Cristiana De Filippis (University of Parma): "Surfing Regularity on Nonlinear Potentials"

The representation formula for the Poisson equation gives an explicit expression of solutions in terms of the data, yielding zero- and first-order pointwise bounds via convolution with appropriate Riesz potentials. The mapping properties of these potentials provide sharp regularity transfer from data to solutions, giving a complete description of the regularity features of solutions. I will outline key aspects of nonlinear potential theory that reproduce this behavior for nonlinear elliptic PDEs, where representation formulae are unavailable, and trace their regularity theory back to that of the Poisson equation up to the C^{1} level. I will then present a novel potential-theoretic approach, altering a century old paradigm in nonlinear regularity theory, that resolves the longstanding problem of the validity of Schauder theory in nonuniformly elliptic PDEs.

Justin Salez (Université Paris Dauphine): "An invitation to the cutoff phenomenon"

The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity. Discovered four decades ago in the context of card shuffling, this surprising phenomenon has since then been observed in a variety of models, from random walks on groups or complex networks to Glauber dynamics for high-temperature spin systems. It is now believed to be universal among fast-mixing high-dimensional processes. Yet, current proofs are heavily model-dependent, and identifying the general conditions that trigger a cutoff remains one of the biggest challenges in the quantitative analysis of finite Markov chains. In this talk, I will provide a self-contained introduction to this fascinating question, and then describe a recent partial answer based on entropy and curvature.

Tatjana Tchumatchenko (University of Bonn): How mathematical concepts can help understand neuronal dynamics

Some of the most fascinating open scientific topics lie at the interface between disciplines, often between biology, medicine, and math. I will show how standard mathematical concepts such as linear algebra, topology, and differential equations can help understand numerous processes inside neurons, from molecular trafficking inside individual cells to multi-cell activity patterns. I will discuss our recent work showing how activity manifolds can jointly encode behavioral variables and how molecular dynamics can be traced down to optimality concepts and multi-dimensional diffusion equations.