Jan
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01/07/25
Gil Goffer - UCSD
Analysis of relations in groups
AbstractI’ll demonstrate how careful analysis of group relations yields unexpected constructions, addressing several central questions in group theory. These include a question by Elliott, Jonusas, Mesyan, Mitchell, Morayne, and Peresse regarding Zariski topologies on groups and semigroups, a series of questions by Amir, Blachar, Gerasimova, and Kozma concerning algebraic group laws, and a longstanding question by Wiegold on invariably generated groups.
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01/07/25
Dr. Akihiro Miyagwa - UC San Diego
Q-deformation of independent Gaussian random variables in non-commutative probability
AbstractIn 1970, Frisch and Bourret introduced a q-deformation of independent Gaussian random variables (say "q-Gaussian system"). In one-variable case, q-Gaussian is the distribution whose orthogonal polynomials are q-Hermite polynomials, and this distribution interpolates between Rademacher (q=-1), semicircle (q=0), Gaussian (q=1) distribution. In multivariable case, q-Gaussian system is represented as a tuple of operators (which are non-commutative in general) on the q-deformed Fock space introduced by Bożejko and Speicher.
In this talk, I will explain related combinatorics (pair partitions and number of crossings) and analysis for q-Gaussian system.
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01/09/25
Luke Jeffreys - University of Bristol
Local dimension in the Lagrange and Markov spectra
AbstractInitially studied by Markov around 1880, the Lagrange spectrum, $L$, and the Markov spectrum, $M$, are complicated subsets of the real line that play a crucial role in the study of Diophantine approximation and binary quadratic forms. Perron's 1920s description of the spectra in terms of continued fractions allowed powerful dynamical machinery to come to bear on many problems. In this talk, I will discuss recent work with Harold Erazo and Carlos Gustavo Moreira investigating the function $d_\textrm{loc}(t)$ that determines the local Hausdorff dimension at a point $t$ in $L'$.
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01/09/25
John Treuer - UCSD
Holomorphic mapping problems
AbstractBiholomorphic mapping problems for domains in complex Euclidean space and in complex manifolds will be discussed.
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01/09/25
Professor Mitchell Luskin - School of Mathematics, University of Minnesota
Continuum Models for Twisted 2D Moiré Materials
AbstractPlacing a two-dimensional lattice on another with a small rotation gives rise to periodic “moiré” patterns on a superlattice scale much larger than the original lattice. The Bistritzer-MacDonald (BM) model attempts to capture the electronic properties of twisted bilayer graphene (TBG) by an effective periodic continuum model over the bilayer moiré pattern. We use the mathematical techniques developed to study waves in inhomogeneous media to identify a regime where the BM model emerges as the effective dynamics for electrons modeled as wave-packets spectrally concentrated at the monolayer Dirac points of linear dispersion, up to error that we rigorously estimate. Using measured values of relevant physical constants, we argue that this regime is realized in TBG at the first “magic" angle where the group velocity of the wave packet is zero and strongly correlated electronic phases (superconductivity, Mott insulators, etc.) are observed.
We are working to develop models of TBG which account for the effects of mechanical relaxation and to couple our relaxed BM model with interacting TBG models. We are also extending our approach to essentially arbitrary moirématerials such as twisted multilayer transition metal dichalcogenides (TMDs) or even twisted heterostructures consisting of layers of distinct 2D materials.
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01/10/25
Nathan Wenger - UCSD
Remarks on Ultrafilters
AbstractUltrafilters show up in many places, including logic, topology, and analysis. Despite this, the concept does not seem to be well-known among mathematicians (indeed, the speaker completed several years of graduate school without learning about them). The goal of this talk is to present a friendly introduction to ultrafilters and to highlight a few of their various manifestations. If all goes well, the talk will include a topological proof of Arrow’s Impossibility Theorem, a classic result from political science.
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01/10/25
Shubham Saha - UCSD
The Chow ring of the universal moduli space of (semi)stable bundles over smooth curves
AbstractWe will discuss some ongoing work on the subject, specifically in the rank $2$, genus $2$ case. The talk will start with a quick review of existing literature on $M_2$ and some of its étale covers, along with results and constructions involving moduli of rank $2$ bundles. We will go over their generalizations to the universal setting and outline the usage of these tools for computing the Chow ring. Lastly, we shall go over some ideas to relate the generators to tautological classes.
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01/13/25
Dr. Yizhen Zhao - UC San Diego
Symbol Length Problem and Restricted Lie Algebra
AbstractThe symbol length problem is a longstanding question concerning the Brauer group of a field. In the case of fields of positive characteristic, every Brauer class is split by a finite extension of height 1. This observation suggests a connection between the symbol length problem and the Galois theory of purely inseparable extensions, where the restricted Lie algebra naturally arise. In this talk, we will explore how various symbol length problems in Brauer groups relate to restricted Lie algebras and introduce a moduli-theoretic description of restricted subspaces in a restricted Lie algebra.
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01/14/25
Yupei Huang - Duke University
Classification of the analytic steady states of 2D Euler equation
AbstractClassification of the steady states for 2D Euler equation is a classical topic in fluid mechanics. In this talk, we consider the rigidity of the analytic steady states in bounded simply-connected domains. By studying an over-determined elliptic problem in Serrin type, we show the stream functions for the steady state are either radial functions or solutions to semi-linear elliptic equations. This is the joint work with Tarek Elgindi, Ayman Said and Chunjing Xie.
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01/14/25
Ishan Ishan - University of Nebraska - Lincoln
Von Neumann orbit equivalence
AbstractI will introduce the notion of a new equivalence relation on the class of countable discrete groups, called von Neumann orbit equivalence (vNOE). I will also discuss the stability of vNOE under the operations of taking free products and graph products of groups. This is based on a joint work with Aoran Wu.
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01/14/25
Dr. Sam Spiro - Rutgers University
The Random Turan Problem
AbstractLet $G_{n,p}$ denote the random $n$-vertex graph obtained by including each edge independently and with probability $p$. Given a graph $F$, let $\mathrm{ex}(G_{n,p},F)$ denote the size of a largest $F$-free subgraph of $G_{n,p}$. When $F$ is non-bipartite, the asymptotic behavior of $\mathrm{ex}(G_{n,p},F)$ is determined by breakthrough work done independently by Conlon-Gowers and by Schacht, but the behavior for bipartite $F$ remains largely unknown.
