In this talk, I will discuss semi-parametric estimation when nuisance parameters cannot be estimated consistently, focusing in particular on the estimation of average treatment effects, conditional correlations, and linear effects under high-dimensional GLM specifications. In this challenging regime, even standard doubly-robust estimators can be inconsistent. I describe novel approaches which enjoy consistency guarantees for low-dimensional 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 high-dimensional prediction models.

Quantum graphs are a non-commutative 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 two-colorability 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.

Let $\mathrm{ex}(G_{n,p}^r,F)$ 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 non-trivial 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.

 How to understand the Galois group of Q-bar 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 simply-connected 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.

I 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 dbar-Neumann problem.

The latter is joint work with Samuele Mongodi (Univ. Milano-Bicocca, Italy).

An {\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.

The 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.

We will introduce a local notion of Hilbert-Schmidt stability (HS-stability), 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 HS-stability of amenable groups, by analogy with the character criterion for HS-stability of Hadwin and Shulman. We will then discuss further examples of (flexible versions of) local HS-stability. 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 Fournier-Facio and Maria Gerasimova.

Estimation 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 theory---smoothness and convexity---rarely 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, semi-algebraic geometry, and high dimensional probability and statistics.

Representations of p-adic groups and Hecke algebras Abstract: An explicit understanding of the category of all (smooth, complex) representations of p-adic 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 p-adic 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 depth-zero 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 p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

There 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 Coxeter-Cayley 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 Hurwitz-Cayley graph of the symmetric group and explain their relation to gauge-string/dualities in Yang-Mills 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.    

This 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

Heuristically, 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.

The Weierstrass p-function plays a great role in the classic theory of complex elliptic curves. A related function, the Weierstrass zeta-function, is used by Guerzhoy to construct preimages under the xi-operator of newforms of weight 2, corresponding to elliptic curves.  In this talk, I will discuss a generalization of the Weierstrass zeta-function and an application to harmonic Maass forms. More precisely, I will describe a construction of a preimage of the xi-operator of a newform of weight k for k>2. This is based on joint work with C. Alfes-Neumann, J. Funke and M. Mertens.

 Traditional numerical methods based on expensive matrix factorizations struggle with the scale of modern scientific applications. For example, kernel-based 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 low-rank. 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 speed-up factor from $N^3$ to $N k^2$ operations can be 3 million. The newest algorithms achieve this speed-up factor while guaranteeing performance across a broad range of input matrices.

Given 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 Chi-Kang Chang.

Matroids combinatorially abstract the ubiquitous notion of "independence" in various contexts such as linear algebra and graph theory.  Recently, an algebro-geometric 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.

The 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 long-held 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. 

Algebraic 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 K-stability, which is an algebraic definition introduced by differential geometers to characterize when a Fano variety admits a Kahler-Einstein metric.

The 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.

We 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 well-known problems in the area about estimating on off-diagonal ordered Ramsey numbers of ordered matchings versus a triangle. This is a joint work with Marian Poljak.

I 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 80s-90’s that solutions can exhibit bubbling along a well-chosen sequence of times — the solution decouples into a superposition of well-separated 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 multi-bubbles 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. 

Bayesian 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 non-convex region, the user can find local minimizers in non-convex, strongly sparsity-promoting regimes that are consistent with a convex relaxation drawn from the same family of posteriors. We show experimentally that these solutions are less error-prone than direct optimization of the non-convex problem. The same relaxation approach allows sampling from highly non-convex multi-modal posteriors in high dimension via a variational Bayesian formulation. We demonstrate predictor-corrector methods for estimator and sample continuation.

Classical 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. 

Translation 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 well-studied, 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.

Ligand-receptor binding and unbinding are fundamental molecular processes, whereas water fluctuations impact strongly their thermodynamics and kinetics. We develop a variational implicit-solvent model (VISM) and a fast binary level-set method to calculate the potential of mean force and the molecule-water 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 dry-wet transitions and the mean first-passage times for the ligand-pocket binding and unbinding. Without any explicit description of individual water molecules, our predictions are in a very good, qualitative and semi-quantitative, agreement with existing explicit-water 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. 

I 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.

