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 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.

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.

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.