We will discuss some recent developments that have been made for bipartite $F$, with a particular emphasis on the case of theta graphs. Based on joint work with Gwen McKinley.
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01/15/25
Suhan Zhong - Texas A&M University (suzhong@tamu.edu)
Polynomial Optimization in Data Science Under Uncertainty
AbstractOptimization models that incorporate uncertainty and hierarchical structures have attracted much attention in data science. Recent advances in polynomial optimization offer promising methods to certify global optimality for these complex models. In this talk, I will use two-stage stochastic optimization as a major model to demonstrate how polynomial optimization can be efficiently applied to data science optimization under uncertainty.
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01/15/25
Arijit Chakraborty - UC San Diego
A Power-Saving Error Term in Counting C2 ≀ H Number Fields
AbstractOne of the central problems in Arithmetic Statistics is counting number field extensions of a fixed degree with a given Galois group, parameterized by discriminants. This talk focuses on C2 ≀ H extensions over an arbitrary base field. While Jürgen Klüners has established the main term in this setting, we present an alternative approach that provides improved power-saving error terms for the counting function.
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01/16/25
Professor Naveen Vaidya - SDSU
HIV Infection in Drug Abusers: Mathematical Modeling Perspective
AbstractDrugs of abuse, such as opiates, have been widely associated with enhancing susceptibility to HIV infection, intensifying HIV replication, accelerating disease progression, diminishing host-immune responses, and expediting neuropathogenesis. In this talk, I will present a variety of mathematical models to study the effects of the drugs of abuse on several aspects of HIV infection and replication dynamics. The models are parameterized using data collected from simian immunodeficiency virus infection in morphine-addicted macaques. I will demonstrate how mathematical modeling can help answer critical questions related to the HIV infection altered due to the presence of drugs of abuse. Our models, related theories, and simulation results provide new insights into the HIV dynamics under drugs of abuse. These results help develop strategies to prevent and control HIV infections in drug abusers.
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01/16/25
Rishabh Dixit - UCSD (ridixit@ucsd.edu)
Nonconvex first-order optimization: When can gradient descent escape saddle points in linear time?
AbstractMany data-driven problems in the modern world involve solving nonconvex optimization problems. The large-scale nature of many of these problems also necessitates the use of first-order optimization methods, i.e., methods that rely only on the gradient information, for computational purposes. But the first-order optimization methods, which include the prototypical gradient descent algorithm, face a major hurdle for nonconvex optimization: since the gradient of a function vanishes at a saddle point, first-order methods can potentially get stuck at the saddle points of the objective function. And while recent works have established that the gradient descent algorithm almost surely escapes the saddle points under some mild conditions, there remains a concern that first-order methods can spend an inordinate amount of time in the saddle neighborhoods. It is in this regard that we revisit the behavior of the gradient descent trajectories within the saddle neighborhoods and ask whether it is possible to provide necessary and sufficient conditions under which these trajectories escape the saddle neighborhoods in linear time. The ensuing analysis relies on precise approximations of the discrete gradient descent trajectories in terms of the spectrum of the Hessian at the saddle point, and it leads to a simple boundary condition that can be readily checked to ensure the gradient descent method escapes the saddle neighborhoods of a class of nonconvex functions in linear time. The boundary condition check also leads to the development of a simple variant of the vanilla gradient descent method, termed Curvature Conditioned Regularized Gradient Descent (CCRGD). The talk concludes with a convergence analysis of the CCRGD algorithm, which includes its rate of convergence to a local minimum of a class of nonconvex optimization problems.
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01/16/25
Professor Xiaohua Zhu - Peking University
Limit and singularities of Kaehler-Ricci flow
AbstractAs we know, Kaehler-Ricci flow can be reduced to solve a class of parabolic complex Monge-Amp\`ere equations for Kaehler potentials and the solutions usually depend on the Kaehler class of initial metric. Thus there gives a degeneration of Kaehler metrics arising from the Kaehler-Ricci flow. For a class of $G$-spherical manifolds, we can use the local estimate of Monge-Amp\`ere equations as well as the H-invariant for $C^*$-degeneration to determine the limit of Kaehler-Ricci flow after resales. In particular, on such manifolds, the flow will develop the singularities of type II.
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01/17/25
Yiyun He - UCI
Differentially Private Algorithms for Synthetic Data
AbstractWe present a highly effective algorithmic approach, PMM, for generating differentially private synthetic data in a bounded metric space with near-optimal utility guarantees under the 1-Wasserstein distance. In particular, for a dataset in the hypercube [0,1]^d, our algorithm generates synthetic dataset such that the expected 1-Wasserstein distance between the empirical measure of true and synthetic dataset is O(n^{-1/d}) for d>1. Our accuracy guarantee is optimal up to a constant factor for d>1, and up to a logarithmic factor for d=1. Also, PMM is time-efficient with a fast running time of O(\epsilon d n). Derived from the PMM algorithm, more variations of synthetic data publishing problems can be studied under different settings.
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01/17/25
Srikiran Poreddy - UCSD
Nash’s C1 Isometric Embedding Theorem
AbstractRiemannian geometry, the study of smooth manifolds and how to define distances and angles on them, can be viewed either intrinsically or extrinsically. In this talk, we discuss how Nash unified these views starting with his 1954 paper “C1 Isometric Imbeddings,” where the isometric embedding and the solution to the corresponding system of partial differential equations is constructed as the limit of iteratively defined subsolutions. This technique is cited as one of the first instances of what is now known as convex integration, and is used to construct solutions to many problems in geometry and PDE.
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01/21/25
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01/21/25
Rolando De Santiago - CSU Long Beach
Bounding quantum chromatic numbers of quantum graphs
AbstractIn this talk we will discuss extensions of the 4 fundamental products of graphs (cartesian, categorical, lexicographical, and strong products) to quantum graphs, and provide bounds on the resulting graphs akin to those for products of classical graphs. We will pay particular attention to the lexicographical product, discussing our notion of a quantum b-fold chromatic number as a tool for computing the quantum chromatic number of the lexicographical products.
