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. 

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. 

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.

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.

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]

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. 

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.

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. 

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.  

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.

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.

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.

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.

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.

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.

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.

 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.

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.

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.

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.

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.

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.

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

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.

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.

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.
 

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.

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.

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.

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).

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.

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. 

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 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.

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.

In this talk, I'll first give a quick introduction to the theory of Drinfeld modules and talk about an equivariant $L$-function associated to Drinfeld modules as defined by Ferrara-Higgins-Green-Popescu in their work on the ETNC. As is usual, these $L$-functions are defined as an infinite product of Euler factors, and the main focus of this talk is a result relating these Euler factors to a certain quotient of Fitting ideals of some algebraically relevant modules. This is joint work with Cristian Popescu.

We consider the mixed-strategy zero-sum game such that each player’s objective function is quadratic in its own variables. By considering each player’s value function and duality, the bi-quadratic games are reformulated as linear programs over the cone of copositive (COP) and completely positive (CP) matrices. We apply moment and SOS relaxations for the conic constraints of CP and COP matrices, respectively, and obtain a hierarchy of semidefinite relaxations. Under certain conditions, the finite convergence for this hierarchy is guaranteed, and the tightness can be checked via flat truncation. We present numerical experiments to show the effectiveness of our approach.

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

We study stochastic interacting particle methods with and without field coupling for high dimensional concentration and singularity formation phenomena. In case of entropy production of reverse-time diffusion processes, the method computes concentrated invariant measures mesh-free up to dimension 16 at a linear complexity rate based on solving a principal eigenvalue problem of non-self-adjoint advection-diffusion operators. In case of fully parabolic chemotaxis nonlinear dynamics in 3D, the method captures critical mass for finite time singularity formation and blowup time at low costs through a smoother field without relying on self-similarity.

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.

Inspired by the classical random graph process introduced by Erdos and Renyi in 1960, we discuss two analogous processes for random representable matroids, one introduced by Kelly and Oxley in 1982 and the other one introduced by Cooper, Frieze and Pegden in 2019. In the talk we address the evolution of the rank, circuits, connectivity, and the critical number (corresponding to the logarithm of the chromatic number of graphs) of the first random matroid, and then we focus on the minors in both matroid models. 

This talk discusses the challenging problem of finding global optimal solutions for two-stage stochastic programs with continuous decision variables and nonconvex recourse functions. We introduce a two-phase approach, which does not only generate global lower bounds for the nonconvex stochastic program but also simplifies the computation of the expected value of the recourse function by using moments of random vectors. This makes our overall algorithm particularly suitable for the case where the random vector follows a continuous distribution or when dealing with many scenarios. Numerical experiments are given to demonstrate the effectiveness of our proposed approach.

We will consider the behavior of the Kolmogorov-Sinai entropies of amenable group actions under a Shannon orbit equivalence. Although dynamical entropy is in general not invariant under orbit equivalences, recent works have shown that various notions of restricted orbit equivalences will preserve entropy. We focus on the case where the orbit equivalence is Shannon, and both groups are finitely generated amenable. In this talk, we will present a proof for our main result.

As the shortest path metric on a weighted tree can be embedded isometrically into a finite $\ell_1$ space, a Lipschitz embedding of a given graph into $\ell_1$ can be obtained by constructing a low distortion embedding into a tree. Conversely, while there are various topological properties of graphs that guarantee controlled distortion Lipschitz embeddings into $\ell_1$ ($k$-outerplanar, series-parallel, low Euler characteristic), it is still often the case that such a graph embeds quite poorly into a tree.

By introducing the notion of a stochastic embedding into a family of trees one can find more general concrete embeddings into $\ell_1$ then those limited by a single tree. In fact, it is known that every graph with n vertices embeds stochastically into trees with distortion O(log(n)). Nevertheless, this upper bound is sharp for graphs such as expanders, grids and, by a recent joint work with Schlumprecht, a large class of "fractal-like" series-parallel graphs called slash powers. 

In this talk we introduce an equivalent characterization of stochastic distortion called expected distortion and after proving a mild extension of a result of Gupta regarding poor tree embeddings of a cycle, inductively lower bound the expected distortion of generalized Laakso graphs found in most nontrivial slash power families.

We will discuss questions surrounding automorphic L-functions, particularly Deligne’s conjecture about critical values of motivic L-functions. In particular, we study the adjoint L-function of U(2,1).

Hundley showed that a certain integral, involving an Eisenstein series on the exceptional group G_2, computes this L-function at unramified places. We discuss the computation of this integral at the archimedean place for quaternionic modular forms, and how this relates to Deligne's
conjecture.

Given an automorphism of a variety $X$, there is an induced ''natural'' automorphism on $X^{[n]}$, the Hilbert scheme of $n$ points of $X$. While unnatural automorphisms of $X^{[n]}$ are known to exist for certain varieties $X$ and integers $n$, all previously known examples could be shown to be unnatural because they do not preserve multiplicities. Belmans, Oberdieck, and Rennemo thus asked if an automorphism of a Hilbert scheme of points of a surface is natural if and only if it preserves the diagonal of non-reduced subschemes.

We give an answer in the negative for all $n\ge 2$ by constructing explicit counterexamples on certain abelian surfaces $X$. These surfaces are not generic, and hence we prove a partial converse statement that all automorphisms of the Hilbert scheme of two points on a very general abelian surface are natural for certain polarization types (including the principally polarized case).

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

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.

We propose a quantum analog of the Mycielski transformation of graphs and iwe study relations between quantum automorphism groups of a quantum graph and its Mycielskian. Based on joint work with A. Bochniak (arXiv:2306.09994) and work in progress with A. Bochniak, P.M. Sołtan, and I. Chełstowski.

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

The List Colouring Conjecture posits that the list edge chromatic number of any graph is equal to the edge chromatic number, and thus is at most D+1 where D is the maximum degree.  This means that if each edge is assigned a list of $D+1$ colours then it is always possible to obtain a proper edge colouring by choosing one colour from each list.

In the 1990's, Khan proved that one can always obtain a proper edge colouring from lists of size $D+o(D)$, then Molloy and Reed obtained $D+D^{1/2}\mathrm{poly}(\log D)$.  We improve that bound to $D+D^{2/5}\mathrm{poly}(\log D)$
 

Saddle connections on a translation surface generalize both diagonals in a polygon and primitive vectors in a 2-dimensional lattice. Their slopes thus contain geometric and algebraic information about the surface. Slopes of saddle connections can be studied using the action of a horocycle subgroup of $\mathrm{SL}_2(\mathbb{R})$ on the moduli space of all translation surfaces. In particular, gaps between slopes are directly related to the return time function of a Poincaré section for the horocycle flow.

In this talk, we will describe a Poincaré section for horocycle flow in the smallest nontrivial stratum $\mathcal{H}(2)$ and see how to compute the return time function. Then we will examine some consequences for gap distributions. This is joint work with Anthony Sanchez.

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. 

Classic mechanisms of spatial pattern formation in developmental biology are characterized by high degrees of multistability and sensitivity to initial conditions. These traits are commonly seen as undermining the capacity of these processes to exhibit robust morphogenesis. However, a growing body of experimental evidence suggests developing organisms can accomplish robust pattern selection in reaction-diffusion processes with relatively simple spatiotemporal forcings. To better understand this phenomenon, we perform a series of systematic investigations into the optimal controllability of a minimal pattern-forming system. Using machine-learning-inspired techniques, we generate simple optimal control protocols to drive an underactuated system to a desired steady state.  We numerically demonstrate the effectiveness of control in two universal scenarios of pattern formation: within a weakly nonlinear regime associated with a supercritical Turing instability and for localized states associated with homoclinic snaking.

