at least two singularities. We will focus on the real Rel flow in the stratum $\mathcal{H}(1,1)$. We will provide some examples of orbits, and their closures. Finally, we will describe the real Rel orbits of tremors of surfaces and provide explicit examples of trajectories that are not recurrent, but do not diverge.

05/30/24
Dr. Riccardo Caniato  California Institute of Technology
Variations of the YangMills Lagrangian in high dimension
AbstractIn this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary smooth bundles and connections under L^2control of their curvature. We will start from the classical theory in critical dimension and then move to the supercritical dimension, making a digression about the so called “abelian” case and thus showing an analogy between pYangMills fields on abelian bundles and a special class of singular vector fields. In the last part, we will show how the previous analysis can be used in order to build a SchoenUhlenbeck type regularity theory for YangMills fields in supercritical dimension.

05/30/24
Prof. David Weisbart  UC Riverside
$p$Adic Brownian Motion is a Scaling Limit
AbstractThe Laplace operator is the infinitesimal generator of Brownian motion with a real state space. The Vladimirov operator, a $p$adic analogue of the Laplace operator, similarly gives rise to Brownian motion with a $p$adic state space. This talk aims to introduce the concept of a $p$adic Brownian motion and demonstrate a further similarity with its real analogue: $p$adic Brownian motion is a scaling limit of a discretetime random walk on a discrete group. Attendees need not have prior knowledge of $p$adic analysis, as the talk will provide a brief review of necessary background information.

05/30/24
Hui Tan  UCSD
Some applications of Shlyakhtenko’s operatorvalued semicircular systems
AbstractI will present several applications of Shlyakhtenko’s operatorvalued semicircular systems, including characterization of Property (T) for II$_1$ factors in terms of spectral gap in inclusions, and on weak containment of bimodules.

05/30/24
Ellen Eischen  University of Oregon
Algebraic and padic aspects of Lfunctions, with a view toward Spin Lfunctions for GSp_6
AbstractI will discuss recent developments and ongoing work for algebraic and padic aspects of Lfunctions. Interest in padic properties of values of Lfunctions originated with Kummer’s study of congruences between values of the Riemann zeta function at negative odd integers, as part of his attempt to understand class numbers of cyclotomic extensions. After presenting an approach to studying analogous congruences for more general classes of Lfunctions, I will conclude by introducing ongoing joint work of G. Rosso, S. Shah, and myself (concerning Spin Lfunctions for GSp_6). I will explain how this work fits into the context of earlier developments, while also indicating where new technical challenges arise. All who are curious about this topic are welcome at this talk, even without prior experience with padic Lfunctions or Spin Lfunctions.

05/30/24
Ellen Eischen  University of Oregon
It’s what you do next that matters.
AbstractIn my experience, successes often arise from circumstances that appear to be less than ideal, or even hopeless. In the AWM Colloquium, I will discuss some key developments along my career path.
The target audience is graduate students and postdocs. Audience engagement is encouraged. In particular, I will allow ample time for questions.

05/30/24
Gongping Niu  UCSD
Singular Isoperimetric Regions and Twisted Jacobi fields on Locally Stable CMC Hypersurfaces with Isolated Singularities
AbstractIn this talk, we will demonstrate that the wellknown singularity Hausdorff dimension estimates for isoperimetric regions are sharp by constructing singular examples in dimensions 8 and higher. Then, to explore the isoperimetric regions under generic Riemannian metrics, we will discuss the twisted Jacobi field of singular constant mean curvature hypersurfaces under certain regularity assumptions.

05/31/24
Jacob Keller  UC San Diego
The Birational Geometry of KModuli Spaces
AbstractFor $C$ a smooth curve and $\xi$ a line bundle on $C$, the moduli space $U_C(2,\xi)$ of semistable vector bundles of rank two and determinant $\xi$ is a Fano variety. We show that $U_C(2,\xi)$ is Kstable for a general curve $C \in \overline{M}_g$. As a consequence, there are irreducible components of the moduli space of Kstable Fano varieties that are birational to $\overline{M}_g$. In particular these components are of general type for $g\geq 22$.

05/31/24
Ryan Schneider  UC San Diego
Pseudospectral DivideandConquer for the Generalized Eigenvalue Problem
AbstractCome find out how to (randomly) diagonalize any $n \times n$ matrix pencil in fewer than $O(n^3)$ operations!

05/31/24
Jiajie Shi  Department of Mathematics, UCSD
Studying Complex Networks via Hyperbolic Random Graph
AbstractThis study delves into the study of complex networks within a hyperbolic latent space model, presenting theoretical analysis of popular link prediction indices on hyperbolic random graphs. We investigate how different degrees of nodes influence link prediction heuristics. By modifying indices like the common neighbor and shortest path index, the study demonstrates theoretical and empirical improvements in both simulated and realworld networks. Additionally, we also explore embedding methods to recover hyperbolic geometry, introducing a modified hyperbolic ordinal embedding method.

05/31/24
Jesse Kim
Webs for flamingo Specht modules

05/31/24
Hugo Jenkins  UCSD
"Universals"
AbstractYou use universal properties everyday. But what are they exactly? We'll give an answer via something called a universal element. It is the nice concrete thing which defines a Yoneda representation. We'll give several examples of the universal elements for common representations.
Prerequisite: Watch this video https://www.youtube.com/
watch?v=mLRgKPwyg4Y 
05/31/24
Dr. Michael McQuillan  University of Rome Tor Vergata
Flattening and algebrisation.
AbstractOften natural moduli problems, e.g. foliated surfaces, come without an ample line bundle, so the algebraisability of formal deformations, and the very existence of a moduli space requires a study of the mermorphic functions on the aforesaid deformations, and the flattening (by blowing up) of the resulting meromorphic maps. In such a context the flattening theorem of Raynaud & Gruson, and derivatives thereof, is close to irrelevant since it systematically uses schemeness to globally extend local centres of blowing up. This was already well understood by Hironaka in his proof of holomorphic flattening, and his ideas are the right ones. Nevertheless, the said ideas can be better organised with a more systematic use of Grothendieck's universal flatifier, and, doing so, leads to a fully functorial, and radically simpler, proof provided the sheaf of nilpotent functions is coherentwhich is true for excellent formal schemes, but, unlike schemes or complex spaces, is false in general.
Jun

06/03/24
Jeb Runnoe  UC San Diego
SecondDerivative SQP Methods for LargeScale Nonconvex Optimization
AbstractThe class of stabilized sequential quadratic programming (SQP) methods for nonlinearly constrained optimization solve a sequence of related quadratic programming (QP) subproblems formed from a twonorm penalized quadratic model of the Lagrangian function subject to shifted, linearized constraints. While these methods have been shown to exhibit superlinear local convergence even when the constraint Jacobian is rank deficient at the solution, they generally have no global convergence theory. To address this, primaldual SQP methods (pdSQP) employ a certain primaldual augmented Lagrangian merit function and solve a subproblem that consists of boundconstrained minimization of a quadratic model of the merit function. The model of the merit function is constructed in a particular way so that the resulting primaldual subproblem is equivalent to the stabilized SQP subproblem. Together with a flexible linesearch, the use of the merit function guarantees convergence from any starting point, while the connection with the stabilized subproblem allows pdSQP to retain the superlinear local convergence that is characteristic of stabilized SQP methods.
A new dynamic convexification framework is developed that is applicable for nonconvex general standard form, stabilized, and primaldual boundconstrained QP subproblems. Dynamic convexification involves three distinct stages: preconvexification, concurrent convexification and postconvexification. New techniques are derived and analyzed for the implicit modification of symmetric indefinite factorizations and for the imposition of temporary artificial constraints, both of which are suitable for preconvexification. Concurrent convexification works synchronously with the activeset method solving the subproblem, and computes minimal modifications needed to ensure the QP iterates are uniformly bounded. Finally, postconvexification defines an implicit modification that ensures the solution of the subproblem yields a descent direction for the merit function.
A new exact secondderivative primaldual SQP method (dcpdSQP) is formulated for largescale nonconvex optimization. Convergence analysis is presented that demonstrates guaranteed global convergence. Extensive numerical testing indicates that the performance of the proposed method is comparable or better than conventional full convexification while significantly reducing the number of factorizations required.