[pre-talk at 1:20PM]

Groups 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. 

A 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 non-arithmetic hyperbolic manifolds. Aspects of this talk represent different works with L. Kumandari and J. Wang, and with K. Ohm.

Constrained estimation problems are prevalent in statistics, machine learning, and engineering. These problems encompass constrained generalized linear models, constrained deep neural networks, physics-inspired 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 regularization-based 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.

In 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.  

Computing the derivative of long-time-averaged observables with respect to system parameters is a central problem for many numerical applications. Conventionally, there are three straight-forward 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), high-dimensional, and small-noise.

We present the fast response formula, which is a 'Monte-Carlo' type formula for the parameter-derivative of hyperbolic chaos. It is the average of some function of u-many 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 non-hyperbolicity.

The zeroeth complex topological K-theory of a space encodes complex vector bundles up to stabilization. Since complex topological K-theory is highly computable, this is a great place to start when asking bundle-theoretic questions. However, in general, many non-equivalent bundles represent the same K-theory class. Bridging the gap between K-theory 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 r-space that are trivial in K-theory is unknown. In this talk, we will compute the p-primary 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 Weiss-theoretic techniques in tandem with a little chromatic homotopy theory to translate bundle enumeration to a computation of the higher real K-theory of particular simple spectra. The result will involve actual numbers!  This is joint work with Hood Chatham and Yang Hu.

Nonparametric models are of great interest in various scientific and engineering disciplines. Classical kernel methods, while numerically robust and statistically sound in low-dimensional settings, become inadequate in higher-dimensional settings due to the curse of dimensionality.

In this talk, we will introduce a new framework called Variance-Reduced 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 infinite-size matrices, facilitating a new matrix-based bias-variance tradeoff in various nonparametric contexts.

We will demonstrate the robust numerical performance of VRS through a series of simulated experiments and real-world 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.

We 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 non-Archimedean 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 Bruhat-Tits tree $\operatorname{PGL}_2$. In this talk, we introduce the highly-branched Schottky subgroups and steps for the proof of the main theorem.

This is a joint work with Seonhee Lim.

One 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.

A 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 error-prone proxy of the treatment-free 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 post-treatment 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 treatment-free 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 post-treatment data. We illustrate the SPC and SPSC approaches with real-world applications from the Zika virus outbreak in Brazil and the 1907 financial crisis.

In this paper, we address the problem of continuous-time reinforcement learning in scenarios where the dynamics follow a stochastic differential equation. When the underlying dynamics remain unknown and we have access only to discrete-time information, how can we effectively perform policy evaluation? We first demonstrate that the commonly used Bellman equation is a first-order approximation to the true value function. We then introduce a higher order PDE-based Bellman equation called PhiBE. We show that the solution to the i-th order PhiBE is an i-th order approximation to the true value function. Additionally, even the first-order 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 discrete-time trajectory data. Numerical experiments are provided to validate the theoretical guarantees we propose.

We 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 non-empty $k$-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

In 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.

I 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 high-quality solution). Surprisingly, these diverse tasks admit a unified analysis involving the same two main ingredients. The first is the study of algorithms that output low-degree 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 Sherrington-Kirkpatrick spin glass model.

Next, by examining the sum-of-squares 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 long-standing 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.

In the setting of zero-entropy transformations, the class of subshifts--closed shift-invariant 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 probability-preserving 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.

Given 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

$$\mathcal{E}_W(f)=\liminf_{\tilde f\to f}\ \frac12\int_{\mathcal{P}(M)} \big\|\nabla_W \tilde f\big\|^2(\mu)\ d\mathbb{P}^\beta(\mu);$$

• non-degenerate, at least in the case of the $n$-sphere and the $n$-torus.

Since 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 Temperley-Lieb-Jones 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.

The $S$-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.

[pre-talk at 1:20PM, in person only]

 I 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 non-spatial 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.   

Representations 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 Bruhat-Tits building. If time permits, I will discuss open problems.

Have 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 Dirac-Klein-Gordon system.