This is joint work with A. Meenakshi McNamara.
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01/22/25
John Voight - University of Sydney
Hilbert modular forms obtained from orthogonal modular forms on quaternary lattices
AbstractWe make explicit the relationship between Hilbert modular forms and orthogonal modular forms arising from positive definite quaternary lattices over the ring of integers of a totally real number field. Our work uses the Clifford algebra, and it generalizes that of Ponomarev, Bocherer--Schulze-Pillot, and others by allowing for general discriminant, weight, and class group of the base ring. This is joint work with Eran Assaf, Dan Fretwell, Colin Ingalls, Adam Logan, and Spencer Secord.
[pre-talk at 3:00PM]
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01/23/25
Yi Fu - UCSD
Analysis of singularly perturbed stochastic chemical reaction networks motivated by applications to epigenetic cell memory
AbstractEpigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms. It was previously found via simulations of stochastic models that the time scale separation between establishment (fast) and erasure (slow) of chromatin modifications (such as DNA methylation and histone modifications) extends the duration of cell memory, and that different asymmetries between erasure rates of chromatin modifications can lead to different gene expression patterns. We provide a mathematical framework to rigorously validate these computational findings using stochastic models of chemical reaction networks. For our study of epigenetic cell memory, these are singularly perturbed, finite state, continuous time Markov chains. We exploit special structure in our models and extend beyond existing theory to study these singularly perturbed Markov chains when the perturbation parameter is small. We also develop comparison theorems to study how different erasure rates affect the behavior of our chromatin modification circuit. The theoretical tools developed in our work not only allow us to set a rigorous mathematical basis for highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains beyond the applications in this work, especially those associated with chemical reaction networks.
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01/24/25
Sanjoy Dasgupta - UCSD
Recent progress on interpretable clustering
AbstractThe widely-used k-means procedure returns k clusters that have arbitrary convex shapes. In high dimension, such a clustering might not be easy to understand. A more interpretable alternative is to constraint the clusters to be the leaves of a decision tree with axis-parallel splits; then each cluster is a hyperrectangle given by a small number of features.
Is it always possible to find clusterings that are intepretable in this sense and yet have k-means cost that is close to the unconstrained optimum? A recent line of work has answered this in the affirmative and moreover shown that these interpretable clusterings are easy to construct.
I will give a survey of these results: algorithms, methods of analysis, and open problems.
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01/24/25
Haoyu Zhang
Advancement to Candidacy
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01/24/25
Gavin Pettigrew - UCSD
One Approach to the Inverse Galois Problem
AbstractIs every finite group isomorphic to the Galois group of some Galois extension of the rational numbers? Although this question remains open in general, powerful methods have led to an affirmative answer in some cases, including that of solvable groups, symmetric and alternating groups, and most of the sporadic groups. In this talk, we call upon seemingly disconnected areas of algebra, topology, and complex analysis to describe the rigidity method of inverse Galois theory.
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01/27/25
Dr. Harold Jimenez Polo - UC Irvine
A Goldbach Theorem for Polynomial Semirings
AbstractWe discuss an analogue of the Goldbach conjecture for polynomials with coefficients in semidomains (i.e., subsemirings of an integral domain).
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01/28/25
Lijun Ding - UCSD
Flat minima generalize for low-rank matrix recovery
AbstractEmpirical evidence suggests that for a variety of overparameterized nonlinear models, most notably in neural network training, the growth of the loss around a minimizer strongly impacts its performance. Flat minima -- those around which the loss grows slowly -- appear to generalize well. This work takes a step towards understanding this phenomenon by focusing on the simplest class of overparameterized nonlinear models: those arising in low-rank matrix recovery. We analyze overparameterized matrix and bilinear sensing, robust PCA, covariance matrix estimation, and single hidden layer neural networks with quadratic activation functions. In all cases, we show that flat minima, measured by the trace of the Hessian, exactly recover the ground truth under standard statistical assumptions. For matrix completion, we establish weak recovery, although empirical evidence suggests exact recovery holds here as well.
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01/28/25
Akihiro Miyagawa - UCSD
Strong Haagerup inequality for q-circular operators
AbstractThe q-circular system is a tuple of non-commutative random variables (operators with some state) which interpolate independent standard complex Gaussian random variables (q=1) in classical probability and freely independent circular random variables (q=0) in free probability. One of the interesting results on q-deformed probability is that -1<q<1 case has similar properties to free case (q=0). Haagerup inequality is one of such properties, which was originally proved for generators of free groups with respect to the left regular representation.
In this talk, I will explain the strong version of Haagerup inequality for the q-circular system, which was originally proved by Kemp and Speicher for q=0. This talk is based on a joint project with T. Kemp.
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01/29/25
Masato Wakayama - Kyushu University
Quantum interactions and number theory
AbstractQuantum interaction models discussed here are the (asymmetric) quantum Rabi model (QRM) and non-commutative harmonic oscillator (NCHO). The QRM is the most fundamental model describing the interaction between a photon and two-level atoms. The NCHO can be considered as a covering model of the QRM, and recently, the eigenvalue problems of NCHO and two-photon QRM (2pQRM) are shown to be equivalent. Spectral degeneracy can occur in models, but correspondingly there is a hidden symmetry relates geometrical nature described by hyperelliptic curves. In addition, the analytical formula for the heat kernel (propagator)/partition function of the QRM is described as a discrete path integral and gives the meromorphic continuation of its spectral zeta function (SZF). This discrete path integral can be interpreted to the irreducible decomposition of the infinite symmetric group $\mathfrak{S}_\infty$ naturally acting on $\mathbb{F}_2^\infty$, $\mathbb{F}_2$ being the binary field. Moreover, from the special values of the SZF of NCHO, an analogue of the Apéry numbers is naturally appearing, and their generating functions are, e.g., given by modular forms, Eichler integrals of a congruence subgroup. The talk overviews those above and present questions which are open.