06/04/24
Dr. Zhenghui Huo  Duke Kunshan University
Weighted estimates of the Bergman projection and some applications.
AbstractIn harmonic analysis, the Muckenhoupt $A_p$ condition characterizes weighted spaces on which classical operators are bounded. An analogue $B_p$ condition for the Bergman projection on the unit ball was given by Bekolle and Bonami. As the development of the dyadic harmonic analysis techniques, people have made progress on weighted norm estimates of the Bergman projection for various settings. In this talk, I will discuss some of these results and outline the main ideas behind the proof. I will also mention the application of these results in analyzing the $L^p$ boundedness of the projection. This talk is based on joint work with Nathan Wagner and Brett Wick.

06/04/24
Prof. Lutz Warnke  UC San Diego
Extreme local statistics in random graphs: maximum tree extension counts
AbstractWe consider a generalization of the maximum degree in random graphs. Given a rooted tree $T$, let $X_v$ denote the number of copies of T rooted at v in the binomial random graph $G_{n,p}$. We ask the question where the maximum $M_n = max \{X_1, ..., X_n\}$ of $X_v$ over all vertices is concentrated. For edgeprobabilities $p=p(n)$ tending to zero not too fast, the maximum is asymptotically attained by the vertex of maximum degree. However, for smaller edge probabilities $p=p(n)$, the behavior is more complicated: our large deviation type optimization arguments reveal that the behavior of $M_n$ changes as we vary $p=p(n)$, due to different mechanisms that can make the maximum large.
Based on joint work with Pedro Araújo, Simon Griffiths and Matas Šileikis; see arXiv:2310.11661

06/04/24
Myeonghun Yu  UCSD
Nonparametric estimation and inference of expected shortfall regression

06/04/24
Bochao Kong  UCSD
On the moduli space of elliptic K3 surfaces
AbstractI will present the Poincare polynomial, the Chow ring, and some tautological relations on the moduli space of elliptic K3 surfaces.

06/05/24
Nicholas Zhao
Advancement to Candidacy

06/05/24
Felipe CastellanoMacias
Advancement to Candidacy

06/06/24
SaiKee Yeung  Purdue University
Aspects of Carathéodory geometry
AbstractThe goal of the talk is to explain some geometric results on quasiprojective manifolds from the perspective of Carathéodory metrics and distances. We will study some conjectures of Lang on manifolds which satisfied some Carath\'eodory conditions. The results are also used to study hyperbolicity of suitable compactifications of the noncompact manifolds involved. As applications, we also prove some statements to the effect of nonexistence of level structures on manifolds such as abelian varieties over function fields, as well as the socalled volume estimates for mapping of curves into the manifolds involved. Most of the results to be presented are joint work with KwokKin Wong.

06/06/24
Prof. Lutz Warnke  UC San Diego
PDEs in random graph theory: Analyticity of scaling limits
AbstractIn this talk we discuss a problem in Combinatorial Probability, that concerns some finer details of the socalled 'giant component' phase transition in random graphs. More precisely, it is wellknown that the size $L_1(G_{n,p})$ of the largest component of the binomial random graph $G_{n,p}$ has a scaling limit for $p=c/n$, i.e., that $L_1(G_{n,p})/n$ converges in probability to some limiting function $\rho(c)$. It is of interest to understand finer details of this limiting function, in particular if $\rho(c)$ is wellbehaved for some range of $c$, say analytic. Analyticity can be shown directly for the binomial random graph $G_{n,p}$, since explicit descriptions and formulas for $\rho(c)$ are available. In this talk I will outline a somewhat more robust approach, that also works in models where explicit formulas are not available. Our approach combines tools from random graph theory (multiround exposure arguments), stochastic processes (differential equation approximation), generating functions, and partial differential equations (CauchyKovalevskaya Theorem).

06/06/24
Dr. SiaoHao Guo  National Taiwan University
Level set flow and the set of singular points
AbstractIn this talk we will introduce the level set flow. Then we will talk about the relation between the rate of curvature blowup near a singularity of the flow and the distribution of surrounding singular points.

06/06/24
Chris Xu  UC San Diego
Rational points on modular curves via the moduli interpretation
AbstractIn theory, ChabautyColeman provides an explicit method to obtain rational points on any curve, so long as its genus exceeds its MordellWeil rank. In practice, when applied to modular curves, we often encounter difficulties in finding a suitable plane model, which only worsens as the genus increases. In this talk we describe how to skip this step and instead work directly with the coarse moduli space. This is joint work with Steve Huang and Jun Bo Lau.

06/06/24
Prof. Yoichiro Mori  Department of Mathematics and Department of Biology, University of Pennsylvania
Stability of Planar Fronts of the Bidomain AllenCahn Equation
AbstractThe bidomain model is the standard model describing electrical activity of the heart. We discuss the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other with respect to medium‐wavelength perturbations. Interestingly, whether the front is stable or unstable under long‐wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate‐wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate‐wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions. Time permitting, I will also discuss properties of the bidomain FitzHugh Nagumo equations. This is joint work with Hiroshi Matano, Mitsunori Nara and Koya Sakakibara.

06/06/24
Mohsen Aliabadi
Minimal algebra of the fundamental theorem of algebra
AbstractThe algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First every polynomial with odd degree and real coefficients has a real root. Second every nonnegative real number has a square root. It is proved in characteristic zero that the assumption about odddegree polynomials is stronger than necessary any field of characteristic zero in which polynomials of prime degree have roots is algebraically closed. In this talk we show that this result is the case for all fields regardless of their characteristics.

06/07/24
Yanyi Wang  UC San Diego
On the Analysis of HighDimensional Time Series Clustering and Classification

06/07/24
Dr. Fanjung Meng  John Hopkins University
Wall crossing for moduli of stable pairs.
AbstractHassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.