A large area of modern number theory (the Langlands program) studies a deep correspondence between the representation theory of Galois groups, algebraic varieties and certain analytic objects (automorphic forms). Many spectacular theorems have come from this area, for example the key insight in Wiles' proof of Fermat's last theorem was a connection between elliptic curves, modular forms and Galois representations.

The goal of this talk is to explain how geometric constructions, particularly related to Shimura varieties, arise naturally in the Langlands program. I will then talk about joint work with Stefan Patrikis, stating that Galois representations arising from certain Shimura varieties satisfy the properties predicted by the correspondence introduced above.

In 2015, Marcus, Spielman, and Srivastava realized that expected characteristic polynomials of sums and products of randomly rotated matrices behave like finite versions of Voiculescu's free convolution operations. In 2022, I obtained a similar result for commutators of such random matrices; one feature of this result is the special role of even polynomials, in parallel with the situation in free probability.

It turns out that a certain family of special polynomials, called hypergeometric polynomials, arises naturally in relation to convolution of even polynomials and finite free commutators. I will explain how these polynomials can be used to approach questions of real-rootedness and asymptotics for finite free commutators. Based on arXiv:2209.00523 and ongoing joint work with Rafael Morales and Daniel Perales.

It is well known that any directed graph induces an undirected graph by forgetting the direction of the edges and keeping the underling structure. In fact, this assignment can be extended to consider graph morphisms, thus obtaining a functor from the category of simple directed graphs and directed graph morphism, to the category of undirected graphs and undirected graph morphisms. This particular functor is known as a “forgetful” functor, since it forgets the notion of direction.

In this talk, I will present a bijective functor that relates the category of simple directed graphs with a particular category of undirected graphs, whose objects we call “prime graphs”. Intuitively, prime graphs are undirected bipartite graphs endowed with a label that evokes a notion of direction. As an application, we use two undirected graph techniques to study directed graphs: spectral clustering and network alignment.

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

We introduce repetition noise into quantum error-correcting codes with a tensor network structure. Our approach employs a quantum channel, which is a superposition of exact encodings and repetition encoding with a small probability. The boundary states of our models capture key features of conformal field theory states, particularly the power law of the two-point function and logarithmic entanglement, which are precisely obeyed in many cases. The noisy holographic quantum error-correcting codes on trees and tilings of two-dimensional hyperbolic space preserve the bulk/boundary duality in AdS/CFT, and their boundary states exhibit the features of conformal field theory accordingly.

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

The Laplace operator is the infinitesimal generator of Brownian motion with a real state space.  The Vladimirov operator, a $p$-adic analogue of the Laplace operator, similarly gives rise to Brownian motion with a $p$-adic state space.  This talk aims to introduce the concept of a $p$-adic Brownian motion and demonstrate a further similarity with its real analogue: $p$-adic Brownian motion is a scaling limit of a discrete-time random walk on a discrete group.  Attendees need not have prior knowledge of $p$-adic analysis, as the talk will provide a brief review of necessary background information.

In this talk we will present some analysis aspects of gauge theory in high dimension. First, we will study the completion of the space of arbitrary smooth bundles and connections under L^2-control of their curvature. We will start from the classical theory in critical dimension and then move to the supercritical dimension, making a digression about the so called “abelian” case and thus showing an analogy between p-Yang-Mills fields on abelian bundles and a special class of singular vector fields. In the last part, we will show how the previous analysis can be used in order to build a Schoen-Uhlenbeck type regularity theory for Yang-Mills fields in supercritical dimension.

I will present several applications of Shlyakhtenko’s operator-valued semicircular systems, including characterization of Property (T) for II$_1$ factors in terms of spectral gap in inclusions, and on weak containment of bimodules.

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

In my experience, successes often arise from circumstances that appear to be less than ideal, or even hopeless.  In the AWM Colloquium, I will discuss some key developments along my career path.

The target audience is graduate students and postdocs.  Audience engagement is encouraged.  In particular, I will allow ample time for questions.
 

In this talk, we will demonstrate that the well-known singularity Hausdorff dimension estimates for isoperimetric regions are sharp by constructing singular examples in dimensions 8 and higher. Then, to explore the isoperimetric regions under generic Riemannian metrics, we will discuss the twisted Jacobi field of singular constant mean curvature hypersurfaces under certain regularity assumptions.

For $C$ a smooth curve and $\xi$ a line bundle on $C$, the moduli space $U_C(2,\xi)$ of semistable vector bundles of rank two and determinant $\xi$ is a Fano variety. We show that $U_C(2,\xi)$ is K-stable for a general curve $C \in \overline{M}_g$. As a consequence, there are irreducible components of the moduli space of K-stable Fano varieties that are birational to $\overline{M}_g$. In particular these components are of general type for $g\geq 22$.

Come find out how to (randomly) diagonalize any $n \times n$ matrix pencil in fewer than $O(n^3)$ operations!

This study delves into the study of complex networks within a hyperbolic latent space model, presenting theoretical analysis of popular link prediction indices on hyperbolic random graphs. We investigate how different degrees of nodes influence link prediction heuristics. By modifying indices like the common neighbor and shortest path index, the study demonstrates theoretical and empirical improvements in both simulated and real-world networks. Additionally, we also explore embedding methods to recover hyperbolic geometry, introducing a modified hyperbolic ordinal embedding method. 

You use universal properties everyday. But what are they exactly? We'll give an answer via something called a universal element. It is the nice concrete thing which defines a Yoneda representation. We'll give several examples of the universal elements for common representations.

Prerequisite: Watch this video https://www.youtube.com/watch?v=mLRgKPwyg4Y

Often natural moduli problems, e.g. foliated surfaces, come without an ample line bundle, so the algebraisability of formal deformations, and the very existence of a moduli space requires a study of the mermorphic functions on the aforesaid deformations, and the flattening (by blowing up) of the resulting meromorphic maps. In such a context the flattening theorem of Raynaud & Gruson, and derivatives thereof, is close to irrelevant since it systematically uses schemeness to globally extend local centres of blowing up. This was already well understood by Hironaka in his proof of holomorphic flattening, and his ideas are the right ones. Nevertheless, the said ideas can be better organised with a more systematic use of Grothendieck's universal flatifier, and, doing so, leads to a fully functorial, and radically simpler, proof provided the sheaf of nilpotent functions is coherent-which is true for excellent formal schemes, but, unlike schemes or complex spaces, is false in general.

The class of stabilized sequential quadratic programming (SQP) methods for nonlinearly constrained optimization solve a sequence of related quadratic programming (QP) subproblems formed from a two-norm penalized quadratic model of the Lagrangian function subject to shifted, linearized constraints. While these methods have been shown to exhibit superlinear local convergence even when the constraint Jacobian is rank deficient at the solution, they generally have no global convergence theory. To address this, primal-dual SQP methods (pdSQP) employ a certain primal-dual augmented Lagrangian merit function and solve a subproblem that consists of bound-constrained minimization of a quadratic model of the merit function. The model of the merit function is constructed in a particular way so that the resulting primal-dual subproblem is equivalent to the stabilized SQP subproblem. Together with a flexible line-search, the use of the merit function guarantees convergence from any starting point, while the connection with the stabilized subproblem allows pdSQP to retain the superlinear local convergence that is characteristic of stabilized SQP methods. 

A new dynamic convexification framework is developed that is applicable for nonconvex general standard form, stabilized, and primal-dual bound-constrained QP subproblems. Dynamic convexification involves three distinct stages: pre-convexification, concurrent convexification and post-convexification. New techniques are derived and analyzed for the implicit modification of symmetric indefinite factorizations and for the imposition of temporary artificial constraints, both of which are suitable for pre-convexification. Concurrent convexification works synchronously with the active-set method solving the subproblem, and computes minimal modifications needed to ensure the QP iterates are uniformly bounded. Finally, post-convexification defines an implicit modification that ensures the solution of the subproblem yields a descent direction for the merit function.