[pre-talk at 3:00PM]
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01/29/25
Prof. Lei Huang
Finite Convergence of the Matrix Moment-SOS Hierarchy
AbstractThis talk discusses the matrix Moment-SOS hierarchy for solving polynomial matrix optimization problems. We first establish the finite convergence of this hierarchy under the Archimedean property, provided the nondegeneracy condition, strict complementarity condition, and second-order sufficient condition hold at every minimizer. Furthermore, we also prove that every minimizer of the moment relaxation must exhibit a flat truncation when the relaxation order is sufficiently large.
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01/30/25
Pengfui Guan - McGill University
Entropy of anisotropic Gauss curvature flow and $L^p$ Minkowski problem
AbstractThere is a special entropy quantity associated to the Gauss curvature flow which plays an important rule for the convergence of the flow. Similar entropy can also be defined for a class of generalized Gauss curvature flows, in particular for anisotropic flows. One crucial property is monotonicity of the associated entropy along the flow. Another is the fact that critical point of entropy associated to the anisotropic flow under volume constraint is a solution to the $L^p$-Minkowski problem. This provides a flow approach to the $L^p$-Minkowski problem. The main question is under what condition entropy can control the diameter, as to obtain non-collapsing estimate for the flow. We will discuss the main steps of the approach, and open problems related to inhomogeneous type flows.
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01/30/25
Sri Kunnawalkam Elayavalli - UCSD
Strict comparison for C* algebras
AbstractI will prove strict comparison of C* algebras associated to free groups and then use it to solve the C* version of Tarski's problem from 1945 in the negative. It is joint work with Amrutam, Gao and Patchell and another joint work with Schafhauser.
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01/30/25
Prof. Daniel Tataru - UC Berkeley
The small data global well-posedness conjectures for dispersive flows
AbstractThe key property of linear dispersive flows is that waves with different frequencies travel with different group velocities, which leads to the phenomena of dispersive decay. Nonlinear dispersive flows also allow for interactions of linear waves, and their long time behavior is determined by the balance of linear dispersion on one hand, and nonlinear effects on the other hand.
The first goal of this talk will be to present a new set of conjectures which aim to describe the global well-posedness and the dispersive properties of solutions in the most difficult case when the nonlinear effects are dominant, assuming only small initial data. This covers many interesting physical models, yet, as recently as a few years ago, there was no clue even as to what one might reasonably expect. The second objective of the talk will be to describe some very recent results in this direction. This is joint work with Mihaela Ifrim.
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01/31/25
Rahul Parhi - UCSD
Function-Space Models for Deep Learning
AbstractDeep learning has been wildly successful in practice and most state-of-the-art artificial intelligence systems are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep neural networks. In this talk, I present a new mathematical framework that provides the beginning of a deeper understanding of deep learning. This framework precisely characterizes the functional properties of trained neural networks. The key mathematical tools which support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory. This framework explains the effect of weight decay regularization in neural network training, the importance of skip connections and low-rank weight matrices in network architectures, the role of sparsity in neural networks, and explains why neural networks can perform well in high-dimensional problems. At the end of the talk we shall conclude with a number of open problems and interesting research directions.
This talk is based on work done in collaboration with Rob Nowak, Ron DeVore, Jonathan Siegel, Joe Shenouda, and Michael Unser.
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01/31/25
Finn Southerland - UCSD
An Informal Talk on Formal Mathematics
AbstractCoq is a programming language and "proof assistant", where one can state and prove theorems which are checked for soundness by Coq itself. Looking at an example formalization of the hypernatural numbers, we'll explore what makes such a tool useful, interesting, and even fun! At the end of this talk attendees will hopefully have reasons to consider using Coq or similar tools themselves, and incidentally be able to construct a non-standard model of arithmetic (whenever the need arises).
Feb
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02/03/25
Shihao Zhang - UCSD
Advancement to Candidacy
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02/04/25
Robert Webber - UCSD
Randomized least-squares solvers
AbstractMany data science problems require solving a least-squares problem min_x || A x - b ||^2. Efficiently solving this problem becomes a challenge when A has millions of rows, or even higher. I am developing solutions based on randomized numerical linear algebra:
1. If A is small enough to fit in working memory, an efficient solution is conjugate gradient with randomized preconditioning.
2. If A is too large to fit in working memory but x fits in memory, an intriguing possibility is randomized Kaczmarz.
3. If x is too large to fit in working memory, the final possibility is randomly sparsified Richardson iteration.
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02/04/25
Isaac M. Goldbring - UC Irvine
Elementary equivalence for group von Neumann algebras
AbstractTwo tracial von Neumann algebras are elementarily equivalent if they cannot be distinguished by first-order sentences or, more algebraically, if they have isomorphic ultrapowers. The same definition can be made for (countable, discrete) groups, and it is natural to wonder whether or not there is a connection between two groups being elementarily equivalent and their corresponding group von Neumann algebras being elementarily equivalent. In the first part of the talk, I will give examples to show that, in general, there is no connection in either direction. In the second part of the talk, I will introduce a strengthening of elementary equivalence, called back-and-forth equivalence (in the sense of computability theory) and show that back-and-forth equivalent groups have back-and-forth equivalent group von Neumann algebras. I will also discuss how the same is true for the group measure space von Neumann algebra associated to the Bernoulli action of a group on an arbitrary tracial von Neumann algebra. The latter half of the talk represents joint work with Matthew Harrison-Trainor.
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02/04/25
Sutanay Bhattacharya - UCSD
Hilbert Series of the type B Superspace Coinvariant Ring
AbstractThe superspace ring of rank $n$ is defined as the tensor product of the polynomial ring over $n$ variables and the exterior product of $n$ additional variables. This carries an action of the symmetric group, as well as the hyperoctahedral group (the group of signed permutations). For each of these actions, we define the coinvariant ideal as the ideal generated by invariants under the action with vanishing constant term. We explore some results on bases and Hilbert series of the quotient rings cut out by these ideals.