06/11/24
Tolson Bell  Carnegie Mellon University
Random Hypergraphs and O(1) Insertion for Cuckoo Hashing
AbstractA hash table is a data structure that efficiently stores objects in a way that allows for fast access, insertion, and deletion. Cuckoo hashing is a method of creating and maintaining hash tables that has been widely used in both theory and practice. Random walk dary cuckoo hashing is a natural generalization of cuckoo hashing with low space overhead, guaranteed fast access, and fast in practice insertion time. In this presentation, I will explain this algorithm and my work proving a bound on its insertion time. More precisely, we show that for four or more hash functions and load factors up to the optimal threshold, the expectation of the random walk insertion time is O(1), that is, a constant depending only on the number of hash functions and the load factor, not the size of the table.
The study of cuckoo hashing is directly connected to the study of ErdősRényi random hypergraphs, and I will emphasize these connections during my presentation. This presentation is based on https://arxiv.org/abs/2401.
14394 , joint work with Alan Frieze. 
06/14/24
Dr. Morgan Brown  University of Miami
Birational geometry and Berkovich spaces
AbstractBerkovich spaces give a formalism for constructing spaces of valuations on varieties over nonarchimedean fields. As such they encode a great deal of information from birational geometry. The most notable invariant is the essential skeleton, a subset of the Berkovich space corresponding to the valuations monomial on strata of a dlt minimal model of 𝑋. Inspired by Mori's conjecture in birational geometry, we conjecture that the essential skeleton is the complement of the images of all transcendental disks, which are analytic objects analogous to families of rational curves. I will present some progress on this conjecture in joint work with Jiachang Xu and Muyuan Zhang.
Aug

08/27/24
Tianyi Yu
Tableaux formulas for Lascoux polynomials
AbstractLascoux polynomials simultaneously generalize two famous families of polynomials arising from geometry and representation theory: They are nonsymmetric analogs of Grassmannian stable Grothendieck polynomials, which represent Schubert classes in the connective Ktheory of Grassmannians. Additionally, they serve as nonhomogeneous analogs of key polynomials, the characters of Demazure modules. Both of these families have classical combinatorial formulas involving tableaux. We further generalize several of these formulas by establishing two combinatorial formulas for Lascoux polynomials.
Sep

09/04/24
Shanshan Hua  University of Oxford
Classification of approximately finitedimensional *homomorphisms
AbstractA nuclear C*algebra is quasidiagonal if there is a unital embedding into the ultraproduct of matrices. The concept is essential in the Elliott Classification Program. The existence of such embeddings is shown by the TikusisWhiteWinter theorem (2015) for any stably finite, simple, separable, nuclear, unital C*algebra satisfying the UCT. Thus an interesting question is to understand how unique are such embeddings given the existence.
We are able to answer this question by obtaining the corresponding KKuniqueness theorem and applying the abstract classification approach. The major difficulty appears since the codomain of the map does not (separably) tensorially absorb the JiangSu algebra, and thus not covered by previous classification techniques. We will explain how to tackle the difficulty by looking at Ktheoretical properties of a special C*algebra, the Paschke Dual algebra, associated to the map.

09/17/24
Prof. Hongchao Zhang  Louisiana State University (zhc@lsu.edu)
Proximal Gradient Method for Nonconvex Composite Optimization
AbstractThis talk introduces a unified proximal gradient method with extrapolation (UPGE) to solve a class of nonconvex and nonsmooth composite optimization. UPGE provides unified treatment to both convex and nonconvex problems, and adaptively estimates the nonconvexity modulus of the possibly nonconvex component function in the objective function. In this talk, we will discuss the global convergence and convergence rate of UPGE for solving both convex and nonconvex composite optimization. We will also show the promising numerical performance of UPGE compared with other wellestablished proximal gradient methods in the literature.

09/26/24
Professor Vishal Patil  UCSD
Topology and coordination of living filamentous matter
AbstractFilaments and fibers represent a fundamental unit of biological matter, from chromosomal DNA and microtubule networks to cilia carpets and worm collectives. How topology and elasticity mediate coordination in biological filaments remains poorly understood. To uncover the topological principles at play in living matter, we first examine the adaptive topological dynamics exhibited by California blackworms, which form tangled aggregates in minutes but can rapidly untangle in milliseconds. By constructing stochastic trajectory equations, we build a predictive model which explains how the dynamics of individual active filaments controls their emergent topological state. We then examine the elastodynamics and coordination of individual filaments more closely, and exhibit mathematical principles underlying knotformation in filamentous organisms across length scales. By identifying how topology and elasticity produce stable yet responsive structures, these results have applications in understanding organization and selfoptimization in broad classes of biological systems.

09/27/24
Dr. Francois Thilmany  UCLouvain
Finding pingpong partners for finite subgroups of linear groups
AbstractIn his paper on free subgroups of linear groups, Tits proved his famous alternative: a linear group is either virtually solvable or contains a free subgroup. Since then, Tits’ work has been generalized and applied in many different ways.
One remaining open question in this field was the one asked by de la Harpe: let $G$ be a semisimple Lie group without compact factors and with trivial center and let $\Gamma$ be a Zariskidense subgroup of $G$. Given a prescribed finite subset $F$ of $G$, is it always possible to find an element $\gamma \in \Gamma$ such that for any $h \in F$, the subgroup generated by $h$ and $\gamma$ is freely generated (in that case, we say $h$ and $\gamma$ are pingpong partners).
In the talk, we will discuss a variant of the question of de la Harpe, where $F$ is a finite set of finite subgroups $H_i$ of $G$. Using careful refinements of the main steps of Tits’ proof of the alternative (which we will recall), we give sufficient conditions for the existence of pingpong partners for the $H_i$ in any Zariskidense subgroup $\Gamma$.We will also show that these conditions are satisfied for products of copies of $\mathrm{SL}_n$ over division $\mathbb{R}$algebras.
The existence of such free products has applications in the theory of integral group rings of finite groups, which will be briefly mentioned.
Joint work with G. Janssens and D. Temmerman. 
09/30/24
Itai Maimon  UCSD
Homological Codes, Their Extensions, and Certain Unfair Games
Oct

10/01/24
Scott Atkinson  Elder Research
A discussion of machine learning: neural networks and free probability
AbstractIn this talk we will begin by discussing some of the main concepts of machine learning at a high level. Then we will take a closer look at neural networks. And finally we will discuss an interesting application of free probability to deep learning observed by PenningtonSchoenholzGanguli.
The goal of this talk is to introduce mathematicians without much background in statistics to the fundamentals of machine learning while highlighting an interesting application of noncommutative probability. Graduate students are highly encouraged to attend. 
10/03/24
Morris Ang  UC San Diego
Proof of the DelfinoViti conjecture for percolation
AbstractFor critical percolation on the 2D triangular lattice, consider the probability that three points lie in the same cluster. The DelfinoViti conjecture predicts that in the fine mesh limit, under suitable normalization, this probability converges to the imaginary DOZZ formula from conformal field theory. We prove the DelfinoViti conjecture, and more generally, obtain the cluster connectivity threepoint function of the conformal loop ensemble. Our arguments depend on the coupling between Liouville quantum gravity and the conformal loop ensemble.
Based on joint work with Gefei Cai, Xin Sun, and Baojun Wu. 
10/03/24
Professor Pearson Miller
Continuum approaches to condensatedriven morphogenesis
AbstractEvolution has predisposed developing organisms to rely heavily on the repeated application of select biomechanical motifs to achieve robust growth. Within vertebrates, one prevalent architecture is the dermal condensate: these dense bundles of mesenchymal cells that as organizing centers for developing organs such as teeth, hair follicles, glands, and limbs. Though long recognized as significant, the study of these protoorgans has historically focused on biochemical regulation. This talk, however, will emphasize the mechanical role of these structures, illustrating how they can be viewed as selforganized actuators that drive tissue deformation. This discussion will be split into two parts: first, we will highlight some recently published results based on early feather morphogenesis, demonstrating how dermal condensates coordinate tissuescale flows by localizing active contractile stresses. Second, we will use a model of the mammalian gliding membrane to examine the formation of condensates more closely, with an eye on clarifying the role of cellmatrix interactions and better understanding where condensate mechanics fits among competing hypotheses of limb growth.

10/04/24
JJ Garzella
Graduate Student Survival Guide
AbstractGraduate school can feel like trekking through the jungle... and every adventurer needs a good survival guide! In this talk, we'll give you a few life hacks that will help along your journey through the rainforest. Older grad students who wish to attend are invited to bring their own survival strategies. Maybe we'll even see a toucan! Or a poisonous spider!

