Random multiplicative functions are probabilistic models for multiplicative arithmetic functions, such as Dirichlet characters or the Liouville function. In this talk, I will first quickly give an overview of the area, and then focus on some of the recent works on proving central limit theorems, connections to additive combinatorics, as well as some other deterministic applications. Part of the talk is based on joint work with Soundararajan, with Harper and Soundararajan (in progress) and with Angelo and Soundararajan (in progress).

The twisted Gan--Gross--Prasad conjectures consider the restriction of representations from GL_n to a unitary group over a quadratic extension E/F. In this talk, I will explain the relative trace formula approach to the global twisted GGP conjecture. In particular, I will discuss how the fundamental lemma that arises can be reduced to the Jacquet--Rallis fundamental lemma, which allows us to obtain the global twisted GGP conjecture under some unramifiedness assumptions and local conditions.

I'll discuss work on shift invariance in a half space setting. These are non-trivial symmetries allowing certain observables of integrable models with a boundary to be shifted while preserving their joint distribution. The starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure, both in a half space setting. An application to the asymptotics of a half space analogue of the oriented swap process is also given.

In a world of polarized opinions on many cultural issues, we propose a model for the evolution of opinions on a large complex network. Our model is akin to the popular Friedkin-Johnsen model, with the added complexity of vertex-dependent media signals and confirmation bias, both of which help explain some of the most important factors leading to polarization. The analysis of the model is done on a directed random graph, capable of replicating highly inhomogeneous real-world networks with various degrees of assortativity and community structure. Our main results give the stationary distribution of opinions on the network, including explicitly computable formulas for the conditional means and variances for the various communities. Our results span the entire range of inhomogeneous random graphs, from the sparse regime, where the expected degrees are bounded, all the way to the dense regime, where a graph having n vertices has order n^2 edges.

We consider an ensemble of N interacting particles modeled by a system of N stochastic differential equations (SDEs). The coefficients of the SDEs are taken to be such that as N approaches infinity, the system undergoes Kac’s propagation of chaos, and is well-approximated by the solution to a McKean-Vlasov Equation. Rare but possible deviations of the behavior of the particles from this limit may reflect a catastrophe, and computing the probability of such rare events is of high interest in many applications. In this talk, we design an importance sampling scheme which allows us to numerically compute statistics related to these rare events with high accuracy and efficiency for any N. Standard Monte Carlo methods behave exponentially poorly as N increases for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field control. This HJB Equation is seen to be connected to the large deviations rate function for the empirical measure on the ensemble of particles. We identify conditions under which our scheme is provably asymptotically optimal in N in the sense of log-efficiency. We also provide evidence, both analytical and numerical, that with sufficient regularity of the solution to the HJB Equation, our scheme can have vanishingly small relative error as N increases.

We discuss new methods for using the Heegaard Floer homology of hypersurfaces to distinguish between smooth closed 4-manifolds that are homeomorphic but non-diffeomorphic. Specifically, for a 4-manifold X with b_1(X)=1, the minimum rank of the reduced Heegaard Floer homology of any embedded 3-manifold X representing a generator of H_1(X) gives a diffeomorphism invariant of X. We use this invariant to distinguish certain infinite families of exotic 4-manifolds that cannot be distinguished by previously known techniques. Using related ideas, we also provide the first known examples of (non-simply-connected) exotic 4-manifolds with negative definite intersection form. This is joint work with Tye Lidman and Lisa Piccirillo.

We will explore the role of demographic stochasticity in triggering extinction events in models of large finite populations. While prior works have focused on large fluctuations from quasi-stationary distributions, we instead consider extinction events occurring before entering a metastable state. Since such extinction events require only slight deviations from the mean-field trajectories, we can derive the approximating extinction probability PDE with a modified Robin-type boundary condition. We then investigate the utility of this approximation by comparing to the Lotka-Volterra model as well as the Lotka-Volterra model with logistic growth.

Schedule (approximate times): 1:00-2:00: An, Deng, McPhail-Snyder research groups 2:00-2:20: break 2:20-3:00: Robles, Stokols research groups

Satellite operations are an valuable method of constructing complicated knots from simpler ones, and much work has gone into understanding how knot invariants change under these operations. We describe a new way of computing the (UV=0 quotient of the) knot Floer complex using an immersed Heegaard diagram obtained from a Heegaard diagram for the pattern and the immersed curve representing the knot Floer complex of the companion. This is particularly useful for (1,1)-patterns, since in this case the resulting immersed diagram is genus one. In some cases the immersed curve representing the satellite knot Floer complex can be obtained directly by deforming the diagram, generalizing earlier work with Watson on cables. This is joint work with Wenzhao Chen.

While establishing various versions of the h-principle for contact distributions (Eliashberg (1989) in dimension 3, Borman-Eliashberg-Murphy (2015) in arbitrary dimension, and even-contact contact (D. McDuff, 1987) distributions are among the most remarkable advances in differential topology in the last four decades, very little is known about analogous results for other classes of distributions, e.g. generic distributions of corank 2 or higher. The smallest dimensional nontrivial case of corank 2 distributions is Engel distributions, i.e. the maximally nonholonomic rank 2 distributions on $4$-manifolds. This case is highly nontrivial and was treated recently by Casals-Pérez-del Pino-Presas (2017) and Casals-Pérez-Presas (2017). In my talk, I will show how to use the method of convex integration in order to establish all versions of the h-principle for corank 2 distributions of arbitrary odd rank satisfying a natural generic assumption on the associated pencil of skew-symmetric forms. During the talk, I will try to give all the necessary background related to the method of convex integration in principle. This is the joint work with Milan Jovanovic, Javier Martinez-Aguinaga, and Alvaro del Pino.

In the first part of the talk I will describe a general context which, in some specific situations, permits us to give a cohomological interpretation to the Langlands-Shahidi theory of L-functions. In the second part of the talk, I will specialize to the context of the general linear group over a totally imaginary base field F, and discuss some recent results of mine on the special values of Rankin-Selberg L-functions for GL(n) x GL(m) over such an F. The talk is based on my preprint: https://arxiv.org/abs/2207.03393