A new exact second-derivative primal-dual SQP method (dcpdSQP) is formulated for large-scale nonconvex optimization. Convergence analysis is presented that demonstrates guaranteed global convergence. Extensive numerical testing indicates that the performance of the proposed method is comparable or better than conventional full convexification while significantly reducing the number of factorizations required.

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

We consider a generalization of the maximum degree in random graphs. Given a rooted tree $T$, let $X_v$ denote the number of copies of T rooted at v in the binomial random graph $G_{n,p}$. We ask the question where the maximum $M_n = max \{X_1, ..., X_n\}$ of $X_v$ over all vertices is concentrated. For edge-probabilities $p=p(n)$ tending to zero not too fast, the maximum is asymptotically attained by the vertex of maximum degree. However, for smaller edge probabilities $p=p(n)$, the behavior is more complicated: our large deviation type optimization arguments reveal that the behavior of $M_n$ changes as we vary $p=p(n)$, due to different mechanisms that can make the maximum large.

Based on joint work with Pedro Araújo, Simon Griffiths and Matas Šileikis; see arXiv:2310.11661 

I will present the Poincare polynomial, the Chow ring, and some tautological relations on the moduli space of elliptic K3 surfaces.

The goal of the talk is to explain some geometric results on quasi-projective manifolds from the perspective of Carathéodory metrics and distances.  We will study some conjectures of Lang on manifolds which satisfied some Carath\'eodory conditions. The results are also used to study hyperbolicity of suitable compactifications of the non-compact manifolds involved.  As applications, we also prove some statements to the effect of non-existence of level structures on manifolds such as abelian varieties over function fields, as well as the  so-called volume estimates for mapping of curves into the manifolds involved. Most of the results to be presented are joint work with Kwok-Kin Wong.

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

In this talk we will introduce the level set flow. Then we will talk about the relation between the rate of curvature blow-up near a singularity of the flow and the distribution of surrounding singular points.

In theory, Chabauty-Coleman provides an explicit method to obtain rational points on any curve, so long as its genus exceeds its Mordell-Weil rank. In practice, when applied to modular curves, we often encounter difficulties in finding a suitable plane model, which only worsens as the genus increases. In this talk we describe how to skip this step and instead work directly with the coarse moduli space. This is joint work with Steve Huang and Jun Bo Lau.

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

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

Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.

A hash table is a data structure that efficiently stores objects in a way that allows for fast access, insertion, and deletion. Cuckoo hashing is a method of creating and maintaining hash tables that has been widely used in both theory and practice. Random walk d-ary cuckoo hashing is a natural generalization of cuckoo hashing with low space overhead, guaranteed fast access, and fast in practice insertion time. In this presentation, I will explain this algorithm and my work proving a bound on its insertion time. More precisely, we show that for four or more hash functions and load factors up to the optimal threshold, the expectation of the random walk insertion time is O(1), that is, a constant depending only on the number of hash functions and the load factor, not the size of the table.

The study of cuckoo hashing is directly connected to the study of Erdős-Rényi random hypergraphs, and I will emphasize these connections during my presentation. This presentation is based on https://arxiv.org/abs/2401.14394, joint work with Alan Frieze.

Berkovich spaces give a formalism for constructing spaces of valuations on varieties over nonarchimedean fields. As such they encode a great deal of information from birational geometry. The most notable invariant is the essential skeleton, a subset of the Berkovich space corresponding to the valuations monomial on strata of a dlt minimal model of 𝑋. Inspired by Mori's conjecture in birational geometry, we conjecture that the essential skeleton is the complement of the images of all transcendental disks, which are analytic objects analogous to families of rational curves. I will present some progress on this conjecture in joint work with Jiachang Xu and Muyuan Zhang.

Lascoux polynomials simultaneously generalize two famous families of polynomials arising from geometry and representation theory: They are non-symmetric analogs of Grassmannian stable Grothendieck polynomials, which represent Schubert classes in the connective K-theory of Grassmannians. Additionally, they serve as non-homogeneous analogs of key polynomials, the characters of Demazure modules. Both of these families have classical combinatorial formulas involving tableaux. We further generalize several of these formulas by establishing two combinatorial formulas for Lascoux polynomials.

A nuclear C*-algebra is quasi-diagonal if there is a unital embedding into the ultraproduct of matrices. The concept is essential in the Elliott Classification Program. The existence of such embeddings is shown by the Tikusis-White-Winter theorem (2015) for any stably finite, simple, separable, nuclear, unital C*-algebra satisfying the UCT. Thus an interesting question is to understand how unique are such embeddings given the existence.

We are able to answer this question by obtaining the corresponding KK-uniqueness theorem and applying the abstract classification approach. The major difficulty appears since the codomain of the map does not (separably) tensorially absorb the Jiang-Su algebra, and thus not covered by previous classification techniques. We will explain how to tackle the difficulty by looking at K-theoretical properties of a special C*-algebra, the Paschke Dual algebra, associated to the map.

This talk introduces a unified proximal gradient method with extrapolation (UPG-E) to solve a class of nonconvex and nonsmooth composite optimization. UPG-E provides unified treatment to both convex and nonconvex problems, and adaptively estimates the nonconvexity modulus of the possibly nonconvex component function in the objective function. In this talk, we will discuss the global convergence and convergence rate of UPG-E for solving both convex and nonconvex composite optimization. We will also show the promising numerical performance of UPG-E compared with other well-established proximal gradient methods in the literature.

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

In his paper on free subgroups of linear groups, Tits proved his famous alternative: a linear group is either virtually solvable or contains a free subgroup. Since then, Tits’ work has been generalized and applied in many different ways.

One remaining open question in this field was the one asked by de la Harpe: let $G$ be a semisimple Lie group without compact factors and with trivial center and let $\Gamma$ be a Zariski-dense subgroup of $G$. Given a prescribed finite subset $F$ of $G$, is it always possible to find an element $\gamma \in \Gamma$ such that for any $h \in F$, the subgroup generated by $h$ and $\gamma$ is freely generated (in that case, we say $h$ and $\gamma$ are ping-pong partners).

In the talk, we will discuss a variant of the question of de la Harpe, where $F$ is a finite set of finite subgroups $H_i$ of $G$. Using careful refinements of the main steps of Tits’ proof of the alternative (which we will recall), we give sufficient conditions for the existence of ping-pong partners for the $H_i$ in any Zariski-dense subgroup $\Gamma$.

We will also show that these conditions are satisfied for products of copies of $\mathrm{SL}_n$ over division $\mathbb{R}$-algebras.

The existence of such free products has applications in the theory of integral group rings of finite groups, which will be briefly mentioned.

Joint work with G. Janssens and D. Temmerman.

In this talk we will begin by discussing some of the main concepts of machine learning at a high level. Then we will take a closer look at neural networks. And finally we will discuss an interesting application of free probability to deep learning observed by Pennington-Schoenholz-Ganguli.  The goal of this talk is to introduce mathematicians without much background in statistics to the fundamentals of machine learning while highlighting an interesting application of noncommutative probability.  Graduate students are highly encouraged to attend.

For critical percolation on the 2D triangular lattice, consider the probability that three points lie in the same cluster. The Delfino-Viti conjecture predicts that in the fine mesh limit, under suitable normalization, this probability converges to the imaginary DOZZ formula from conformal field theory. We prove the Delfino-Viti conjecture, and more generally, obtain the cluster connectivity three-point function of the conformal loop ensemble. Our arguments depend on the coupling between Liouville quantum gravity and the conformal loop ensemble. 
Based on joint work with Gefei Cai, Xin Sun, and Baojun Wu.