10/07/24
Professor B Sury  Indian Statistical Institute, Bangalore
Cyclic cubic extensions of Q, binary cubic forms and Sylvester’s conjecture
AbstractThe classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history, from early works of Sylvester, Satge, Selmer etc. and, up to recent work of AlpogeBhargavaShnidman. A conjecture attributed to Sylvester predicts that the primes >2 in the residue classes 2,5 mod 9 are not sums of two rational cubes, while those in the residue classes 4,7 or 8 mod 9 are. Primes which are 1 mod 9 may or may not be sums of two rational cube sums. We rephrase the problem in terms of elliptic curves, and use certain integral binary cubic forms to prove unconditionally that there are infinitely many primes in each of the residue classes 1 mod 9 and 8 mod 9 that are expressible as sums of two rational cubes. More generally, we prove that every nonzero residue class a (mod q), for any prime q, contains infinitely many primes which are sums of two rational cubes. Among other results, we show that corresponding to any positive integer n, there are infinitely many imaginary quadratic fields in which n is a sum of two cubes. These results represent joint work with Somnath Jha and Dipramit Majumdar. The starting point of this work was an accidental encounter in earlier work with Dipramit Majumdar when we classified all cubic cyclic extensions of Q.

10/08/24
Kai Toyosawa  Vanderbilt University
Weak exactness and amalgamated free product of von Neumann algebras
AbstractWe show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of ToeplitzPimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weakly exact von Neumann algebras.

10/08/24
Youngho Yoo  Texas A&M University (yyoo@tamu.edu)
ErdősPósa property in grouplabelled graphs
AbstractErdős and Pósa proved in 1965 that every graph contains either $k$ vertexdisjoint cycles or a set of at most $O(k \log k)$ vertices intersecting every cycle. Such an approximate duality does not hold for odd cycles due to certain projectiveplanar grids, as pointed out by Lovász and Schrijver, and Reed showed in 1999 that these grids are the only obstructions to this duality. In this talk, we generalize these results by characterizing the obstructions in grouplabelled graphs. Specializing to the group $\mathbb{Z}/m\mathbb{Z}$ gives a characterization of when cycles of length $\ell \bmod m$ satisfy this approximate duality, resolving a problem of Dejter and NeumannLara from 1988. We discuss other applications and analogous results for $A$paths.
Based on joint work with Pascal Gollin, Kevin Hendrey, Ojoung Kwon, and Sangil Oum. 
10/09/24
Prof. Lijun Ding  UCSD (l2ding@ucsd.edu)
Optimization for statistical learning with low dimensional structure: regularity and conditioning
AbstractMany statistical learning problems, where one aims to recover an underlying lowdimensional signal, are based on optimization, e.g., the linear programming approach for recovering a sparse vector. Existing work often either overlooked the high computational cost in solving the optimization problem, or required casespecific algorithm and analysis  especially for nonconvex problems. This talk addresses the above two issues from a unified perspective of conditioning. In particular, we show that once the sample size exceeds the intrinsic dimension of the signal, (1) a broad range of convex problems and a set of key nonsmooth nonconvex problems are wellconditioned, (2) wellconditioning, in turn, inspires new algorithms and ensures the efficiency of offtheshelf optimization methods.

10/10/24
Professor Rodolfo GutiérrezRomo  Universidad de Chile (gr@rodol.fo)
The diagonal flow detects the topology of strata of quadratic differentials
AbstractA halftranslation surface is a collection of polygons on the plane with side identifications by translations or halfturns in such a way that the resulting topological surface is closed and orientable. We also assume that the total Euclidean area of the polygons is finite. Two halftranslations are equivalent if a sequence of cutandpaste operations takes one to the other. From the view of complex geometry, an equivalent definition is a Riemann surface endowed with a meromorphic quadratic differential with poles of order at most one.
A stratum of halftranslation surfaces consists of those with prescribed cone angles at the vertices of the polygons. Strata are, in general, not connected. A natural flow, the diagonal or Teichmüller flow, acts on stratum components.
In this talk, we investigate some topological properties of stratum components. We show that the (orbifold) fundamental group of such a component is “detected” by the diagonal flow in that every loop is homotopic to a concatenation of closed geodesics (coned to a basepoint). Using this result, we show that the Lyapunov spectrum of the homological action of the diagonal flow is simple, thus establishing the Kontsevich–Zorich conjecture for quadratic differentials.
This is a joint work with Mark Bell, Vincent Delecroix, Vaibhav Gadre, and Saul Schleimer.

10/10/24
Professor Chris Lee  UCSD
Addressing cellular complexity by advancing multiscale biophysical modeling
AbstractThe sequencestructurefunction relationship for proteins is well established, but are there corresponding relationships at larger scales, from organelles to specialized subcellular structures? My research seeks to address how molecular organization influences the shape of cellular membranes. In this talk I will discuss our recent progress towards developing approaches to enable the incorporation of biological complexity in models of cellular membrane mechanics. This includes a new simulation engine called Mem3DG which uses concepts from discrete differential geometry to model the coupled mechochemical feedback of inplane membrane components interacting with membrane geometry. I will show biological problems we have been able to address and give a perspective on possible directions for future mathematical and computational development.

10/10/24
Francois Thilmany  UCLouvain <francois.thilmany@uclouvain.be>
Using hyperbolic Coxeter groups to construct highly regular expander graphs
AbstractA graph $X$ is defined inductively to be $(a_0, . . . , a_{n−1})$regular if $X$ is $a_0$regular and for every vertex $v$ of $X$, the sphere of radius 1 around $v$ is an $(a_1, . . . , a_{n−1})$regular graph. A family $F$ of graphs is said to be an expander family if there is a uniform lower bound on the Cheeger constant of all the graphs in $F$.
After briefly (re)introducing Coxeter groups and their geometries, we will describe how they can be used to construct very regular polytopes, which in turn can yield highly regular graphs. We will then use the superapproximation machinery, whenever the Coxeter group is hyperbolic, to obtain the expansion of these family of graphs. As a result, we obtain interesting infinite families of highly regular expander graphs, some of which are related to the exceptional groups.
The talk is based on work joint with Conder, Lubotzky and Schillewaert. 
10/11/24
Gavin Pettigrew  UCSD PhD Student
Winning Ways for Your Impartial Plays
AbstractA game of Nim is traditionally played with heaps of objects called counters. In this game, two players take turns choosing a heap and then removing any positive number of counters from that heap. Play continues until the final counter is removed, at which point the player responsible for this is declared the winner. In search of a general winning strategy for Nim, we encounter and prove several core principles of combinatorial game theory.

10/14/24
Dr. Keller VandeBogert  University of Notre Dame
From Total Positivity to Pure Free Resolutions
AbstractPolya frequency sequences are ubiquitous objects with a surprising number of connections to many different areas of mathematics. It has long been known that such sequences admit a "duality" operator that mimics the duality of a Koszul algebra and its quadratic dual, but the precise connection between these notions turns out to be quite subtle. In this talk, we will see how the equivariant analogue of Polya frequency is closely related to the problem of constructing Schur functors "with respect to" an algebra. We will moreover see how these ideas come together to understand the problem of extending BoijSoederberg theory to other classes of rings, with particular attention given to the case of quadric hypersurfaces. This is based on joint work with Steven V. Sam.

