I will describe a newly introduced toolbox that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh.

I will discuss the problem of understanding the long-time behavior of Ricci flow on a compact Kähler manifold, assuming that a solution exists for all positive time. Inspired by an analogy with the minimal model program in algebraic geometry, Song and Tian posed several conjectures which describe this behavior. I will report on recent work (joint with Hein and Lee) which confirms these conjectures.

Homogenization is the approximation of a complex, “disordered” system by a simpler, “ordered” one. Picture a walker on a grid. In each step, the walker chooses to walk along a neighboring edge with equal probability. At large scales, the walker approximates Brownian motion. But what if some edges are more likely to be traversed than others? I will discuss recent advances in the theory of quantitative homogenization which make it possible to analyze random walk with drift and other models in probability. Joint work with Scott Armstrong and Tuomo Kuusi.

The categorical p-adic local Langlands correspondence has been established for the group GL_2(Q_p) in joint work of the speaker with Andrea Dotto and Toby Gee. In this talk I will describe some aspects of this categorical correspondence. I hope to indicate the relationship to existing ideas in the subject: particularly to Taylor--Wiles--Kisin patching, but also to the work of Colmez and Paskunas, and to recent work of Johansson--Newton--Wang-Erickson. But more than this, I hope to indicate some of the underlying philosophy of the correspondence: what it means to represent the category of representations of a group geometrically, and why stacks (rather than just varieties) play a key role.

Weight Filtrations are mysterious: they record some shadow of how a variety might be recovered from smooth and projective ones. Some of the information recorded by weight filtrations can be understood via the motivic measures they define, i.e. group homomorphisms from the Grothendieck ring of varieties. With Zakharevich’s discovery of the higher K groups of the category of varieties, there is an ongoing project to understand these groups by lifting motivic measures (on the level of K_0) to so-called "derived" ones, i.e. on the level of K_i for all i. I will describe some of this work, which shows that if one closely studies how the Gillet-Soulé weight complex is constructed, then one can also obtain derived motivic measures to non-additive categories as well, such as the compact objects in the category of motivic spaces, along with that of compact objects in the motivic stable homotopy category. These new derived motivic measures allow us to answer questions in the literature, providing new ways to understand the higher K groups of varieties, and relating them to other interesting algebro-geometric objects in the literature.

We study the Brownian evolution of large non-Hermitian matrices and show that their log-determinant converges to a 2+1-dimensional Gaussian field in the Edwards-Wilkinson regularity class, i.e. logarithmically correlated for the parabolic distance. This gives a dynamical extension of the celebrated result by Rider and Virag (2006) proving that the fluctuations of the eigenvalues of Gaussian non-Hermitian matrices converge to a 2-dimensional log-correlated field. Our result, previously not known even in the Gaussian case, holds out of equilibrium for general matrices with i.i.d. entries. We also study the extremal values of these fields and demonstrate their logarithmic dependence on the matrix dimension.

Let π be a cuspidal automorphic representation of Sp_2n over Q which is holomorphic discrete series at infinity, and χ a Dirichlet character. Then one can attach to π an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is irreducible, that π is ordinary at p, and that p does not divide the conductor of χ. I will describe work in progress which aims to prove that the Bloch--Kato Selmer group attached to the twist of ρ by χ vanishes, under some mild ramification assumptions on π; this is what is predicted by the Bloch--Kato conjectures. The proof uses "ramified Eisenstein congruences" by constructing p-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of nonclassical weight, and using these families to construct ramified Galois cohomology classes for the Tate dual of the twist of ρ by χ.

The Square Peg Problem was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve in the plane contains the vertices of a square, and it is still open to this day. I will survey the approaches to this problem and its relatives using symplectic geometry. This talk is based on joint work with Andrew Lobb.

The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. We give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, corresponding to a lower bound on the geometric entanglement of random quantum states. The proof is based on spin-glass methods, the Kac—Rice formula, and recent progress coming from random matrices. Joint work with Stéphane Dartois.

Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre-)stabilization of relative trace formulas.