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02/05/25
A. Raghuram - Fordham University
Congruences and the special values of L-functions
AbstractThere is an idea in number theory that if two objects are congruent modulo a prime p, then the congruence can also be seen for the special values of L functions attached to the objects. Here is a context explicating this idea: Suppose f and f' are holomorphic cuspidal eigenforms of weight k and level N, and suppose f is congruent to f' modulo p; suppose g is another cuspidal eigenform of weight l; if the difference k - l is large then the Rankin-Selberg L function L(s, f x g) has enough critical points; same for L(s, f' x g); one expects then that there is a congruence modulo p between the algebraic parts of L(m, f x g) and L(m, f' x g) for any critical point m. In this talk, after elaborating on this idea, I will describe the results of some computational experiments where one sees such congruences for ratios of critical values for Rankin-Selberg L-functions. Towards the end of my talk, time-permitting, I will sketch a framework involving Eisenstein cohomology for GL(4) over Q which will permit us to prove such congruences. This is joint work with my student P. Narayanan.
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02/05/25
Prof. Long Chen - UC Irvine
Accelerated Gradient Methods through Variable and Operator Splitting
AbstractIn this talk, we present a unified framework for accelerated gradient methods through the variable and operator splitting. The operator splitting decouples the optimization process into simpler subproblems, and more importantly, the variable splitting leads to acceleration. The key contributions include the development of strong Lyapunov functions to analyze stability and convergence rates, as well as advanced discretization techniques like Accelerated Over-Relaxation (AOR) and extrapolation by the predictor-corrector (EPC) methods. The framework effectively handles a wide range of optimization problems, including convex problems, composite convex optimization, and saddle point systems with bilinear coupling. A dynamic updating parameter, which serves as a rescaling of time, is introduced to handle the weak convex cases.
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02/06/25
Prof. Rahul Parhi - UC San Diego (ECE Department)
Characteristic Functionals and the Innovations Approach to Stochastic Processes With Applications to Random Neural Networks
AbstractMany stochastic processes (such as the full family of Lévy processes) can be linearly and deterministically transformed into a white noise process. Consequently these processes can be viewed as the deterministic "mixing" of a white noise process. This formulation is the so-called "innovation model" of Bode, Shannon, and Kailath (ca. 1950-1970), where the white noise represents the stochastic part of the process, called its innovation. This allows for a conceptual decoupling between the correlation properties of a process (which are imposed by the whitening operator L) and its underlying randomness, which is determined by its innovation. This reduces the study of a stochastic process to the study of its underlying innovation. In this talk, we will introduce the innovations approach to studying stochastic processes and adopt the beautiful formalism of generalized stochastic processes of Gelfand (ca. 1955), where stochastic processes are viewed as random tempered distributions (more generally, random variables that take values in the dual of a nuclear space). This formulation uses the so-called characteristic functional (infinite-dimensional analog of the characteristic function of a random variable) of a stochastic process in lieu of more traditional concepts such as random measures and Itô integrals. A by-product of this formulation is that the characteristic functional of any stochastic process that satisfies the innovation model can be readily derived, providing a complete description of its law. We will then discuss some of my recent work where we have derived the characteristic functional of random neural networks to study their properties. This setting will reveal the true power of the characteristic functional: Any property of a stochastic process can be derived with short and simple proofs. For example, we will show that, as the "width" of these random neural networks tends to infinity, these processes can converge in law not only to Gaussian processes, but also to non-Gaussian processes depending on the law of the parameters. Our asymptotic results provide a new take on several classical results that have appeared in the machine learning community (wide networks converge to Gaussian processes) as well as some new ones (wide networks can converge to non-Gaussian processes). This talk is based on joint work with Pakshal Bohra, Ayoub El Biari, Mehrsa Pourya, and Michael Unser from our recent preprint arxiv:2405.10229.
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02/06/25
Koji Shimizu and Gyujin Oh - Tsinghua University/Columbia University
Moduli stack of isocrystals and counting local systems
AbstractTo a smooth projective curve over a finite field, we associate rigid-analytic moduli stacks of isocrystals together with the Verschiebung endomorphism. We develop relevant foundations of rigid-analytic stacks, and discuss the examples and properties of such moduli stacks. We also illustrate how such moduli can be used to count p-adic coefficient objects on the curve of rank one.
The main talk will be given by Oh. In the pre-talk, Shimizu will introduce integrable connections and isocrystals, which will be the key objects in the main talk.
[pre-talk at 1:00PM]
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02/07/25
Li Wang - University of Minnesota
Learning-enhanced structure preserving particle methods for nonlinear PDEs
AbstractIn the current stage of numerical methods for PDE, the primary challenge lies in addressing the complexities of high dimensionality while maintaining physical fidelity in our solvers. In this presentation, I will introduce deep learning assisted particle methods aimed at addressing some of these challenges. These methods combine the benefits of traditional structure-preserving techniques with the approximation power of neural networks, aiming to handle high dimensional problems with minimal training. I will begin with a discussion of general Wasserstein-type gradient flows and then extend the concept to the Landau equation in plasma physics.
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02/07/25
Sutanay Bhattacharya - UCSD
Is the set of all binary trees equal to a complex number?
AbstractNo, it's not; that question doesn't even make sense. But pretending it is for a minute lets us construct a special class of bijections involving sets of binary trees (known in the literature as "particularly elementary" bijections, or sometimes "very explicit" bijections), and even deduce nice equivalent conditions for when such a bijection exists. Based on the paper "Seven Trees in One" by Andreas Blass, this talk explores whether we can ever "solve for" the set of binary trees, and whether we should.
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02/07/25
Dr. Jose Yanez - UCLA
Polarized endomorphism of log Calabi-Yau pairs
AbstractAn endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to qA, for some integer q>1. Examples of these endomorphisms are naturally found in toric varieties and abelian varieties. Indeed, it is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. In this talk, we will restrict to the case of log Calabi-Yau pairs (X,B). We prove that if (X,B) admits a polarized endomorphism that preserves the boundary structure, then (X,B) is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. This is joint work with Joaquin Moraga and Wern Yeong.
