Georgia Institute of Technology College of Sciences

Brascamp-Lieb inequalities are estimates for certain multilinear forms on functions on Euclidean spaces. They generalize several classical inequalities, such as Hoelder's inequality or Young's convolution inequality. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions in the Brascamp-Lieb inequality is replaced by a singular integral kernel. Examples include multilinear singular integral forms such as paraproducts or the multilinear Hilbert transform. We survey some results in the area. 

(This is Part 2, continuation of Tuesday's lecture.)

A fundamental tool used in sampling, counting, and inference problems is the Markov Chain Monte Carlo method, which uses random walks to solve computational problems. The main parameter defining the efficiency of this method is how quickly the random walk mixes (converges to the stationary distribution). The goal of these talks is to introduce a new approach for analyzing the mixing time of random walks on high-dimensional discrete objects. This approach works by directly relating the mixing time to analytic properties of a certain multivariate generating polynomial. As our main application we will analyze basis-exchange random walks on the set of bases of a matroid. We will show that the corresponding multivariate polynomial is log-concave over the positive orthant, and use this property to show three progressively improving mixing time bounds: For a matroid of rank r on a ground set of n elements:

- We will first show a mixing time of O(r^2 log n) by analyzing the spectral gap of the random walk (based on related works on high-dimensional expanders).

- Then we will show a mixing time of O(r log r + r log log n) based on the modified log-sobolev inequality (MLSI), due to Cryan, Guo, Mousa.

- We will then completely remove the dependence on n, and show the tight mixing time of O(r log r), by appealing to variants of well-studied notions in discrete convexity.

Time-permitting, I will discuss further recent developments, including relaxed notions of log-concavity of a polynomial, and applications to further sampling/counting problems.

Based on joint works with Kuikui Liu, Shayan OveisGharan, and Cynthia Vinzant.

We study the mean field limit of large stochastic systems of interacting particles. To treat more general and singular kernels, we propose a modulated free energy combination of the method that we had previously developed and of the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the most singular terms involving the divergence of the flow. Our modulated free energy allows to treat singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller-Segel system in the subcritical regimes, is obtained. This is a joint work with D. Bresch and Z. Wang.

We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. Using the tropical Torelli theorem (due to Caporaso and Viviani), this characterization may be phrased in terms of 3-edge connectiviations. We show that being of hyperelliptic type is independent of the edge lengths and is preserved when passing to genus ≥2 connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type.