Consider a uniformly random k-SAT instance with a fixed ratio of the number of clauses to the number of variables. As the clause-variable 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 k-SAT, including various algorithmic approaches, the connections to statistical physics, Friedman's sharp threshold theorem, and the determination of the k-SAT threshold for large k by Ding, Sun, and Sly in 2014.

A 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.

I 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 fact-check 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 multi-modal 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.

In this talk, we address the problem of continuous-time reinforcement learning in scenarios where the dynamics follow a stochastic differential equation. When the underlying dynamics remain unknown and we have access only to discrete-time information, how can we effectively conduct policy evaluation? We first demonstrate that the commonly used Bellman equation is a first-order approximation to the true value function. We then introduce higher order PDE-based Bellman equation called PhiBE. We show that the solution to the i-th order PhiBE is an i-th order approximation to the true value function. Additionally, even the first-order 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 discrete-time trajectory data. Numerical experiments are provided to validate the theoretical guarantees we propose. 

Can 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 pseudo-Riemannian 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 Erdős-Renyi random graph and is an interesting toy model for a physical dynamics of discrete space-time.

We 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 d-1. This complements Timar's work who showed the possible nonexistence of growth below this threshold. The proof goes as follows. By Benjamini-Lyons-Schramm, we can realize (T,o) as the cluster of the root for some invariant percolation on the d-regular 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 2-3-method, as the usual pointwise ergodic theorems do not seem to work here. We then define and prove the Cohen-Grigorchuk co-growth 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.

Interpretable machine learning has been widely deployed for scientific discoveries and decision-making, 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 large-scale graph learning and UQ for model-agnostic 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 low-confidence 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, assumption-light, and statistically powerful inference approach for occlusion-based 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 Wisconsin-Madison, 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://lili-zheng-stat.github.io

Tensor 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 low-rank 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 signal-to-noise ratio condition for polynomial-time algorithms. The second part of the talk focuses on uncertainty quantification for tensor learning. Under a classical tensor PCA model, I will present a two-iteration 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.

Quantum 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.

[pre-talk at 1:20PM]

For the last 3.8 billion years, the large-scale 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 so-called phylogenetic or evolutionary trees exhibit two unexplained broad structural features which are consistent across evolutionary time. The first is that phylogenetic trees exhibit scale-invariant 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 coarse-grained 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 scale-invariant topology and the bursty pattern of diversification in time. These results show in principle how dynamical scaling laws of phylogenetic trees on long time-scales 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 large-scale 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 open-ended 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 co-operative agreement NNA13AA91A through the NASA Astrobiology Institute for Universal Biology.
 

Emergent 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. 

In this talk, we will present the development of Corona problem in sereval complex variables and discuss its relation to the solution of the sup-norm estimates for the Cauchy-Riemann 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.

In 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 so-called G.C.D. method as the applications of my recent work with Paul Vojta.  

The 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 two-pointed curve neighborhood of opposite Schubert varieties $X_u$ and $X^v$ in the minimal degree $d$. We establish a Deodhar-like decomposition for tilted Richardson varieties, leveraging it to prove several results. This is joint work with Shiliang Gao and Yibo Gao.

The use of randomness in Ramsey Theory has been a key tool in non-constructive 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 long-standing open problems in the area.

The 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 number-theoretic criterion of Mirzakhani and Wright to classify such triangles with a large obtuse angle. This work is joint with Anne Larsen and Chaya Norton.

Consider 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.

The 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 sub-Riemannian 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.

Each 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 Yang-Baxter equation along with techniques involving Gibbs measures to extract the space-time 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 Kardar-Parisi-Zhang universality class, namely the Airy sheet, directed landscape and KPZ fixed point. This is joint work with Amol Aggarwal and Milind Hegde.

I’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.

Given 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 Grothendieck-Katz p-curvature conjecture. Probably the main beauty of the subject is its completely elementary nature, which makes it accessible to a first year student.

 This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS 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.

The 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.

The 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 multi-type''. The finiteness of the tower multi-type 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 multi-type, each covered locally by strongly pseudoconvex submanifolds of the boundary. The existence of such stratification implies Catlin's celebrated potential-theoretic ''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.

Louis 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.

I 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.

he 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.