Evolution has predisposed developing organisms to rely heavily on the repeated application of select biomechanical motifs to achieve robust growth. Within vertebrates, one prevalent architecture is the dermal condensate: these dense bundles of mesenchymal cells that as organizing centers for developing organs such as teeth, hair follicles, glands, and limbs. Though long recognized as significant, the study of these proto-organs has historically focused on biochemical regulation. This talk, however, will emphasize the mechanical role of these structures, illustrating how they can be viewed as self-organized actuators that drive tissue deformation. This discussion will be split into two parts: first, we will highlight some recently published results based on early feather morphogenesis, demonstrating how dermal condensates coordinate tissue-scale flows by localizing active contractile stresses. Second, we will use a model of the mammalian gliding membrane to examine the formation of condensates more closely, with an eye on clarifying the role of cell-matrix interactions and better understanding where condensate mechanics fits among competing hypotheses of limb growth.

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

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

We show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weakly exact von Neumann algebras.

Erdős and Pósa proved in 1965 that every graph contains either $k$ vertex-disjoint cycles or a set of at most $O(k \log k)$ vertices intersecting every cycle. Such an approximate duality does not hold for odd cycles due to certain projective-planar grids, as pointed out by Lovász and Schrijver, and Reed showed in 1999 that these grids are the only obstructions to this duality. In this talk, we generalize these results by characterizing the obstructions in group-labelled graphs. Specializing to the group $\mathbb{Z}/m\mathbb{Z}$ gives a characterization of when cycles of length $\ell \bmod m$ satisfy this approximate duality, resolving a problem of Dejter and Neumann-Lara from 1988. We discuss other applications and analogous results for $A$-paths.

Based on joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum.

Many statistical learning problems, where one aims to recover an underlying low-dimensional signal, are based on optimization, e.g., the linear programming approach for recovering a sparse vector. Existing work often either overlooked the high computational cost in solving the optimization problem, or required case-specific algorithm and analysis -- especially for nonconvex problems. This talk addresses the above two issues from a unified perspective of conditioning. In particular, we show that once the sample size exceeds the intrinsic dimension of the signal, (1) a broad range of convex problems and a set of key nonsmooth nonconvex problems are well-conditioned, (2) well-conditioning, in turn, inspires new algorithms and ensures the efficiency of off-the-shelf optimization methods.

A half-translation surface is a collection of polygons on the plane with side identifications by translations or half-turns in such a way that the resulting topological surface is closed and orientable. We also assume that the total Euclidean area of the polygons is finite. Two half-translations are equivalent if a sequence of cut-and-paste operations takes one to the other. From the view of complex geometry, an equivalent definition is a Riemann surface endowed with a meromorphic quadratic differential with poles of order at most one.

A stratum of half-translation surfaces consists of those with prescribed cone angles at the vertices of the polygons. Strata are, in general, not connected. A natural flow, the diagonal or Teichmüller flow, acts on stratum components.

In this talk, we investigate some topological properties of stratum components. We show that the (orbifold) fundamental group of such a component is “detected” by the diagonal flow in that every loop is homotopic to a concatenation of closed geodesics (coned to a base-point). Using this result, we show that the Lyapunov spectrum of the homological action of the diagonal flow is simple, thus establishing the Kontsevich–Zorich conjecture for quadratic differentials.

This is a joint work with Mark Bell, Vincent Delecroix, Vaibhav Gadre, and Saul Schleimer.

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

A graph $X$ is defined inductively to be $(a_0, . . . , a_{n−1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius 1 around $v$ is an $(a_1, . . . , a_{n−1})$-regular graph. A family $F$ of graphs is said to be an expander family if there is a uniform lower bound on the Cheeger constant of all the graphs in $F$. 

After briefly (re)introducing Coxeter groups and their geometries, we will describe how they can be used to construct very regular polytopes, which in turn can yield highly regular graphs. We will then use the super-approximation machinery, whenever the Coxeter group is hyperbolic, to obtain the expansion of these family of graphs. As a result, we obtain interesting infinite families of highly regular expander graphs, some of which are related to the exceptional groups. 

The talk is based on work joint with Conder, Lubotzky and Schillewaert.

A game of Nim is traditionally played with heaps of objects called counters. In this game, two players take turns choosing a heap and then removing any positive number of counters from that heap. Play continues until the final counter is removed, at which point the player responsible for this is declared the winner. In search of a general winning strategy for Nim, we encounter and prove several core principles of combinatorial game theory.

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

A coarse structure is a way of talking about "large-scale" properties.  It is encoded in a family of relations that often, but not always, come from a metric.  A coarse structure naturally gives rise to Hilbert space operators that in turn generate a so-called uniform Roe algebra.

In work with Bruno Braga and Joe Eisner, we use ideas of Weaver to construct "quantum" coarse structures and uniform Roe algebras in which the underlying set is replaced with an arbitrary represented von Neumann algebra.  The general theory immediately applies to quantum metrics (suitably defined), but it is much richer.  We explain another source based on measure instead of metric, leading to the new, large, and easy-to-understand class of support expansion C*-algebras.

I will present the big picture: where uniform Roe algebras come from, how Weaver's framework facilitates our definitions.  I will focus on a few illustrative examples and will not presume familiarity with coarse structures or von Neumann algebras.

This research project studies the sparse Moment-SOS hierarchy of relaxations for solving sparse polynomial optimization problems. We show that this sparse hierarchy is tight if and only if the objective can be written as a sum of sparse nonnegative polynomials, each of which belongs to the sum of the ideal and quadratic module generated by the corresponding sparse constraints. Based on this characterization, we give several sufficient conditions for the sparse Moment-SOS hierarchy to be tight. In particular, we show that this sparse hierarchy is tight under some assumptions such as convexity, optimality conditions or finiteness of constraining sets.

The chromatic number of a very dense random graph $G(n,p)$, with $p \ge 1 - n^{-c}$ for some constant $c > 0$, was first studied by Surya and Warnke, who  conjectured that the typical deviation of $\chi(G(n,p))$ from its mean is of order $\sqrt{\mu_r}$, where $\mu_r$ is the expected number of independent sets of size $r$, and $r$ is maximal such that $\mu_r > 1$, except when $\mu_r = O(\log n)$. They moreover proved their conjecture in the case $n^{-2} \ll 1 - p = O(n^{-1})$.

In this talk, we study $\chi(G(n,p))$ in the range $n^{-1}\log n \ll 1 - p \ll n^{-2/3}$, that is, when the largest independent set of $G(n,p)$ is typically of size 3. We prove in this case that $\chi(G(n,p))$ is concentrated on some interval of length $O(\sqrt{\mu_3})$, and for sufficiently `smooth' functions $p = p(n)$, that there are infinitely many values of $n$ such that $\chi(G(n,p))$ is not concentrated on any interval of size $o(\sqrt{\mu_3})$. We also show that $\chi(G(n,p))$ satisfies a central limit theorem in the range $n^{-1} \log n \ll 1 - p \ll n^{-7/9}$.

This talk is based on arXiv:2405.13914

The purpose of dimension reduction methods for data visualization is to project high dimensional data to 2 or 3 dimensions so that humans can understand some of its structure. In this talk, we will give an overview of some of the most popular and powerful methods in this active area. We will then focus on two algorithms: Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP). Here, we will present new rigorous results that establish an equilibrium distribution for these methods when the number of data points diverge in the presence of pure noise or with a planted signal.