10/15/24
David Sherman  University of Virginia
A quantization of coarse structures and uniform Roe algebras
AbstractA coarse structure is a way of talking about "largescale" properties. It is encoded in a family of relations that often, but not always, come from a metric. A coarse structure naturally gives rise to Hilbert space operators that in turn generate a socalled uniform Roe algebra.
In work with Bruno Braga and Joe Eisner, we use ideas of Weaver to construct "quantum" coarse structures and uniform Roe algebras in which the underlying set is replaced with an arbitrary represented von Neumann algebra. The general theory immediately applies to quantum metrics (suitably defined), but it is much richer. We explain another source based on measure instead of metric, leading to the new, large, and easytounderstand class of support expansion C*algebras.
I will present the big picture: where uniform Roe algebras come from, how Weaver's framework facilitates our definitions. I will focus on a few illustrative examples and will not presume familiarity with coarse structures or von Neumann algebras.

10/15/24
Linghao Zhang  UCSD
A Characterization for Tightness of the Sparse MomentSOS Hierarchy
AbstractThis research project studies the sparse MomentSOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse MomentSOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.

10/15/24
Zhifei Yan  IMPA, Rio de Janeiro (zhifei.yan@impa.br)
The chromatic number of very dense random graphs
AbstractThe chromatic number of a very dense random graph $G(n,p)$, with $p \ge 1  n^{c}$ for some constant $c > 0$, was first studied by Surya and Warnke, who conjectured that the typical deviation of $\chi(G(n,p))$ from its mean is of order $\sqrt{\mu_r}$, where $\mu_r$ is the expected number of independent sets of size $r$, and $r$ is maximal such that $\mu_r > 1$, except when $\mu_r = O(\log n)$. They moreover proved their conjecture in the case $n^{2} \ll 1  p = O(n^{1})$.
In this talk, we study $\chi(G(n,p))$ in the range $n^{1}\log n \ll 1  p \ll n^{2/3}$, that is, when the largest independent set of $G(n,p)$ is typically of size 3. We prove in this case that $\chi(G(n,p))$ is concentrated on some interval of length $O(\sqrt{\mu_3})$, and for sufficiently `smooth' functions $p = p(n)$, that there are infinitely many values of $n$ such that $\chi(G(n,p))$ is not concentrated on any interval of size $o(\sqrt{\mu_3})$. We also show that $\chi(G(n,p))$ satisfies a central limit theorem in the range $n^{1} \log n \ll 1  p \ll n^{7/9}$.
This talk is based on arXiv:2405.13914 
10/15/24
Antonio Auffinger  Northwestern University (tuca@northwestern.edu)
Dimension Reduction Methods for Data Visualization
AbstractThe purpose of dimension reduction methods for data visualization is to project high dimensional data to 2 or 3 dimensions so that humans can understand some of its structure. In this talk, we will give an overview of some of the most popular and powerful methods in this active area. We will then focus on two algorithms: Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP). Here, we will present new rigorous results that establish an equilibrium distribution for these methods when the number of data points diverge in the presence of pure noise or with a planted signal.

10/16/24
Jon Aycock  UC San Diego
Jacobians of Graphs via Edges and Iwasawa Theory
AbstractThe Jacobian (or sandpile group) is an algebraic invariant of a graph that plays a similar role to the class group in classical number theory. There are multiple recent results controlling the sizes of these groups in Galois towers of graphs that mimic the classical results in Iwasawa theory, though the connection to the values of the Ihara zeta function often requires some adjustment. In this talk we will give a new way to view the Jacobian of a graph that more directly centers the edges of the graph, construct a module over the relevant Iwasawa algebra that nearly corresponds to the interpolated zeta function, and discuss where the discrepancy comes from.

10/16/24
Prof. Hongchao Zhang  UCSD (zhc@lsu.edu)
A smoothing Newton algorithm for weighted complementarity problems
AbstractThe Weighted Complementarity Problem (WCP) significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this talk, we will introduce a smoothing Newton algorithm by using a oneparametric class of smoothing functions. We will discuss the global convergence and local superlinear or quadratic convergence of this algorithm without nonsingularity assumption on the Jacobian. Some numerical results of this algorithm will be also discussed.

10/17/24
Sidhanth Raman  UC Irvine (svraman@uci.edu)
Uniform Central Limit Theorems on Lie Groups
AbstractRandom walks on groups have been utilized to study a wide array of mathematics, e.g. number theory, the spectral theory of Schrodinger operators, and homogeneous dynamics. Under sufficiently nice dynamical assumptions, these random walks obey central limit theorems. We will discuss some joint work with Omar Hurtado in which we introduce a natural family of topologies on spaces of probability measures, and study continuity and stability of statistical properties of random walks on linear groups over local fields. We are able to extend large deviation results known in the Archimedean case to nonArchimedean local fields and also demonstrate certain large deviation estimates for heavy tailed distributions unknown even in the Archimedean case. Time permitting, we will discuss applications to Schodinger operators (an Anderson localization result) and hyperbolic geometry (a stable geodesic counting result).

10/17/24
Ilka Agricola  University of Marburg
Generalizations of 3Sasaki manifolds and skew torsion
AbstractWe define and investigate new classes of almost 3contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3contact metric manifolds, and second, the newly defined classes should admit ’good’ metric connections with skew torsion with interesting applications: these include a wellbehaved metric cone, the existence of a generalized Killing spinor, and remarkable curvature properties. This is joint work with Giulia Dileo (Bari) and Leander Stecker (Marburg).

10/17/24
Professor Christina Hueschen  UCSD (chueschen@ucsd.edu)
Actin SelfOrganization in Gliding Parasitic Cells
AbstractEukaryotic parasites have evolved striking biomechanical and morphogenetic abilities that (1) enable successful infection of billions of human bodies and (2) present an exciting frontier for biophysics. My work explores cellular dynamics, mechanobiology, and morphogenesis through key phenomena in the lives of parasites: motility, penetration of host tissue, and organismal shape change. This talk will focus on gliding motility, the unique form of cell locomotion used during host infection by unicellular apicomplexan parasites like Toxoplasma gondii and the Plasmodium species that cause malaria. Gliding is powered by a thin layer of flowing actin filaments at the parasite surface. How does the collective motion of surface actin filaments emerge, and how does it drive the varied parasite gliding movements that we observe experimentally? I will present findings based on singlemolecule tracking of actin and myosin in Toxoplasma gondii parasites and develop a continuum model of emergent Factin flow within the confines provided by Toxoplasma geometry. Our actin filament flocking model enables the exploration of distinct selforganized states tuned by filament lifetime, which can account for the diversity of observed Toxoplasma gliding motions. This theoryexperiment interplay illustrates how different forms of gliding motility can arise as an intrinsic consequence of selforganized actin filament flows at a cell surface.

10/18/24
Kristin DeVleming  (kedevleming@ucsd.edu)
The HassettKeel program in genus 4
AbstractStudying the minimal model program with scaling on the moduli space of genus g curves and interpreting the steps in a modular way is known as the HassettKeel program. The first few steps are wellunderstood yet the program remains quite incomplete in general. We complete the HassettKeel program in genus 4 using wallcrossing. This is joint work with Kenneth Ascher, Yuchen Liu, and Xiaowei Wang.

10/21/24
Dr. Gil Goffer  UC San Diego
Can group laws be learned using random walks?
AbstractIn various cases, a law that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law \([x,y]=1\) holds with probability larger than \(5/8\), must be abelian. In the talk I'll discuss a probabilistic approach to laws on infinite groups, using random walks, and present results, joint with Greenfeld and Olshanskii, answering a few questions of Amir, Blachar, Gerasimova, and Kozma.