Quantum 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‘’ Inexact-Feasible IPMs (IF-IPM) for LO and SDO problems, using novel Newton systems to generate inexact but feasible steps. We show that IF-QIPMs enjoys the to-date 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 IF-IPM’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 Editor-in-Chief 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.

Eco-Evolutionary 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 near-equilibrium 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.

Originally, 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 Bruhat-Tits building associated to PGL(3, F) where F is a non-archimedean local field. This may be seen as both a higher rank analogue of the regular graphs setting as well as a non-archimedean analogue of the symmetric space setting.

Locally 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 Grosse-Klönne. 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´echet-Stein algebra, i.e., it is equipped with a desirable algebraic structure. This is joint work with Claus Sorensen and Matthias Strauch.

The 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 large-scale 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 Editor-in-Chief 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.

Symmetry 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.

Predicting 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 patient-specific 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 Physics-Informed Neural Networks (PINNs) to estimate patient-specific parameters of a reaction-diffusion 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 non-dimensional parameters, a pre-training step that utilizes the non-dimensional parameters and a fine-tuning 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 real-time parametric inference in the clinical setting for personalized GBM treatment.

We say that an integer-valued 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. 2-point 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.

Neural networks are highly non-linear 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 data-driven 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.  

In 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 multi-variable 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.

Margulis'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.

Recent 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.

Do 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:  p-adic 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.

We 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.

In 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.

The 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 quasi-Newton approach, which maintains a positive-definite approximation of the Hessian of the Lagrangian, second-derivative 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 primal-dual SQP methods.  Extensive numerical results will be presented.

Weighted Hurwitz numbers were introduced by Harnad and Guay-Paquet 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 Chekhov--Eynard--Orantin: 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 rationally-weighted Hurwitz numbers was recently proved by Bychkov--Dunin-Barkowski--Kazarian--Shadrin. 

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 so-called $W$-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.

For 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_p-points 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 n-punctured 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.

Hyperkahler 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 non-constant 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.

In 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.

Inspired 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.

This talk discusses computationally feasible algorithms to uncover low-dimensional 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 low-dimensional 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 speed-up 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.

Evolutionary dynamics permeates life and life-like 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.

In 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!

Compact hyper-Kähler manifolds and their Lagrangian fibrations are higher-dimensional 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 hyper-Kähler manifolds and the boundedness problem.

Originally appearing in string theory, the Boson-Fermion 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 H-modules, 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 back-stable symmetric functions represents the Bosonic space.

Around 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. 

Cellular 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.

The motion of undulating or rotating elastic `tails’ 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 closed-loop 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.

I will discuss how novel ideas from non-abelian  Brill-Noether 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

Many 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, so-called 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 surface-based geometry representations into structured and easy-to-generate 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 Navier-Stokes equations and fluid-structure interaction problems with large deformations. In the second part of the talk, we show that the CutFEM framework can also be used to discretize surface-bound PDEs as well as mixed-dimensional problems where PDEs are posed on domains of different topological dimensionality.

As a particular example, we consider the so-called Extracellular-Membran-Intracellular (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.

It 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 high-degree 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.

We will consider the dynamics of automorphisms acting on highly-symmetric 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.

Asymmetric 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.

We 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.

I 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 semi-simple Lie groups and their lattices, obtaining interesting applications to operator algebras and representation theory.
 

Schur-Weyl 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, `Schur-Weyl-Jones 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 $w : S_n \times \cdots  \times S_n \rightarrow S_n$.

We study a general class of finite-player stochastic games with mean-field interaction where the linear-quadratic 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 finite-player Nash equilibrium to the mean-field equilibrium in the infinite player limit. Our general framework includes examples of stochastic Volterra linear-quadratic 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.
 

Cell 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 shape-changing 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.

This 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.

In the early years of the XX century, Weyl initiated study of compact perturbations of pseudo-differential operators. The Weyl-von Neumann theorem asserts that two self-adjoint 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. 

The 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 BW-Del 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 under-represented 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. 

 We 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 Au-C{\'e}bron-Dahlqvist-Gabriel-Male.  In particular, this gives a new probabilistic proof of Ciobanu-Holt-Rees'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.