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

The Weighted Complementarity Problem (WCP) significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this talk, we will introduce a smoothing Newton algorithm by using a one-parametric class of smoothing functions. We will discuss the global convergence and local superlinear or quadratic convergence of this algorithm without nonsingularity assumption on the Jacobian. Some numerical results of this algorithm will be also discussed.

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

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

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

Studying the minimal model program with scaling on the moduli space of genus g curves and interpreting the steps in a modular way is known as the Hassett-Keel program.  The first few steps are well-understood yet the program remains quite incomplete in general.  We complete the Hassett-Keel program in genus 4 using wall-crossing.  This is joint work with Kenneth Ascher, Yuchen Liu, and Xiaowei Wang. 

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

A sequence of tuples of random matrices $(X_1^N, \dots, X_d^N)$ converges strongly to a tuple of operators $ (x_1, . . . , x_d)$ in a $C^*$-algebra if for any noncommutative polynomial $P$, $\|P(X_1^N, \dots, X_d^N)\|$ converges (say, in probability) to $\|P(x_1, . . . , x_d)\|$ as $N$ goes to infinity. This phenomenon plays a central role in breakthrough results  in operator algebras, as well as in the construction of: expander graphs, hyperbolic surfaces with nearly optimal spectral gaps and minimal surfaces. Given its far-reaching implications, it is no surprise that strong convergence is notoriously difficult to prove and has generally required delicate problem-specific methods.
In this talk I will discuss recent joint work with Chi-Fang Chen, Joel Tropp and Ramon van Handel, where we introduce a new flexible and elementary technique for proving strong convergence. This technique can be applied to random matrix models that have a lot of symmetry, for example, random permutation matrices, classical Gaussian and unitary matrices (i.e. GOE, GUE, GSE, $O(N)$, $U(N)$, and $Sp(N)$), and some others, constructed via representations of the symmetric and unitary group, for which other methods seem to break. In all of these models, the technique yields the sharpest quantitative results known so far. 

A real number $\sigma$ is called a \textit{linear threshold} of a bipartite graph $H$ if every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $|V| \gtrsim |U|^\sigma$ and without a copy of $H$ must have a linear number of edges $|E| \lesssim |V|$. We prove that $\sigma_s = 2 - 1/s$ is a linear threshold of the \textit{complete bipartite subdivision} graph $K_{s,t}'$. Moreover, we show that any $\sigma < \sigma_s$ is not a linear threshold of $K_{s,t}'$ for sufficiently large $t$ (depending on $s$ and $\sigma$). Some applications of our result in incidence geometry are discussed.

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

[pre-talk at 3:00pm]

How do linear projections affect the dimensions of subsets of Cartesian space? Marstrand's result from 1954 demonstrates that each Borel set behaves generically under projections onto almost every linear subspace. Recent developments in Fourier analysis have permitted these results to be expanded significantly to apply to much more restricted families of projections, and even effectively bound the dimension of the set of exceptional projections.

We discuss the special case of projecting subsets of three dimensions onto lines, working over non-Archimedean local fields of characteristic not equal to 2. We will briefly discuss the relevancy to polynomial effective equidistribution in homogeneous dynamics. The main technical input will be a refined decoupling theorem for non-Archimedean local fields. This work mirrors the work in the real setting by the authors Gan, Guo, Guth, Harris, Iosevich, Maldague, Ou, and Wang, and builds on previous work by the speaker and Zuo Lin.

Phase separation underpins a wide range of phenomena from the formation of membraneless intracellular compartments known as biomolecular condensates to the collective behavior of bacteria.  Unlike simpler systems like oil and water, however, phase separation in these systems is often complicated by mechanical interactions, nonequilibrium activities, and heterogeneity.

In my talk, I will delve into how I navigated these complexities to uncover new insights into three distinct systems. I will first address biomolecular condensates within chromatin-packed cell nuclei, highlighting how the competition between elastic and capillary forces crucially shapes the structure and mechanics of the chromatin networks. Next, I will share my discovery of emergent phenomena in active systems such as bacteria and active colloids, due to the interplay between movements along chemical gradients and motility-induced phase separation. Lastly, I'll discuss how image-based learning uncovered the physics of phase-separating particles driven out of equilibrium by electrochemical reactions, revealing their reaction kinetics, free energy landscape, and heterogeneity.

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

Let $X$ be a normal variety with a projective contraction $f:X\to S$. Assume $U$ is an open subset of $S$ and $L_U$ is a Cartier divisor on $X_U:=X\times_S U$ such that $L_U$ is numerical trivial over $U$.

We will discuss about when it is possible and how to extend $L_U$ to a global Cartier divisor $L$ on $X$ such that $L\equiv_f 0$.

Let $D$ be a relatively compact $C^2$ domain in a complex manifold $X$ of dimension $n$. Assume that $H^1(D,\Theta)$ vanishes, where $\Theta$ is the sheaf of germs of holomorphic tangent fields of $D$. Suppose that the Levi-form of the boundary $b D$ has at least $3$ negative eigenvalues or at least $n-1$ positive eigenvalues pointwise. We will show that if a formally integrable  almost complex structure $H$ of the Holder class $C^r$ with $r>5/2$ on $D$ is sufficiently close to the complex structure on $D$, there is a embedding $F$ from $D$ into $X$ that transforms the almost complex structure into the complex structure on $F(D)$, where  $F $ has class $C^s$ for all $s<r+1/2$. This result was due to R. Hamilton in the 1970s when both $b D$ and $H$ are of class $C^\infty$.

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

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

Real-world networks frequently exhibit a clustering phenomenon where the friends of friends are likely to be friends. We show that the clustering effect is highly correlated with Ricci curvatures of a graph for random clustering graphs with given degree distributions. In particular, we show that for a random clustering graph with certain power-law degree distributions the Ricci curvature (in the sense of Lin, Lu, and Yau) is concentrated around the clustering coefficient.

Based on joint work with Fan Chung (UCSD), Michael Rawson (PNNL), Zhaiming Shen (University of Georgia), and Murong Xu (University of Scranton).

Results on the regularity of operators on -spaces are often proved by means of interpolation operators applied to estimates at the endpoints. A classical example is that of the Hibert transform on the real line, the -behavior of which can be deduced from a weak  type (1,1) estimate and the Marcinkiewicz interpolation theorem. Attempts to apply this idea to the Bergman projection on certain domains such as the Hartogs triangle  in  lead to some unexpected endpoint behavior. In particular, we show that for the Hartogs triangle, at the left endpoint  of the interval of -boundedness, the Bergman projection  on this domain is of restricted strong type  in the sense of Stein-Weiss, that is, for a characteristic function  of a measurable subset , we have

for a constant  independent of . This now determines the -behavior of the Bergman projection via classical interpolation results. We discuss several generalizations of this result to other domains. This is ongoing joint work with Zhenghui Huo of Duke Kunshan University, China.

Crystalline deformation rings play an important role in Kisin's proof of the Fontaine-Mazur conjecture for GL2 in most cases. One crucial step in the proof is to prove the Breuil-Mezard conjecture on the Hilbert-Samuel multiplicity of the special fiber of the crystalline deformation ring. In pursuit of formulating a horizontal version of the Breuil-Mezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. Our approach, based on reverse-engineering the Taylor-Wiles-Kisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally.

[pre-talk at 3:00PM]

We show that the random degree constrained process (a time-evolving random graph model with degree constraints) has a local weak limit, provided that the underlying host graphs are high degree almost regular. We, moreover, identify the limit object as a multi-type branching process, by combining coupling arguments with the analysis of a certain recursive tree process. Using a spectral characterization, we also give an asymptotic expansion of the critical time when the giant component emerges in the so-called random $d$-process, resolving a problem of Warnke and Wormald for large $d$.

