I'll describe a new rigidity phenomenon of lattices in higher rank semisimple groups. Specifically, I'll explain why the theories of such groups can't have (finitely generated) deformations, why these groups have a very rich collection of definable subgroups, and finish by discussing a conjecture saying that being a higher rank arithmetic lattice is a first-order property. Based on joint works with Alex Lubotzky and Chen Meiri.

It has been known for a long time that as their size grow to infinity, many models of random matrices behave as free operators. This link was first explicited by Voiculescu in 1991 in a paper in which he proved that the trace of polynomials in independent GUE matrices converges towards the trace of the same polynomial evaluated in free semicircular variables. In 2005, Haagerup and Thorbjornsen proved the convergence of the norm instead of the trace. The main difficulty of their proof was to prove a sharp enough upper bound of the difference between the trace of random matrices and their free limit. They managed to do so with the help of the so-called linearization trick which allows to relate the spectrum of a polynomial of any degree with scalar coefficients with a polynomial of degree 1 with matrix coefficients. A drawback of this method is that it does not give easily good quantitative estimates. In arXiv:1912.04588, we introduced a new strategy to approach those questions which does not rely on the linearization trick and instead is based on free stochastic calculus. In this talk, I will first focus on the paper arXiv:2011.04146, in which we proved an asymptotic expansion for traces of smooth functions evaluated in independent GUE random matrices, whose coefficients are defined through free probability. And then I will talk about arXiv:2005.1383, in which we adapted the previous method to the case of Haar unitary matrices.

In a large class of reaction-diffusion equations, the solution starting from a compactly supported initial datum develops a transition between two rest states, that moves at an asymptotically linear rate in time, and whose thickness remains asymptotically bounded in time. The issue is its precise location in time, that is, up to terms that are o(1) as time goes to infinity. This question is well understood in one space dimension; I will discuss what happens in the less well settled multi-dimensional framework. Joint works with L. Rossi and V. Roussier.

What makes the digits 645751311064590590501615753639260425710 and 1010100101010011111111010100111010010111 so special? These digits look as if they were chosen at random, yet they are entirely deterministic (take the fractional part of the square root of 7). In this talk, I will explore the theory of quasi-randomness, which characterizes ``random-like" sequences, graphs, sets, and many other objects. In particular, I will present a theory of quasi-randomness for Boolean functions and show how random Boolean functions lead to a very challenging open problem: the Inverse Theory of the Gowers Norms.

In this talk, we discuss the problem of finding generalized Nash equilibria (GNE) in the viewpoint of sparse polynomials. To obtain optimality conditions for GNE, we consider the Karush-Kuhn-Tucker (KKT) system using the Lagrange multiplier. We discuss that if all objectives and constraints polynomials are generic, the number of solutions of the KKT system equals its mixed volume, and so the polyhedral homotopy method can be optimal for finding GNEs. Lastly, comparisons with existing methods will be given.

This talk is concerned with the inverse problem of recovering a discrete measure on the torus given a finite number of its noisy Fourier coefficients. We focus on the diffraction limited regime where at least two atoms are closer together than the Rayleigh length. We show that the fundamental limits of this problem and the stability of subspace (algebraic) methods, such as ESPRIT and MUSIC, are closely connected to the minimum singular value of non-harmonic Fourier matrices. We provide novel bounds for the latter in the case where the atoms are located in clumps. We also provide an analogous theory for a statistical model, where the measure is time-dependent and Fourier measurements are collected over at various times. Joint work with Wenjing Liao, Albert Fannjiang, Zengying Zhu, and Weiguo Gao.

What does it mean to pick a ``random'' hyperbolic surface, and how does one even go about ``picking'' one? Mirzakhani gave an inductive answer to this question by gluing together smaller random surfaces along long curves; this is equivalent to studying the equidistribution of certain sets inside the moduli space of hyperbolic surfaces. Starting from first concepts, in this talk I’ll explain a new method for building random hyperbolic surfaces by building random \emph{flat} ones. As time permits, we will also discuss the application of this technique to Mirzakhani’s ``twist torus conjecture.'' This is joint work (in progress) with James Farre.

Chemical Reaction Network theory is an area of mathematics that analyzes the behaviors of chemical processes. One major problem concerns the stability of steady states of these networks. Does a given chemical reaction network have the capacity for Hopf bifurcations (an important unstable steady-state that yields periodic oscillations)? Our first contribution is a novel procedure for constructing a Hopf bifurcation of a chemical reaction network. This algorithm -- our Newton-polytope method -- gives an easy-to-check condition for the existence of a Hopf bifurcation and explicitly constructs one if it exists. Another important invariant of a chemical reaction network is its maximum number of steady states. This number, however, is in general difficult to compute, as it translates to counting positive real solutions of parametrized polynomial systems. To this end, we introduce an upper bound on this number -- namely, a network's mixed volume -- that is easy to compute. In this talk, we apply our two new tools to an important biological-signaling network, called the ERK network. Rubinstein et al. (2016) showed that the ERK network exhibits multiple steady states, bistability, and periodic oscillations (for a very particular choice of initial conditions). Conradi and Shiu (2015) proved that when certain reactions are omitted, the ERK network reduces to the processive dual-site phosphorylation network, which has a unique, stable steady-state (for any initial conditions). This stark contrast in dynamics prompted Rubinstein et al.'s question, "How are bistability and oscillations lost as reactions are removed from the ERK network?" By analyzing subnetworks of the ERK network, we systematically answer this question and demonstrate that bistability and oscillations persist even after we greatly simplify the model (by making reactions irreversible and removing intermediate species).

Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a
connected $p$-adic reductive group $G/L$, and a smooth irreducible
representation $\pi$ of $G(L)$, Fargues-Scholze recently attached a
semisimple Weil parameter to such $\pi$, giving a general candidate for
the local Langlands correspondence. It is natural to ask whether this
construction is compatible with known instances of the correspondence
after semisimplification. For $G = GL_{n}$ and its inner forms,
Fargues-Scholze and Hansen-Kaletha-Weinstein show that the
correspondence is compatible with the correspondence of
Harris-Taylor/Henniart. We verify a similar compatibility for $G =
GSp_{4}$ and its unique non-split inner form $G = GU_{2}(D)$, where $D$
is the quaternion division algebra over $L$, assuming that
$L/\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case, the local
Langlands correspondence has been constructed by Gan-Takeda and
Gan-Tantono. Analogous to the case of $GL_{n}$ and its inner forms, this
compatibility is proven by describing the Weil group action on the
cohomology of a local Shimura variety associated to $GSp_{4}$, using
basic uniformization of abelian type Shimura varieties due to Shen,
combined with various global results of Kret-Shin and Sorensen on Galois
representations in the cohomology of global Shimura varieties associated
to inner forms of $GSp_{4}$ over a totally real field. After showing the
parameters are the same, we apply some ideas from the geometry of the
Fargues-Scholze construction explored recently by Hansen, to give a more
precise description of the cohomology of this local Shimura variety,
verifying a strong form of the Kottwitz conjecture in the process.

Mnev's universality theorem asserts that every singularity type over the ring of integers appears in some thin Schubert cell of the Grassmannian Gr(3,E) for some vector space E. We construct sequential blowups of Gr(3,E) such that certain induced birational transforms of all thin Schubert cells become smooth over prime fields. This implies that every singular variety X defined over a prime field admits local resolutions. For a singular variety X over a general perfect field k, we spread it out and deduce that X/k admits local resolution as well.