The unadjusted Langevin algorithm is widely used for sampling from complex high-dimensional distributions. It is well known to be biased, with the bias typically scaling linearly with the dimension when measured in squared Wasserstein distance. However, a recent paper of Chen, Cheng, Niles-Weed, and Weare identified an intriguing new "delocalization" phenomenon: For certain classes of target distributions, they proved that the bias between low-dimensional marginals scales only with the lower dimension, not the full dimension. This talk will explain this new phenomenon and our recent results which strengthen those of Chen et al. in several directions. The proofs are based on a hierarchical analysis of the marginal relative entropies (a.k.a. KL-divergence), inspired by our recent work on propagation of chaos for mean field models. A byproduct of our analysis, of independent interest, is a new kind of convergence estimate for low-dimensional marginals of continuous-time Langevin diffusions. Based on joint work with Fuzhong Zhou.

The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified stochastic heat equation, specifically at the critical dimension of two and near the critical temperature. I will present an axiomatic formulation of the SHF as well as its construction based on its moments, and discuss how this formulation can be applied to solve a range of problems.

For every natural number $n$, if we start with sufficiently many points in $\mathbb{R}^d$ in general position there will always exist n points in convex position. The problem of determining quantitative bounds for this statement is known as the Erdős-Szekeres problem, and is one of the oldest problems in extremal combinatorics (sometimes also called the “Happy Ending Problem”). We will discuss some of its history, some recent developments in the plane and in higher dimensions, as well as some connections with a few other topics in combinatorics and beyond. 

In a joint work with Diederik van Engelenburg, Romain Panis and Franco Severo, we study the probability that the origin is connected to the boundary of the box of size $n$ (the one-arm probability) in several percolation models related to the Ising model. We prove that different universality classes emerge at criticality and that the FK-Ising model has upper-critical dimension equal to 6, in contrast to the Ising model, where it is known to be (less or) equal to 4. I will start the talk with a short introduction on the Ising model on $\mathbb{Z}^d$.