Based on joint work with Balázs Ráth and Lutz Warnke; see arXiv:2409.11747

Entire critical points of the abelian Higgs functional are known to blow down to generalized minimal submanifolds (of codimension 2). In this talk we prove an Allard type large-scale regularity result for the zero set of solutions. In the "multiplicity one" regime, we show the uniqueness of blow-downs and classify entire solutions in low dimensions and minimizers in all dimensions; thus obtaining an analogue of Savin's theorem in codimension two. This is based on a joint work with Guido de Philippis and Alessandro Pigati.

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

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

In this talk I will show you how to maximize your a-mean-ability. More precisely, I will survey some results about maximal amenable subgroups and subalgebras and share a new result which states that it is possible for the space of maximal amenable extensions to be any k-simplex (such as a triangle!). While this talk will technically be about operator algebras, it will be accessible to anyone who knows a bit of group theory (semi-direct products, free products, and the left regular representation).

I will discuss recent results describing the Chow rings and the tautological classes of the moduli space of quasi-polarized K3 surfaces of degree 2. This is based on joint work with Samir Canning and Rahul Pandharipande.

SDSU Climate Informatics Lab has developed a suite of computer code and Apps for visualizing and delivering real climate data for the general public, such as school classrooms. This presentation will specifically demonstrate the following tools:

1.  4-dimensional visual delivery (4DVD) of big climate data: www.4dvd.org.

2. Statistics, machine learning, and data visualization for climate science with R and Python: www.climatestatistics.org

3. Climate mathematics with R and Python:  www.climatemathematics.org

4. 4DVD Rural Heat Island for a California Climate Action project: www.4dvdrhi.sdsu.edu

We will also discuss our proprietary database optimization algorithms for fast queries. Using cutting-edge database technologies and 3D video games, we will outline our product development for the NSF program of AI Institutes and NOAA National Centers for Environmental Information.

Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings. On the other hand, enveloping algebras of infinite-dimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will summarize what is known about the noetherianity of enveloping algebras, with a focus on Lie algebras of derivations of associative algebras.

Using amalgamated free products of lamplighter groups, we construct masas that have exactly k maximal amenable extensions which are factors. Moreover, in this setting we are able to identify the k-simplex with the space of all maximal amenable extensions. The proof uses the concepts of simultaneous relative asymptotic orthogonality, an extension of relative asymptotic orthogonality, and the uniform flattening strategy, which was introduced by the authors in a previous paper to compute lower bounds on sequential commutation path lengths. This work is joint with Srivatsav Kunnawalkam Elayavalli.

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

The canonical van der Waerden theorem states that, for large enough $n$, any colouring of $[n]$ gives rise to monochromatic or rainbow $k$-APs. In joint work with Alvarado, Kohayakawa, Morris and Mota, we study sparse random versions of this result. More concretely, we determine the threshold at which the binomial random set $[n]_p$ inherits the canonical van der Waerden properties of $[n]$, using the container method.

We will discuss an inductive approach to determining the asymptotic number of G-extensions of a number field with bounded discriminant, and outline the proof of Malle's conjecture in numerous new cases. This talk will include discussions of several examples demonstrating the method.

[pre-talk at 3:00PM]

We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics. This is a joint work with Subhajit Jana.

In Bernoulli percolation, events such as "two vertices are connected by an open path" naturally emerge. In this talk, I will explore the dependencies between these events for various vertex pairs and derive key FKG-type inequalities governing their probabilities and explain the relevance of these inequalities to the recent disproof of the Bunkbed Conjecture.

Self-expanders are a special class of solutions to the mean curvature flow, in which a later time slice is a scale-up copy of an earlier one. They are also critical points for a suitable weighted area functional. Self-expanders model the asymptotic behavior of a mean curvature flow when it emerges from a cone singularity. The nonuniqueness of self-expanders presents challenges in the study of cone-like singularities in the flow. In this talk, I will discuss some recent development on a variational theory for self-expanders and an application to the question on lower density bounds for minimal cones.

In the 1960s Thoma developed a theory of characters which generalizes the classical Fourier/Pontryagin theory of abelian groups, and at the same time Frobenius' theory on finite (and compact) groups.

After presenting the general theory, I will focus on arithmetic groups, or similarly, lattices in (semi)simple Lie groups, and tell about my work with Levit and Slutsky regarding the geometry/topology of the space of characters of such groups. Our main result is that for lattices in higher rank simple Lie groups (e.g for the group SL3(Z)), any sequence of distinct characters must converge pointwise to the dirac character at the identity.  This implies character bounds of finite groups of Lie type (e.g SL3(Fp)).

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

Not the factorizations you're thinking of! A 1-factorization of a graph is a partition of its edges into perfect matchings. In this talk I hope to share some of the many questions about 1-factorizations that I find interesting, explain how I came to care about a rather obscure fact, and prove it. Along the way we will draw some pretty pictures, of course. This talk should be totally accessible to anyone who has ever heard of graphs, and is based on collaboration with Michael Orrison and Rohan Chauhan.

We propose a scalable preconditioned primal-dual gradient algorithm for solving partial differential equations (PDEs). We multiply the PDE with a dual function to obtain an inf-sup problem whose loss function involves lower-order differential operators. The Primal-Dual hybrid gradient (PDHG) algorithm is leveraged for this saddle point problem. By introducing suitable precondition operators to the proximal steps in the PDHG algorithm, we obtain an alternative natural gradient ascent-descent optimization scheme for updating the primal and the adversarial neural network parameters. We apply the Krylov subspace method to evaluate the natural gradients efficiently. A posterior convergence analysis is established for the time-continuous version of the proposed method.

The algorithm is tested on various types of linear and nonlinear PDEs with dimensions ranging from 1 to 50. We compare the performance of the proposed method with several commonly used deep learning algorithms, such as physics-informed neural networks (PINNs), the DeepRitz method, and weak adversarial networks (WANs) using the Adam or L-BFGS optimizers. The numerical results suggest that the proposed method performs efficiently and robustly and converges more stably.

Consider a countable discrete group Γ and its subgroup space-Sub(Γ), the collection of all subgroups of Γ. Sub(Γ) is a compact metrizable space with respect to the Chabauty topology (the topology induced from the product topology on {0, 1} Γ). The normal subgroups of Γ are the fixed points of (Sub(Γ), Γ). Furthermore, the Γ-invariant probability measures of this dynamical system are known as invariant random subgroups (IRSs). Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. They generalize the notion of normal subgroups. Among many other remarkable results, they strengthen the well-known Margulis’s normal subgroup Theorem. More recently, in a joint work with Hartman and Oppelmayer, we introduced the notion of Invariant Random Algebra (IRA), an invariant probability measure on the collection of sub algebras of L(Γ). Motivated by the works of Glasner and Lederle, in ongoing joint work with Yair Glasner, Yair Hartman, and Yongle Jiang, we introduce the notion of Boomerang subalgebras in the context of L(Γ). In this talk, we shall show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group comes from a Boomerang subgroup. We shall also discuss its connection to understanding IRAs in such groups.

One definition of key polynomials is as the weight generating functions of key tableaux. Assaf and Schilling introduced a crystal structure on key tableaux and related it to Morse–Schilling crystals on reduced factorizations for permutations via weak Edelman–Greene insertion. In this talk, we consider generalizations of both crystals depending on a flag. We extend weak EG insertion to a bijection between our flagged objects and show that the recording tableau gives a crystal isomorphism. As an application, we show that flagged key tableaux have a natural Demazure crystal structure, whose characters recover Reiner and Shimozono’s flagged key polynomials.