10/22/24
Jorge Garza Vargas  Caltech
A new approach to strong convergence
AbstractA sequence of tuples of random matrices $(X_1^N, \dots, X_d^N)$ converges strongly to a tuple of operators $ (x_1, . . . , x_d)$ in a $C^*$algebra if for any noncommutative polynomial $P$, $\P(X_1^N, \dots, X_d^N)\$ converges (say, in probability) to $\P(x_1, . . . , x_d)\$ as $N$ goes to infinity. This phenomenon plays a central role in breakthrough results in operator algebras, as well as in the construction of: expander graphs, hyperbolic surfaces with nearly optimal spectral gaps and minimal surfaces. Given its farreaching implications, it is no surprise that strong convergence is notoriously difficult to prove and has generally required delicate problemspecific methods.
In this talk I will discuss recent joint work with ChiFang Chen, Joel Tropp and Ramon van Handel, where we introduce a new flexible and elementary technique for proving strong convergence. This technique can be applied to random matrix models that have a lot of symmetry, for example, random permutation matrices, classical Gaussian and unitary matrices (i.e. GOE, GUE, GSE, $O(N)$, $U(N)$, and $Sp(N)$), and some others, constructed via representations of the symmetric and unitary group, for which other methods seem to break. In all of these models, the technique yields the sharpest quantitative results known so far. 
10/22/24
Ji Zeng  Alfréd Rényi Institute of Mathematics, Budapest (jzeng@ucsd.edu)
Unbalanced Zarankiewicz problem for bipartite subdivisions
AbstractA real number $\sigma$ is called a \textit{linear threshold} of a bipartite graph $H$ if every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $V \gtrsim U^\sigma$ and without a copy of $H$ must have a linear number of edges $E \lesssim V$. We prove that $\sigma_s = 2  1/s$ is a linear threshold of the \textit{complete bipartite subdivision} graph $K_{s,t}'$. Moreover, we show that any $\sigma < \sigma_s$ is not a linear threshold of $K_{s,t}'$ for sufficiently large $t$ (depending on $s$ and $\sigma$). Some applications of our result in incidence geometry are discussed.

10/23/24
Jennifer JohnsonLeung  University of Idaho
Rationality of certain power series attached to paramodular Siegel modular forms
AbstractThe Euler product expression of the Dirichlet series of Fourier coefficients of an elliptic modular eigenform follows from a formal identity in the Hecke algebra for GL(2) with full level. In the case of Siegel modular forms of degree two with paramodular level, the situation is more delicate. In this talk, I will present two rationality results. The first concerns the Dirichlet series of radial Fourier coefficients for an eigenform of paramodular level divisible by the square of a prime. This result is an application of the theory of stable Klingen vectors (joint work with Brooks Roberts and Ralf Schmidt). While we are able to calculate the action of certain Hecke operators on eigenforms, the structure of the Hecke algebra of deep level is not known in general. However, in the case of prime level, there is a robust description of the local Hecke algebra which yields a rationality result for a formal power series of Hecke operators (joint work with Joshua Parker and Brooks Roberts). In both cases, we obtain the expected local Lfactor as the denominator of the rational function.
[pretalk at 3:00pm]

10/24/24
Graduate Student Ben Johnsrude  UCLA (johnsrude@math.ucla.edu)
Exceptional set estimates for projection theorems over nonArchimedean local fields
AbstractHow do linear projections affect the dimensions of subsets of Cartesian space? Marstrand's result from 1954 demonstrates that each Borel set behaves generically under projections onto almost every linear subspace. Recent developments in Fourier analysis have permitted these results to be expanded significantly to apply to much more restricted families of projections, and even effectively bound the dimension of the set of exceptional projections.
We discuss the special case of projecting subsets of three dimensions onto lines, working over nonArchimedean local fields of characteristic not equal to 2. We will briefly discuss the relevancy to polynomial effective equidistribution in homogeneous dynamics. The main technical input will be a refined decoupling theorem for nonArchimedean local fields. This work mirrors the work in the real setting by the authors Gan, Guo, Guth, Harris, Iosevich, Maldague, Ou, and Wang, and builds on previous work by the speaker and Zuo Lin.

10/24/24
Professor Hong Zhao  UCSD
Phase Separation in Living Systems
AbstractPhase separation underpins a wide range of phenomena from the formation of membraneless intracellular compartments known as biomolecular condensates to the collective behavior of bacteria. Unlike simpler systems like oil and water, however, phase separation in these systems is often complicated by mechanical interactions, nonequilibrium activities, and heterogeneity.
In my talk, I will delve into how I navigated these complexities to uncover new insights into three distinct systems. I will first address biomolecular condensates within chromatinpacked cell nuclei, highlighting how the competition between elastic and capillary forces crucially shapes the structure and mechanics of the chromatin networks. Next, I will share my discovery of emergent phenomena in active systems such as bacteria and active colloids, due to the interplay between movements along chemical gradients and motilityinduced phase separation. Lastly, I'll discuss how imagebased learning uncovered the physics of phaseseparating particles driven out of equilibrium by electrochemical reactions, revealing their reaction kinetics, free energy landscape, and heterogeneity.

10/24/24
Karthik Ganapathy  UCSD
GLalgebras and the noetherianity problem
AbstractIn the presence of a large group action, even nonnoetherian rings sometimes behave like noetherian rings. For instance, Cohen proved that every symmetric ideal in the infinite variable polynomial ring is generated by the symmetric group orbit of finitely many polynomials. In this talk, I will provide a brief introduction to GLalgebras—commutative algebras equipped with an action of the infinite general linear group—and recall the rich history of the noetherianity problem for GLalgebras. I will then present recent work where I construct a counterexample to the algebraic noetherianity problem for GLalgebras in characteristic two.

10/25/24
Lingyao Xie  UCSD (l6xie@ucsd.edu)
Extending numerical trivial divisors
AbstractLet $X$ be a normal variety with a projective contraction $f:X\to S$. Assume $U$ is an open subset of $S$ and $L_U$ is a Cartier divisor on $X_U:=X\times_S U$ such that $L_U$ is numerical trivial over $U$.
We will discuss about when it is possible and how to extend $L_U$ to a global Cartier divisor $L$ on $X$ such that $L\equiv_f 0$.

10/25/24
Prof. Xianghong Gong  University of Wisconsin  Madison (gong@math.wisc.edu)
Global NewlanderNirenberg theorem on domains with finite smooth boundary in complex manifolds
AbstractLet $D$ be a relatively compact $C^2$ domain in a complex manifold $X$ of dimension $n$. Assume that $H^1(D,\Theta)$ vanishes, where $\Theta$ is the sheaf of germs of holomorphic tangent fields of $D$. Suppose that the Leviform of the boundary $b D$ has at least $3$ negative eigenvalues or at least $n1$ positive eigenvalues pointwise. We will show that if a formally integrable almost complex structure $H$ of the Holder class $C^r$ with $r>5/2$ on $D$ is sufficiently close to the complex structure on $D$, there is a embedding $F$ from $D$ into $X$ that transforms the almost complex structure into the complex structure on $F(D)$, where $F $ has class $C^s$ for all $s<r+1/2$. This result was due to R. Hamilton in the 1970s when both $b D$ and $H$ are of class $C^\infty$.