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02/11/25
Jason Behrstock - CUNY
Hierarchically hyperbolic groups: an introduction
AbstractHierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view. I'll focus on a few of my favorite "hyperbolic features" and how they manifest in many examples. This talk will include joint work with M. Hagen and A. Sisto, as well as with C. Abbott and M. Durham.
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02/11/25
Sara Billey - University of Washington
Enumerating Quilts of Alternating Sign Matrices and Generalized Rank Functions
AbstractWe present new objects called quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a quilt with respect to two Boolean lattices. When the two posets are chains, a quilt is equivalent to an alternating sign matrix and its corresponding corner sum matrix. Such rank functions are used in the definition of Schubert varieties in both the Grassmannian and the complete flag manifold. Quilts also generalize the monotone Boolean functions counted by the Dedekind numbers, which is known to be a #P-complete problem. Quilts form a distributive lattice with many beautiful properties and contain many classical and well known sublattices, such as the lattice of matroids of a given rank and ground set. While enumerating quilts is hard in general, we prove two major enumerative results, when one of the posets is an antichain and when one of them is a chain. We also give some bounds for the number of quilts when one poset is the Boolean lattice. Several open problems will be given for future development. This talk is based on joint work with Matjaz Konvalinka in arxiv:2412.03236.
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02/12/25
Santiago Arango-Piñeros - Emory University
Counting 5-isogenies of elliptic curves over the rationals
AbstractIn collaboration with Han, Padurariu, and Park, we show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$ action of weights $(4, 4, 2)$.
Pretalk: I will explain how to count isomorphism classes of elliptic curves over the rationals. On the way, I will introduce some basic stacky notions: torsors, quotient stacks, weighted projective stacks, and canonical rings.
[pre-talk at 3:00PM] -
02/13/25
Siyuan Tang - Beijing International Center for Mathematical Research
Effective density of surfaces near Teichmüller curves
AbstractThe study of orbit dynamics for the upper triangular subgroup $P \subset \mathrm{SL}(2, \mathbb R)$ holds fundamental significance in homogeneous and Teichmüller dynamics. In this talk, we shall discuss the quantitative density properties of $P$-orbits for translation surfaces near Teichmüller curves. In particular, we discuss the Teichmüller space $H(2)$ of genus two Riemann surfaces with a single zero of order two, and its corresponding absolute period coordinates, and examine the asymptotic dynamics of $P$-orbits in these spaces.
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02/13/25
Kunal Chawla - Princeton University
The Poisson boundary of hyperbolic groups without moment conditions
AbstractGiven a random walk on a countable group, the Poisson boundary is a measure space which captures all asymptotic events of the markov chain. The Poisson boundary can sometimes be identified with a concrete geometric "boundary at infinity", but almost all previous results relied strongly on moment conditions of the random walk. I will discuss a technique which allows us to identify the Poisson boundary on any group with hyperbolic properties without moment conditions, new even in the free group case, making progress on a question of Kaimanovich and Vershik. This is joint work with Behrang Forghani, Joshua Frisch, and Giulio Tiozzo.
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02/13/25
Harish Kannan - UCSD
Emergent features and pattern formation in dense microbial colonies
AbstractGrowth of bacterial colonies on solid surfaces is commonplace; yet, what occurs inside a growing colony is complex even in the simplest cases. Robust colony expansion kinetics featuring linear radial growth and saturating vertical growth in diverse bacteria indicates a common developmental program, which will be elucidated in this talk using a combination of findings based on modeling and experiments. Agent-based simulations reveal the crucial role of emergent mechanical constraints and spatiotemporal dynamics of nutrient gradients which govern observed expansion kinetics. The consequences of such emergent features will also be examined in the context of pattern formation in multi-species bacterial communities. Future directions and opportunities in theoretical modeling of such pattern formation systems will be discussed.
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02/13/25
Dr. Haoren Xiong - UCLA
Semiclassical asymptotics for Bergman projections
AbstractIn this talk, we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We will first review various approaches to the construction of asymptotic Bergman projections, for smooth weights and for real analytic weights. We shall then explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, we show that Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit. We will also introduce some microlocal analysis tools in the Gevrey setting, including Borel's lemma for symbols and complex stationary phase lemma. This talk is based on joint work with Hang Xu.
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02/13/25
Martin Dindos - University of Edinburgh
The $L^p$ regularity problem for parabolic operators
AbstractIn this talk, I will present a full resolution of the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + {\rm div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a very natural Carleson condition (a parabolic analog of the so-called DKP-condition).
We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known previously even in the "small Carleson case", that is, when the Carleson norm of coefficients is sufficiently small.
In the elliptic case the analogous question was only fully resolved recently (2022) independently by two groups using two very different methods; one involving S. Hofmann, J. Pipher and the presenter, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges. The result is a joint work with L. Li and J. Pipher. -
02/14/25
Lijun Ding - UCSD
On the squared-variable approach for nonlinear (semidefinite) optimization
AbstractConsider minx≥ 0 f(x), where the objective function f: ℝ→ ℝ is smooth, and the variable is required to be nonnegative. A naive "squared variable" technique reformulates the problem to minv∈ ℝ f(v2). Note that the new problem is now unconstrained, and many algorithms, e.g., gradient descent, can be applied. In this talk, we discuss the disadvantages of this approach known for decades and the possible surprising fact of equivalence for the two problems in terms of (i) local minimizers and (ii) points satisfying the so-called second-order optimality conditions, which are keys for designing optimization algorithms. We further discuss extensions to the vector case (where the vector variable is required to have all entries nonnegative) and the matrix case (where the matrix variable is required to be a positive semidefinite) and demonstrate such an equivalence continues to hold.
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02/14/25
Arseniy Kryazhev - UCSD
Chess
AbstractIn this talk, various aspects of the game of chess will be explored. Like true philosophers, we will make random observations, pose rhetorical questions and draw strange parallels, all without claim to the truth, while touching on various topics from the nature of randomness to ways to maximize cognitive performance. Familiarity with the game is not required but will be helpful.