The main objects of study in algebraic geometry are varieties, or solution sets to polynomial equations.  A fundamental goal of the subject is the classification of algebraic varieties.  I will survey some of the steps of classification, including the minimal model program and construction of moduli spaces.  This talk is intended to be accessible to all graduate students, regardless of background in algebraic geometry. 

On the symmetric space for $G_2$, there exist various submanifolds $G_D$ corresponding to the stabilizer of a norm $D$ vector. We show that when a suitable automorphic form is integrated against the $G_D$, the resulting numbers assemble to give a half-integral weight classical modular form. Although this is already implied by a result of Kudla-Millson, we give a simpler proof that avoids the complications in their paper.

Classical results in asymptotic statistics show that the Fisher information matrix controls the difficulty of estimating a statistical model from observed data. In this work, we introduce a companion measure of robustness of an estimation problem: the radius of statistical efficiency (RSE) is the size of the smallest perturbation to the problem data that renders the Fisher information matrix singular. We compute RSE up to numerical constants for a variety of test bed problems, including principal component analysis, generalized linear models, phase retrieval, bilinear sensing, and matrix completion. In all cases, the RSE quantifies the compatibility between the covariance of the population data and the latent model parameter. Interestingly, we observe a precise reciprocal relationship between RSE and the intrinsic complexity/sensitivity of the problem instance, paralleling the classical Eckart–Young theorem in numerical analysis.

Let $G$ be a discrete group. A $G$-space $X$ is called a $G$-boundary if the action $G \curvearrowright X$ is minimal and strongly proximal. In this talk, we shall prove a continuous version of the well-studied Intermediate Factor Theorem in the context of measurable dynamics. When a product group $G = \Gamma_1 \times \Gamma_2$ acts (by a product action) on the product of corresponding $\Gamma_i$-boundaries $\partial \Gamma_i$, we show that every intermediate factor $$X \times (\partial \Gamma_1 \times \partial \Gamma_2) \rightarrow Y \rightarrow X$$ is a product (under some additional assumptions on $X$). We shall also compare it to its measurable analog proved by Bader-Shalom. This is a recent joint work with Yongle Jiang.

Asymptotically conical shrinking Ricci solitons are an important class of potential singularity models for the Ricci flow in dimension four and higher, and give rise to solutions which, near infinity, flow smoothly into a cone at the singular time. We will present some uniqueness and nonexistence results which can be inferred from this latter fact using a unique continuation result valid at the singular time, and discuss their applications to the classification problem for asymptotically conical shrinkers.

Inverse Reinforcement Learning (IRL) is a compelling technique for revealing the rationale underlying the behavior of autonomous agents.  IRL seeks to estimate the unknown reward function of a Markov decision process (MDP) from observed agent trajectories. However, IRL needs a transition function, and most algorithms assume it is known or can be estimated in advance from data. It therefore becomes even more challenging when such transition dynamics is not known a-priori, since it enters the estimation of the policy in addition to determining the system's evolution. When the dynamics of these agents in the state-action space is described by stochastic differential equations (SDE) in It\^{o} calculus, these transitions can be inferred from the mean-field theory described by the Fokker-Planck (FP) equation. We conjecture there exists an isomorphism between the time-discrete FP and MDP that extends beyond the minimization of free energy (in FP) and maximization of the reward (in MDP). We identify specific manifestations of this isomorphism and use them to create a novel physics-aware IRL algorithm, FP-IRL, which can simultaneously infer the transition and reward functions using only observed trajectories.  We employ variational system identification to infer the potential function in FP, which consequently allows the evaluation of reward, transition, and policy by leveraging the conjecture. We demonstrate the effectiveness of FP-IRL by applying it to a synthetic benchmark and a biological problem of cancer cell dynamics, where the transition function is inaccessible. This is joint work with Chengyang Huang, Sid Srivastava, Kenneth Ho, Kathy Luker, Gary Luker and Xun Huan.

We consider properly immersed two-sided stable minimal hypersurfaces of dimension n. We illustrate the validity of curvature estimates for n \leq 6 (and associated Bernstein-type properties with an extrinsic area growth assumption). For n \geq 7 we illustrate sheeting results around "flat points". The proof relies on PDE analysis. The results extend respectively the Schoen-Simon-Yau estimates (obtained for n \leq 5) and the Schoen-Simon sheeting theorem (valid for embeddings).

It is a well-known theorem in combinatorics that everything is counted by the Catalan numbers. These numbers are ubiquitous in mathematics, often appearing where you least expect them — such as in the HOMFLYPT polynomial of torus knots. In this talk, we’ll take a journey into algebraic combinatorics by pursuing the question: why do the torus knots know about Catalan numbers? Along the way, we’ll encounter braid groups, Hecke algebras, complex reflection groups, and more! This talk is based on a paper by Pavel Galashin, Thomas Lam, Minh-Tam Quang Trinh, and Nathan Williams.

We describe a new and elementary proof of the fact that many groups of piecewise-projective transformation of the line are non-amenable by constructing an explicit action without fixed points. One the one hand, such groups provide explicit counter-examples to the Day-von Neumann problem. On the other hand, they illustrate that we can distinguish many "layers" of relative non-amenability between nested subgroups.

The "boundaries" of the title refer to group action on measure spaces that enjoy the amenability property discovered by Furstenberg, Margulis and Zimmer. Over the last half-century these boundaries have been invaluable for group theory, rigidity, and more recently "bounded cohomology" -- which is the same as the cohomology of convolution algebras.

We will see that many familiar groups admit surprisingly strong ergodicity properties for boundaries. This applies to lamplighters, Thompson groups and many transformation groups. As a consequence, we determine the bounded cohomology of some of these groups.

Regularity lemmas are a cornerstone of combinatorics, with many applications in math and CS.  The most famous one is Szemeredi's regularity lemma. It shows that any graph can be partitioned into a "few" parts that mostly "look random". However, there is a caveat - "few" is really a huge number, a tower of exponentials in the error parameter.

Motivated by this, Frieze and Kannan designed a "weak" regularity lemma, sufficient for some applications, where the number of parts is much smaller, only exponential in the error parameter. In this work, we develop an even weaker regularity lemma, called "spread regularity", where the number of parts is even smaller - quasi-polynomial in the error parameter.

I will describe our new notion of regularity and some applications:
1. new lower bound technique in communication complexity, where players partially share information
2. new combinatorial algorithm for boolean matrix multiplication
3. improved bounds for variants of the corners problem in additive combinatorics

Based on joint works with Amir Abboud, Nick Fischer, Michael Jaber, Zander Kelley, Raghu Meka and Anthony Ostuni

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus's (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(C)×SL2(C)→SO4(C), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.

The Birch and Swinnerton-Dyer conjecture relates the leading coefficient of the L-function of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. Beilinson’s conjectures generalize the BSD conjecture to formulas for values of motivic L-functions at non-critical points. In this talk, I will relate motivic cohomology classes, with non-trivial coefficients, of Picard modular surfaces to a non-critical value of the motivic L-function of certain automorphic representations of the group GU(2,1).

We discuss work in progress regarding the following assertion: Let $G$ be a simple higher rank Lie group, then any closed manifold $M$, admitting a smooth locally free action of $G$, with codimension one orbits, is finitely and equivariantly covered by $G/\Gamma \times S^1$, for some cocompact lattice $\Gamma$ of $G$, where $G$ acts by left translations on the first factor, and trivially on $S^1$. This result is in the spirit of the Zimmer program. We will focus on the case $G = \mathrm{SL}(3,\mathbb{R})$ for the talk.