10/28/24
Dr. Teresa Yu  University of Michigan, Ann Arbor
Weighted FImodules and symmetric modules over infinite variable polynomial rings
AbstractA foundational result in equivariant commutative algebra is Cohen's theorem that the infinite variable polynomial ring \(R=\mathbb{C}[x_1,x_2,\ldots]\) is Noetherian up to the action of the infinite symmetric group. This has been applied to prove uniformity results for finitedimensional structures in algebraic geometry, statistics, and algebraic topology, and motivates the study of other aspects of the equivariant commutative algebra of \(R\). In this talk, we explain an approach to developing the local theory of \(R\)modules in this equivariant setting by studying a generalization of FImodules to a "weighted" setting. We introduce these weighted FImodules, and discuss how they too can be studied from the perspective of commutative algebra up to the action of parabolic subgroups of the infinite general linear group.

10/29/24
David Gao  UCSD
Elementary equivalence and disintegration of tracial von Neumann algebras
AbstractWe prove an analog of the disintegration theorem for tracial von Neumann algebras in the setting of elementary equivalence rather than isomorphism, showing that elementary equivalence of two direct integrals implies fiberwise elementary equivalence under mild, and necessary, hypotheses. This verifies a conjecture of Farah and Ghasemi. Our argument uses a continuous analog of ultraproducts where an ultrafilter on a discrete index set is replaced by a character on a commutative von Neumann algebra, which is closely related to Keisler randomizations of metric structures. We extend several essential results on ultraproducts, such as Łoś's theorem and countable saturation, to this more general setting.

10/29/24
Nicolas Sieger  Iowa State University (nsieger@iastate.edu)
Ricci Curvatures in Random Clustering Graphs
AbstractRealworld networks frequently exhibit a clustering phenomenon where the friends of friends are likely to be friends. We show that the clustering effect is highly correlated with Ricci curvatures of a graph for random clustering graphs with given degree distributions. In particular, we show that for a random clustering graph with certain powerlaw degree distributions the Ricci curvature (in the sense of Lin, Lu, and Yau) is concentrated around the clustering coefficient.
Based on joint work with Fan Chung (UCSD), Michael Rawson (PNNL), Zhaiming Shen (University of Georgia), and Murong Xu (University of Scranton). 
10/29/24
Prof. Debraj Chakrabarti  Central Michigan University (chakr2d@cmich.edu)
Interpolation of operators and the Bergman projection
AbstractResults on the regularity of operators on spaces are often proved by means of interpolation operators applied to estimates at the endpoints. A classical example is that of the Hibert transform on the real line, the behavior of which can be deduced from a weak type (1,1) estimate and the Marcinkiewicz interpolation theorem. Attempts to apply this idea to the Bergman projection on certain domains such as the Hartogs triangle in lead to some unexpected endpoint behavior. In particular, we show that for the Hartogs triangle, at the left endpoint of the interval of boundedness, the Bergman projection on this domain is of restricted strong type in the sense of SteinWeiss, that is, for a characteristic function of a measurable subset , we have
for a constant independent of . This now determines the behavior of the Bergman projection via classical interpolation results. We discuss several generalizations of this result to other domains. This is ongoing joint work with Zhenghui Huo of Duke Kunshan University, China.

10/30/24
Chengyang Bao  UCLA
Computing crystalline deformation rings via the TaylorWilesKisin patching method
AbstractCrystalline deformation rings play an important role in Kisin's proof of the FontaineMazur conjecture for GL2 in most cases. One crucial step in the proof is to prove the BreuilMezard conjecture on the HilbertSamuel multiplicity of the special fiber of the crystalline deformation ring. In pursuit of formulating a horizontal version of the BreuilMezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. Our approach, based on reverseengineering the TaylorWilesKisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally.
[pretalk at 3:00PM] 
10/31/24
Márton Szőke  Budapest University of Technology
Local Limit of the Random Degree Constrained Process
AbstractWe show that the random degree constrained process (a timeevolving random graph model with degree constraints) has a local weak limit, provided that the underlying host graphs are high degree almost regular. We, moreover, identify the limit object as a multitype branching process, by combining coupling arguments with the analysis of a certain recursive tree process. Using a spectral characterization, we also give an asymptotic expansion of the critical time when the giant component emerges in the socalled random $d$process, resolving a problem of Warnke and Wormald for large $d$.
Based on joint work with Balázs Ráth and Lutz Warnke; see arXiv:2409.11747

10/31/24
Dr. Aria Halavati  Courant Institute
Decay of excess for the abelian Higgs model
AbstractEntire critical points of the abelian Higgs functional are known to blow down to generalized minimal submanifolds (of codimension 2). In this talk we prove an Allard type largescale regularity result for the zero set of solutions. In the "multiplicity one" regime, we show the uniqueness of blowdowns and classify entire solutions in low dimensions and minimizers in all dimensions; thus obtaining an analogue of Savin's theorem in codimension two. This is based on a joint work with Guido de Philippis and Alessandro Pigati.

10/31/24
Professor Mattia Serra  UCSD
Mathematical Framework for Pattern Formation in Motile Cell Environments
AbstractEmbryogenesisgeneration of functional formsentails coordinated cell motion (morphogenesis), intercellular communications via morphogen patterns, and cell fate decisions. Morphogenesis and patterning have traditionally been studied separately, and how cell movement affects cell fates remains unclear. Traditional models of pattern formation deal mostly with static tissues, preventing the rationalization of increasingly available spatiotemporal data of morphogens and flows in remodeling tissues. We present a theoretical framework for pattern formation in motile cell environments by describing the dynamics of morphogen exposure felt by moving cells (Lagrangian frame) rather than at fixed laboratory coordinates (traditional Eulerian frame). This cell frame description reveals how morphogenetic motifs such as multicellular attractors and repellers (i.e., the Dynamic Morphoskeleton) and convergent extension flows act as barriers and enhancers to diffusive morphogen transport, revealing a robust synergy between morphogenesis and intercellular signaling. We apply our framework to standard models for dynamic cell fate bifurcations and induction and to experimental data from avian gastrulation flows.
Nov

11/01/24
Yuchao Yi  UCSD (yuyi@ucsd.edu)
Bounded inverse scattering problem for nonlinear Dirac equation
AbstractInverse problem is the study of the recovery of parameters or the governing equations of a system based on given observational data. In this talk I will focus on the techniques used in inverse scattering problems for hyperbolic type equations. Scalar wave equation will be used as an example for showing how one can use higher order linearization and microlocal analysis to retrieve information about the unknown nonlinearity. I will also explain the main ideas used in proving that the bounded time scattering map uniquely determines the nonlinearity in the semilinear 4 by 4 Dirac system, with some mild assumptions.

11/01/24
Gregory Patchell  UCSD (gpatchel@ucsd.edu)
How to Maximize your Mean Ability
AbstractIn this talk I will show you how to maximize your ameanability. More precisely, I will survey some results about maximal amenable subgroups and subalgebras and share a new result which states that it is possible for the space of maximal amenable extensions to be any ksimplex (such as a triangle!). While this talk will technically be about operator algebras, it will be accessible to anyone who knows a bit of group theory (semidirect products, free products, and the left regular representation).

11/01/24
Prof. Dragos Oprea  UCSD
The Chow ring of the moduli space of degree 2 K3 surfaces
AbstractI will discuss recent results describing the Chow rings and the tautological classes of the moduli space of quasipolarized K3 surfaces of degree 2. This is based on joint work with Samir Canning and Rahul Pandharipande.