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02/18/25
Félix Parraud
The spectrum of tensor of random and deterministic matrices
AbstractIn this talk, we consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices. I will explain a new strategy to bound its $L^p$-norm, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free semicircular variables as long as $p=o(N^{2/3})$. As a consequence, if the coefficients are $M$-dimensional matrices with $M=exp(o(N^{2/3}))$, then the operator norm of this polynomial converges towards the one of its free counterpart. In particular this provides another proof of the Peterson-Thom conjecture thanks to the result of Ben Hayes. The approach that we take in this paper is based on an asymptotic expansion obtained in a previous paper combined with a new result of independent interest on the norm of the composition of the multiplication operator and a permutation operator acting on a tensor of $C^*$-algebras.
By the way, can I assume that the people attending the seminar will be familiar with notions of free probability?
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02/18/25
John Peca-Medlin - UCSD
Heights of butterfly trees
AbstractBinary search trees (BSTs) are a fundamental data structure, optimized for data retrieval, entry, and deletion as needed for priority queue tasks. A fundamental statistic that controls each of these operations is the height of the tree with $n$ nodes, $h_n$, which returns the maximal depth of a node within the tree. Devroye established the height of a random BST, generated using a uniform permutation of length $n$, has height that limits to $c^*\approx 4.311$ when scaled by $\log n$. We are interested in studying the heights of random block BSTs, $h_{n,m}$, which correspond to uniform permutations combined using Kronecker or wreath products of permutations of lengths $n$ and $m$, that thus arise naturally in the setting of a BST data structure implemented using parallel architecture. We show using one such product of two permutations suffices to increase the asymptotic height of a random BST, while maintaining a logarithmic scaling with respect to the length of the generated permutation. We then explore the question of how much can the height increase when repeatedly using such products. These butterfly trees correspond to block BSTs formed using uniform butterfly permutations, that include a particular 2-Sylow subgroup of the symmetric group of $N = 2^n$ objects formed by taking $n$-iterated wreath products of $S_2$. In this setting, we show the expected heights for the corresponding block BSTs are now polynomial, with a lower bound of $N^\alpha$ for $\alpha \approx 0.585$. We provide exact nonasymptotic and asymptotic distributional descriptions for the case of simple butterfly permutations, which also include connections to other well-studied permutations statistics (e.g., the longest increasing subsequence, number of cycles, left-to-right maxima), while for nonsimple butterfly permutations we provide power-law bounds on the expected heights. This project is joint with Chenyang Zhong.
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02/19/25
Zeyu Liu - UC Berkeley
A stacky approach to prismatic crystals
AbstractNowadays prismatic crystals are gathering an increasing interest as they unify various coefficients in $p$-adic cohomology theories. Recently, attached to any $p$-adic formal scheme $X$, Drinfeld and Bhatt-Lurie constructed certain ring stacks, including the prismatization of $X$, on which quasi-coherent complexes correspond to various crystals on the prismatic site of $X$. While such a stacky approach sheds some new light on studying prismatic crystals, little is known outside of the Hodge-Tate locus. In this talk, we will introduce our recent work on studying quasi-coherent complexes on the prismatization of $X$ via various charts.
[pre-talk at 3:00PM]
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02/20/25
Professor Paul Apisa - University of Wisconsin
$\mathrm{SL}(2, \mathbb R)$-invariant measures on the moduli space of twisted holomorphic $1$-forms and dilation surfaces
AbstractA dilation surface is roughly a surface made up of polygons in the complex plane with parallel sides glued together by complex linear maps. The action of $\mathrm{SL}(2, \mathbb R)$ on the plane induces an action of $\mathrm{SL}(2, \mathbb R)$ on the collection of all dilation surfaces, aka the moduli space, which possesses a natural manifold structure. As with translation surfaces, which can be identified with holomorphic $1$-forms on Riemann surfaces, dilation surfaces can be thought of as “twisted” holomorphic $1$-forms.
The first result that I will present, joint with Nick Salter, produces an $\mathrm{SL}(2, \mathbb R)$-invariant measure on the moduli space of dilation surfaces that is mutually absolutely continuous with respect to Lebesgue measure. The construction fundamentally uses the group cohomology of the mapping class group with coefficients in the homology of the surface. It also relies on joint work with Matt Bainbridge and Jane Wang, showing that the moduli space of dilation surfaces is a $K(\pi,1)$ where $\pi$ is the framed mapping class group.
The second result that I will present, joint with Bainbridge and Wang, is that an open and dense set of dilation surfaces have Morse-Smale dynamics, i.e. the horizontal straight lines spiral towards a finite set of simple closed curves in their forward and backward direction. A consequence is that any fully supported $\mathrm{SL}(2, \mathbb R)$ invariant measure on the moduli space of dilation surfaces cannot be a finite measure. -
02/20/25
Hyuga Ito - Nagoya University
$B$-valued semi-circular system and free Poincaré inequality
AbstractIn 2003, P. Biane characterized a free semi-circular system in terms of free Poincaré inequality, which is an inequality related to the non-commutative L^2-norm of free difference quotients. In this talk, we will generalize his result to $B$-valued semi-circular system using a “natural” $B$-valued free Poincaré inequality. If time permits, we will also give a counterexample to Voiculescu’s conjecture related to $B$-valued free Poincaré inequality.
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02/20/25
Professor Eduardo Sontag - Northeastern University
Some theoretical results about responses to inputs and transients in systems biology
AbstractThis talk will focus on systems-theoretic and control theory tools that help characterize the responses of nonlinear systems to external inputs, with an emphasis on how network structure “motifs” introduce constraints on finite-time, transient behaviors. Of interest are qualitative features that are unique to nonlinear systems, such as non-harmonic responses to periodic inputs or the invariance to input symmetries. These properties play a key role as tools for model discrimination and reverse engineering in systems biology, as well as in characterizing robustness to disturbances. Our research has been largely motivated by biological problems at all scales, from the molecular (e.g., extracellular ligands affecting signaling and gene networks), to cell populations (e.g., resistance to chemotherapy due to systemic interactions between the immune system and tumors; drug-induced mutations; sensed external molecules triggering activations of specific neurons in worms), to interactions of individuals (e.g., periodic or single-shot non-pharmaceutical “social distancing'” interventions for epidemic control). Subject to time constraints, we'll briefly discuss some of these applications.