Non-rigid point cloud registration plays a crucial role in various fields, including robotics, neuroscience, computer graphics, and medical imaging. This process involves determining spatial relationships between different sets of points, typically within a 3D space. In real-world scenarios, complexities arise from non-rigid movements and partial visibility, such as occlusions or sensor noise, making non-rigid registration a challenging problem. Classic non-rigid registration methods are often computationally demanding, suffer from unstable performance, and, importantly, have limited theoretical guarantees.

The talk will focus primarily on a new way to understand non-rigid alignment and manifold learning of point clouds in Euclidean space using Whitney Extensions machinery developed by the author and his collaborators over the last few years. We will possibly  explore relationships of our work to topological data analysis, optimal transport, neural networks and neuroscience.

One method to generate random permutations involves using Gaussian elimination with partial pivoting (GEPP) on a random matrix A and storing the permutation matrix factor P from the resulting GEPP factorization PA=LU. We are interested in exploring properties of random butterfly permutations, which are generated using GEPP on specific random butterfly matrices. We address the questions of the longest increasing subsequence (LIS) and number of cycles for particular uniform butterfly permutations, with full distributional descriptions and limit theorems for simple butterfly permutations. We also establish scaling limit results and limit theorems for nonsimple butterfly permutations, which include certain p-Sylow subgroups of the symmetric group of N=pⁿ elements. For the LIS, we establish power-law bounds on the first moment N^αₚ ≤ E(LIS) ≤ N^βₚ where 1/2 < αₚ < βₚ < 1 and αₚ = 1 - oₚ(1), showing distinction from the typical O(N¹ᐟ²) expected LIS frequently encountered in the study of random permutations (e.g., uniform permutations, colored permutations, recent wreath product permutations studied by Diaconis). For the number of cycles scaled by (2-1/p)ⁿ, we establish a full CLT to a new limiting distribution depending on p with positive support we introduce that is uniquely determined by its positive moments that satisfy explicit recursive formulas; this thus determines a number of cycles CLT for any uniform p-Sylow subgroups of Sₚn. This work is joint with Chenyang Zhong, and continues from a UCSD probability seminar talk I gave last May to now include novel results and proof techniques from our recent preprint arXiv:2410.20952.

In this seminar I aim to show you what happens to a theoretical physicist, who gets to work in Theoretical Immunology and Virology. During the journey the physicist gets to learn mathematics she did not know at the time, and of course, explores the biological universe at different scales.

Since we only have a limited amount of time, I would like to introduce you to a problem that I have recently become interested in, and which has received NIH funding. Co-infection of a single host by different pathogens is ubiquitous in nature. We consider a population of hosts (e.g., small or large vertebrates) and a population of ticks, both of them susceptible to  infection with two different strains of a given virus.  We note that for the purposes of our models, we have Crimean-Congo hemorrhagic fever virus (a segmented Bunyavirus) in mind, as the application system.

First, we focus on the dynamics of a single infection, proposing  a deterministic  model to understand the role of co-feeding in the transmission of the virus.  We then compute the basic reproduction number by making use of the next generation matrix approach.  When considering co-infection by two distinct strains (one resident and one invasive), we make use of differential equations to model the dynamics of susceptible, infected and co-infected species, and we compute the invasion reproduction number of the invasive strain.  I discuss some problems with the calculation, and the solution proposed by Samuel Alizon and Marc Lipsitch. I conclude with a perspective on how the co-infection model can be applied to HIV, and plans for future work and work in progress for tick-borne pathogens. To end the talk, I would like to showcase a number of problems in immunology I have worked on, and which have required, for instance, the theory of stochastic processes, probability generating functions, or the use of a Gröbner basis.

Hopf algebras provide a rich framework for studying symmetry in noncommutative settings, generalizing both group algebras and enveloping algebras of Lie algebras. If one has any hope of understanding the structure of noetherian Hopf algebras, one should first understand when group algebras and enveloping algebras are noetherian. These two famously difficult problems are open to this day. In this talk, I will give a survey of what is known about the noetherianity of group algebras and enveloping algebras, mainly focusing on the Lie algebra side.

Elliptic Curves have been one of the many objects of studying due to their rich structure. We will discuss the group law on elliptic curves and the staggering fact that this group is abelian and finitely generated. Knowing the structure of finitely generated abelian groups, we ask what the torsion piece of the group can be. Mazur’s theorem will give us our answer! In discussion we will encounter modular curves, rational points on a variety of objects and attempt to get a sense of arithmetic geometry. This talk should be accessible to anyone but knowledge of the classification of finitely generated abelian groups and a little complex analysis may be useful in some sections.

We will discuss a natural perspective on stable reduction that extends Deligne--Mumford's stable reduction for curves to higher dimensions. From this perspective, we will outline a new proof of the Hacon--Kovács theorem on the properness of the moduli stack $\overline{\mathscr{M}}_{2,v,k}$ of stable surfaces of volume $v$ defined over $k=\overline{k}$, provided that $\operatorname{char}k>C(v)$, a constant depending only on $v$.

We will survey the class of linear groups, and identify three methods of constructing linear groups: algebraic, geometric, and arithmetic. We will propose that every subgroup comes from one of these types of constructions. There are many interesting consequences all across group theory and geometry if this is indeed the case. Focusing on the case of 2-by-2 matrix groups, the algebraic and arithmetic subgroups are very well-understood. We will fill in some understanding of geometric subgroups in this setting, by showing that for any prime p, there is a closed surface group in \({\rm PSL}_2(\mathbb{Z}[1/p])\). These surface groups will have genus proportional to \(p\), and we speculate that the construction is optimal.

Classical extremal results in graph theory (such as Turán's theorem) concern the maximal size of of a graph of given order and without certain subgraphs. Bollobás, Erdős, and Szemerédi in 1975 studied extremal problems in multipartite graphs. One of their problems (in its complementary form) was determining the maximal degree of a multipartite graph without an independent transversal. This problem has received considerable attention and was settle in 2006 (Szabó--Tardos and Haxell--Szabó). Other questions asked by Bollobás, Erdős, and Szemerédi remain open, such as determining:

(1) the maximum degree in a multipartite graph without a partial independent transversal, and;
(2) the minimum degree that forces an octahedral graph in balanced tripartite graphs.

In this talk I will survey recent progress on these and other related problems.

In this talk, I will discuss some new results on intersection graphs of pseudo-segments in the plane and their applications in graph drawing.  These results are joint work with Jacob Fox and Janos Pach.
 

Micrometer-sized particles made from responsive polymer networks (that is, responsive microgel colloids) are of high potential for the design of functional soft materials due to their adaptive compressibility and stimuli-triggered volume transition. In this talk, I will discuss models and theoretical approaches, such as Langevin simulations and classical (dynamic) density functional theory (DFT), to describe the structural and dynamical behavior of dispersions of these responsive colloids in and out of equilibrium. Moreover, I will argue that chemical fueling and the inclusion of chemomechanical feedback loops may lead to excitable and oscillatory dynamics of the active colloids, establishing first steps to a well-controlled design of artificial cells and their emergent behavior.

Steklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which are introduced by Steklov in 1902 motivated by physics. And there is a deep connection between the extremal Steklov eigenvalue problems and the free boundary minimal surface theory in the unit Euclidean ball as revealed by Fraser and Schoen in 2016. In the talk, we will discuss the question of how rare simple Steklov eigenvalues are on manifolds and its applications in nodal sets and critical points of eigenfunctions.

Aiming to understand the solvability of semilinear elliptic boundary values problems in bounded domains, we will review the best known techniques for establishing the existence of critical points of functionals whose critical points are solutions to such problems. The mountain pass lemma, the Nehari manifold,
the Morse index, and bifurcation analysis will be discussed to conclude the existence of seven solutions for an asymptotically linear elliptic problem.