Invited Talks

Tom Hutchcroft, Caltech

Title: Thick points of 4d branching random walk

Abstract: It is a celebrated theorem of Dembo, Peres, Rosen, and Zeitouni (2001) that if we run a simple random walk on the square grid for n steps, the maximum number of times that a point is visited is asymptotic to (log n)^2/pi. This result is just the start of a long line of work studying the multifractal geometry of random walk "thick points" which is now known to be closely related to two-dimensional quantum gravity. In this talk, I will describe recent joint work with Nathanael Berestycki and Antoine Jego directed at establishing an analogous theory for branching random walk in four dimensions.

Etienne Pardoux, Aix-Marseille Université

Title: Non-Markovian epidemic models and their large population limit

Abstract: Almost 100 years ago, Kermack and McKendrick proposed epidemic models taking into account the fact that the infectivity of infectious individuals varies with their age of infection, and the susceptibility of individuals losing their immunity increases continuously from 0 to a maximum which can be 1 or less. We show that the Kermack - Mc Kendrick models (or more general models) are Law of Large Numbers limits of realistic individual based non Markovian stochastic models, as the population size tends to infinity. We also consider infinite dimensional Markovian versions of our models, where we consider the age of infection as a new variable. We shall describe the PDE LLN limit, and the SPDE CLT limit in this framework. This is joint work with Guodong Pang (Rice University), and in part with also Raphaël Forien (INRAE) and Arsene Brice Zotsa Ngoufack (PhD student at AMU).

Sébastien Roch, University of Wisconsin

Title: Complex Probabilistic Models in Evolutionary Biology: Challenges and Opportunities

Abstract: The reconstruction of species phylogenies from genomic data is a key step in modern evolutionary studies. This task is complicated by the fact that genes evolve under biological phenomena that produce discordant histories. These include horizontal gene transfer, gene duplication and loss, and incomplete lineage sorting, all of which can be modeled using random gene tree distributions building on well-studied stochastic processes (branching processes, the coalescent, etc.). Gene trees are in turn estimated from molecular sequences using Markov models on trees. The rigorous analysis of the resulting complex models can help guide the design of new reconstruction methods with statistical and computational guarantees. I will illustrate the challenges and opportunities in this area via a few recent results. No biology background will be assumed.

Perla Sousi, University of Cambridge

Title: Random walk: cover time and geometry of the uncovered set

Abstract: In this talk we will give a survey of recent progress on fine geometrical properties of random walks. We will focus on the cover time and the structure of the last visited set of points on finite graphs. Along the way we will discuss other related models, such as the Gaussian free field and study its high points.

Ludovic Tangpi, Princeton University

Title: A probabilistic approach to the convergence problem in mean field games

Abstract: In this talk we will discuss the convergence problem in mean field games. After introducing mean field games and motivating the problem from classical statistical physics considerations, we will review recent advances on the convergence problem. We will focus on the method based on “forward-backward propagation of chaos”. The main results discussed will establish that allowing structural conditions such as dissipativity of the drift or displacement monotonicity of the coupling functions in the game allows to derive very general, quantitative convergence theorems that do not require analysis (or even existence) of the master equation.

Yilin Wang, Institut des Hautes Etudes Scientifiques

Title: Onsager-Machlup functional for SLE loop measure

Abstract: Onsager-Machlup functional measures how likely a stochastic process stays close to a given path. SLE is a family of measures on simple paths in the plane introduced by O. Schramm obtained from the Loewner transform of a multiple of Brownian motion. It is well-known that the Onsager-Machlup functional for Brownian motion is the Dirichlet energy. We show that the Onsager-Machlup of the SLE_k loop measure, for any 0 < k \le 4, is expressed using the Loewner energy and the central charge c(k) of SLE_k. Loewner energy is defined as the Dirichlet energy of the Loewner driving function of the loop, but it also has tight links to many other fields of mathematics. Our proof relies on the conformal restriction covariance of SLE. This is based on the joint work (arXiv: 2311.00209) with Marco Carfagnini (UCSD).

Lightning Talks

Talks and abstracts can be downloaded here