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02/21/25
Dr. Joe Foster - University of Oregon
The Lefschetz standard conjectures for Kummer-type hyper-Kähler varieties
AbstractFor a smooth complex projective variety, the Lefschetz standard conjectures of Grothendieck predict the existence of algebraic self-correspondences that provide inverses to the hard Lefschetz isomorphisms. These conjectures have broad implications for Hodge theory and the theory of motives. In this talk, we describe recent progress on the Lefschetz standard conjectures for hyper-Kähler varieties of generalized Kummer deformation type.
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02/24/25
Prof. Carly Klivans - Brown University
The Arboricity Polynomial
AbstractI will introduce a new matroid (graph) invariant: The Arboricity Polynomial. Arboricity is a numerical invariant first introduced by Nash-Williams, Tutte and Edmonds. It captures the minimum number of independent sets (forests) needed to decompose the ground set of a matroid (edges of a graph). The arboricity polynomial enumerates the number of such decompositions. We examine this counting function in terms of scheduling, Ehrhart theory, quasisymmetric functions, matroid polytopes and the permutohedral fan.
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02/24/25
Dr. Aryaman Maithani - University of Utah
Polynomial invariants of ${\rm GL}_2$: conjugation over finite fields
AbstractConsider the conjugation action of \({\rm GL}_2(K)\) on the polynomial ring \(K[X_{2\times 2}]\). When \(K\) is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when \(K\) is a finite field, and show that it is a hypersurface.
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02/26/25
Collin Cranston
Advancement to Candidacy
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02/27/25
Timothée Bénard - Université Sorbonne Paris Nord
Diophantine approximation and random walks on the modular surface
AbstractKhintchine's theorem is a key result in Diophantine approximation. Given a positive non-increasing function f defined over the integers, it states that the set of real numbers that are f-approximable has zero or full Lebesgue measure depending on whether the series of terms (f(n))_n converges or diverges. I will present a recent work in collaboration with Weikun He and Han Zhang in which we extend Khintchine's theorem to any self-similar probability measure on the real line. The argument involves the quantitative equidistribution of upper triangular random walks on SL_2(R)/SL_2(Z).
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02/27/25
Ningchuan Zhang - University of Indiana, Bloomington
Picard groups of quotient ring spectra
AbstractIn classical algebra, the Picard group of a commutative ring R is invariant under quotient by nilpotent elements. In joint work in progress with Ishan Levy and Guchuan Li, we study Picard groups of some quotient ring spectra. Under a vanishing condition, we prove that Pic(R/v^{n+1}) --> Pic(R/v^n) is injective for a ring spectrum R such that R/v is an E_1-R-algebra. This allows us to show Picard groups of quotients of Morava E-theory by a regular sequence in its π_0 are always ℤ/2. Running the profinite descent spectral sequence from there, we prove the Picard group of any K(n)-local generalized Moore spectrum of type n is finite. At height 1 and all primes p, we compute the Picard group of K(1)-local S^0/p^k when k is not too small.
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02/27/25
Jasper Liu
Advancement to Candidacy
Mar
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03/04/25
Minxin Zhang - UCLA
Inexact Proximal Point Algorithms for Zeroth-Order Global Optimization
AbstractThis work concerns the zeroth-order global minimization of continuous nonconvex functions with a unique global minimizer and possibly multiple local minimizers. We formulate a theoretical framework for inexact proximal point (IPP) methods for global optimization, establishing convergence guarantees under mild assumptions when either deterministic or stochastic estimates of proximal operators are used. The quadratic regularization in the proximal operator and the scaling effect of a positive parameter create a concentrated landscape of an associated Gibbs measure that is practically effective for sampling. The convergence of the expectation under the Gibbs measure is established, providing a theoretical foundation for evaluating proximal operators inexactly using sampling-based methods such as Monte Carlo (MC) integration. In addition, we propose a new approach based on tensor train (TT) approximation. This approach employs a randomized TT cross algorithm to efficiently construct a low-rank TT approximation of a discretized function using a small number of function evaluations, and we provide an error analysis for the TT-based estimation. We then propose two practical IPP algorithms, TT-IPP and MC-IPP. The TT-IPP algorithm leverages TT estimates of the proximal operators, while the MC-IPP algorithm employs MC integration to estimate the proximal operators. Both algorithms are designed to adaptively balance efficiency and accuracy in inexact evaluations of proximal operators. The effectiveness of the two algorithms is demonstrated through experiments on diverse benchmark functions and various applications.
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03/06/25
Professor Yen-Hsi Richard Tsai - University of Texas, Austin
Implicit boundary integral methods and applications
AbstractI will review a general framework for developing numerical methods working with non-parametrically defined surfaces for various problems involving. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. Such extension approaches allow us to analyze the well-posedness of the resulting system, develop, systematically and in a unified fashion, numerical schemes for treating a wide range of problems involving differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given.
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03/12/25
Yiran Jia - UCSD
Advancement to Candidacy
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03/13/25
Marie-France Vigneras - Jussieu
Asymptotics of $p$-adic groups, mostly $SL_2$
AbstractLet $p$ be a prime number and $ Q_p$ the field of $p$-adic numbers.
The representations of a cousin of the Galois group of an algebraic closure of $ Q_p$ are related (the {\bf Langlands's bridge}) to the representations of reductive $p$-adic groups, for instance $SL_2(Q_p), GL_n(Q_p) $. The irreducible representations $\pi$ of reductive $p$-adic groups are easier to study than those of the Galois groups but they are rarely finite dimensional. Their classification is very involved but their behaviour around the identity, that we call the ``asymptotics'' of $\pi$, are expected to be more uniform. We shall survey what is known (joint work with Guy Henniart), and what it suggests.
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03/13/25
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03/14/25
Vitor Borges - UCSD
Advancement to Candidacy
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03/27/25
Apr
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04/10/25
Professor Soeren Bartels - University of Freiburg, Germany
Babuska's Paradox in Linear and Nonlinear Bending Theories
AbstractThe plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem. The results are relevant for the construction of curved folding devices.