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.

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.

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.

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

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.