11/01/24
Prof. Samuel Shen  SDSU (sshen@sdsu.edu)
Some visualization tools for big climate data developed at the SDSU Climate Informatics Lab
AbstractSDSU Climate Informatics Lab has developed a suite of computer code and Apps for visualizing and delivering real climate data for the general public, such as school classrooms. This presentation will specifically demonstrate the following tools:
1. 4dimensional visual delivery (4DVD) of big climate data: www.4dvd.org.
2. Statistics, machine learning, and data visualization for climate science with R and Python: www.climatestatistics.org
3. Climate mathematics with R and Python: www.climatemathematics.org
4. 4DVD Rural Heat Island for a California Climate Action project: www.4dvdrhi.sdsu.edu
We will also discuss our proprietary database optimization algorithms for fast queries. Using cuttingedge database technologies and 3D video games, we will outline our product development for the NSF program of AI Institutes and NOAA National Centers for Environmental Information.

11/04/24
Dr. Lucas Buzaglo  UC San Diego
Universal enveloping algebras of infinitedimensional Lie algebras
AbstractUniversal enveloping algebras of finitedimensional Lie algebras are fundamental examples of wellbehaved noncommutative rings. On the other hand, enveloping algebras of infinitedimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will summarize what is known about the noetherianity of enveloping algebras, with a focus on Lie algebras of derivations of associative algebras.

11/05/24
Jasper Liu  UCSD
Matrix loci and orbit harmonics
AbstractLet $\mathrm{Mat}_{n \times n}(\mathbb{C})$ be the affine space of $n \times n$ complex matrices with coordinate ring $\mathbb{C}[{\mathbf x}_{n \times n}]$. We define graded quotients of $\mathbb{C}[{\mathbf x}_{n \times n}]$ where each quotient ring carries a group action. These quotient rings are obtained by applying the orbit harmonics method to matrix loci corresponding to the permutation matrix group $S_n$, the colored permutation matrix group $S_{n,r}$, the collection of all involutions in $S_n$, and the conjugacy classes of involutions in $S_n$ with a given number of fixed points. In each case, we explore how the algebraic properties of these quotient rings are governed by the combinatorial properties of the matrix loci. Based on joint work with Yichen Ma, Brendon Rhoades, and Hai Zhu.

11/05/24
Miquel Ortega  Universitat Politecnica de Catalunya (UPC)
A canonical van der Waerden theorem in random sets
AbstractThe canonical van der Waerden theorem states that, for large enough $n$, any colouring of $[n]$ gives rise to monochromatic or rainbow $k$APs. In joint work with Alvarado, Kohayakawa, Morris and Mota, we study sparse random versions of this result. More concretely, we determine the threshold at which the binomial random set $[n]_p$ inherits the canonical van der Waerden properties of $[n]$, using the container method.

11/06/24
Brandon Alberts  Eastern Michigan University
Inductive methods for counting number fields
AbstractWe will discuss an inductive approach to determining the asymptotic number of Gextensions of a number field with bounded discriminant, and outline the proof of Malle's conjecture in numerous new cases. This talk will include discussions of several examples demonstrating the method.
[pretalk at 3:00PM] 
11/07/24
Shreyasi Datta  University of York (shreyasi.datta@york.ac.uk)
Fourier Asymptotics and Effective Equidistribution
AbstractWe talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics. This is a joint work with Subhajit Jana.

11/07/24
Nikita Gladkov  UCLA (gladkovna@gmail.com)
Inequalities for connectivity events in Bernoulli percolation
AbstractIn Bernoulli percolation, events such as "two vertices are connected by an open path" naturally emerge. In this talk, I will explore the dependencies between these events for various vertex pairs and derive key FKGtype inequalities governing their probabilities and explain the relevance of these inequalities to the recent disproof of the Bunkbed Conjecture.

11/07/24
Itamar Vigdorovich  UCSD
Character limits of arithmetic groups
AbstractIn the 1960s Thoma developed a theory of characters which generalizes the classical Fourier/Pontryagin theory of abelian groups, and at the same time Frobenius' theory on finite (and compact) groups.
After presenting the general theory, I will focus on arithmetic groups, or similarly, lattices in (semi)simple Lie groups, and tell about my work with Levit and Slutsky regarding the geometry/topology of the space of characters of such groups. Our main result is that for lattices in higher rank simple Lie groups (e.g for the group SL3(Z)), any sequence of distinct characters must converge pointwise to the dirac character at the identity. This implies character bounds of finite groups of Lie type (e.g SL3(Fp)).

11/07/24
Denis Osin  Vanderbilt University
Generic Cayley graphs of countable groups
AbstractDoes every infinite group admit a generating set such that the corresponding Cayley graph has infinite diameter? While there are examples of uncountable groups that fail to satisfy this property (e.g., the group of all permutations of the integers), the question for countable groups remains open. After reviewing the necessary background and some known results, I will discuss an attempt to solve this problem by choosing a random generating set. For a wide class of countable groups, this approach answers the question affirmatively and reveals a surprising phenomenon: random generating sets yield the same Cayley graph, independent of the group. Depending on the randomness model, this is either the familiar Rado graph (which has diameter 2) or a certain mysterious graph of infinite diameter.

11/08/24
Finn Southerland  UCSD
Finn's Favorite Factorization Facts
AbstractNot the factorizations you're thinking of! A 1factorization of a graph is a partition of its edges into perfect matchings. In this talk I hope to share some of the many questions about 1factorizations that I find interesting, explain how I came to care about a rather obscure fact, and prove it. Along the way we will draw some pretty pictures, of course. This talk should be totally accessible to anyone who has ever heard of graphs, and is based on collaboration with Michael Orrison and Rohan Chauhan.

11/18/24
Professor Nicolas Monod  EPFL
The fixedpoint property and piecewiseprojective transformations of the line
AbstractWe describe a new and elementary proof of the fact that many groups of piecewiseprojective transformation of the line are nonamenable by constructing an explicit action without fixed points. One the one hand, such groups provide explicit counterexamples to the Dayvon Neumann problem. On the other hand, they illustrate that we can distinguish many "layers" of relative nonamenability between nested subgroups.

11/20/24
Dr. Yee Ern Tan  Auburn University (yzt0060@auburn.edu)
Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states
AbstractWe introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus's (2010) algorithm used the HOSVD to compute normal forms of almost all nqubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(C)×SL2(C)→SO4(C), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all nqubit pure states under the action of the SLOCC group.

11/20/24
Linli Shi  University of Connecticut
On higher regulators of Picard modular surfaces
AbstractThe Birch and SwinnertonDyer conjecture relates the leading coefficient of the Lfunction of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. Beilinson’s conjectures generalize the BSD conjecture to formulas for values of motivic Lfunctions at noncritical points. In this talk, I will relate motivic cohomology classes, with nontrivial coefficients, of Picard modular surfaces to a noncritical value of the motivic Lfunction of certain automorphic representations of the group GU(2,1).

11/26/24
Prof. Yi Zhao  Georgia State University (yzhao6@gsu.edu)
Extremal results in multipartite graphs
AbstractClassical extremal results in graph theory (such as Turán's theorem) concern the maximal size of of a graph of given order and without certain subgraphs. Bollobás, Erdős, and Szemerédi in 1975 studied extremal problems in multipartite graphs. One of their problems (in its complementary form) was determining the maximal degree of a multipartite graph without an independent transversal. This problem has received considerable attention and was settle in 2006 (SzabóTardos and HaxellSzabó). Other questions asked by Bollobás, Erdős, and Szemerédi remain open, such as determining:
(1) the maximum degree in a multipartite graph without a partial independent transversal, and;
(2) the minimum degree that forces an octahedral graph in balanced tripartite graphs.In this talk I will survey recent progress on these and other related problems.