We will overview variants of CUR decompositions for tensors. These are low-rank 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 t-products 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.

Epigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms. Simulation studies have shown how stochastic dynamics and time-scale 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.

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

There 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 non-commutative probability theory. Tracial von Neumann algebras are infinite-dimensional 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.

The Cayley-Bacharach 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.

U(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 Hurwitz-Cayley graph which serves as the Feynman diagram

The 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.  

This talk presents a particle-based optimization method designed for addressing minimization problems with equality constraints, particularly in cases where the loss function exhibits non-differentiability or non-convexity. A rigorous mean-field limit of the particle system is derived, and the convergence of the mean-field limit to the constrained minimizer is established.

A 2015 conjecture of Ross and Yong proposes a K-Kohnert rule for Grothendieck polynomials. In this talk we discuss the utility of Kohnert rules then prove a special case of the Ross-Yong conjecture. We then show the conjecture fails in general.

This talk presents a particle-based optimization method designed for addressing global optimization problems, particularly in cases where the loss function exhibits non-differentiability or non-convexity. Numerically, we show that it outperforms gradient-based method in finding global optimizer. Theoretically, A rigorous mean-field limit of the particle system is derived, and the convergence of the mean-field limit to the global minimizer is established. In addition, we will talk about its application to the constrained optimization problems and federated learning.

We 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

We 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.

In 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 non-amenable). 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. Kazhdan-Margulis minimal covolume theorem.

2. Most hyperbolic manifolds are non-arithmetic (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).

What 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...

Let \(\Gamma\) 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.
 

Quantum graphs and their characteristics are intriguing generalizations of notions and tools known from discrete mathematics into the quantum world. Their non-trivial 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.

This 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.

In 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 rank-one 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.

The classical Coates-Sinnott Conjecture and its refinements predict the dee relationship between the special values of L-functions 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 Coates-Sinnott 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.

I 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. 

Nature-inspired 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 nature-inspired 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 D-optimal 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.  

Let \(G\) 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. 

 In 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.

I 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.

In stochastic modeling we often need to deal with one of two apparently unrelated objects. One is self-similar processes and the other is additive functionals. Self-similar 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 so-called Lamperti transform ensures that any $R^d$-valued self-similar 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 self-similar 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 1980-90 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 self-similar 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.

Simply connected smooth 4-manifolds 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 Bauer-Furuta invariant. The goal of this talk is to provide the context of this area, the background for the families Bauer-Furuta invariant, and some ideas for my future research directions. 

Gaussian 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 2-Sylow subgroup of the symmetric group on a set of size 2ⁿ.

Human 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 multi-disciplinary, data-driven 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, multi-faceted human behavior across diverse domains in critical issues of societal importance, ranging from socio-cognitive biases to pandemic compliance. The talk will also discuss the importance of bottom-up behavior and attitude changes, as well as large-scale human cooperation, in addressing urgent challenges facing our common humanity.

Let $G\stackrel{\alpha}{\curvearrowright}(M,\tau)$ be a trace-preserving 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.

Mean 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 multi-drug chemotherapy schedules that suppress the competitive release of resistant cell populations (to avoid chemo-resistance) by maximizing the Shannon diversity of the competing subpopulations. The models can be extended to networks where spatial connectivity can influence optimal chemotherapy scheduling. 

The famous no-three-in-line 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 non-trivial 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.

Suppose you wish to generate a random graph on vertices $v_1, ..., v_n$ where the degree of each $v_i$ is specified to be $d_i$. Perhaps the most natural approach is: Repeatedly  add an edge joining two vertices chosen uniformly from all non-adjacent pairs that are not yet full (i.e. whose current degrees are less than their specified degrees).

The graph we obtain is not distributed uniformly amongst all graphs with the specified degree sequence (except for some trivial sequences). But is the distribution close to uniform in the sense that every property which holds with high probability in one model holds with high probability in the other?

We answer that question in the negative for bounded degree sequences that are not regular or close to regular.

This is joint work with Erlang Surya and Lutz Warnke; see arXiv:2211.00835

In this talk, I will explain why fractional (or nonlocal) minimal surfaces are ideal objects to which min-max 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.