L3

In the first part of the talk, I will present an overview of recent advances in the description of diffusion-reaction processes and their first-passage statistics, with the special emphasis on the role of the boundary local time and related spectral tools. The second part of the talk will illustrate the use of these tools for the analysis of boundary-catalytic branching processes. These processes describe a broad class of natural phenomena where the population of diffusing particles grows due to their spontaneous binary branching (e.g., division, fission, or splitting) on a catalytic boundary located in a complex environment. We investigate the possibility of the geometric control of the population growth by compensating the proliferation of particles due to catalytic branching events by their absorptions in the bulk or on boundary absorbing regions. We identify an appropriate Steklov spectral problem to obtain the phase diagram of this out-of-equilibrium stochastic process. The principal eigenvalue determines the critical line that separates an exponential growth of the population from its extinction. In other words, we establish a powerful tool for calculating the optimal absorption rate that equilibrates the opposite effects of branching and absorption events and thus results in steady-state behavior of this diffusion-reaction system. Moreover, we show the existence of a critical catalytic rate above which no compensation is possible, so that the population cannot be controlled and keeps growing exponentially. The proposed framework opens promising perspectives for better understanding, modeling, and control of various boundary-catalytic branching processes, with applications in physics, chemistry, and life sciences.

I will introduce the Polchinski dynamics, a general framework to study asymptotic properties of statistical mechanics and field theory models inspired by renormalisation group ideas. The Polchinski dynamics has appeared recently under different names, such as stochastic localisation, and in very different contexts (Markov chain mixing, optimal transport, functional inequalities...) Here I will motivate its construction from a physics point of view and mention a few applications. In particular, I will explain how the Polchinski dynamics can be used to generalise Bakry and Emery’s Γ2 calculus to obtain functional inequalities (e.g. Poincaré, log-Sobolev) in physics models which are typically high-dimensional and non-convex. 

Approximate subrings are subsets $A$ of a ring $R$ satisfying \[ A + A + AA \subset F + A \] for some finite $F \subset R$. They encode the failure of sum-product phenomena, much like approximate subgroups encode failure of growth in groups.

I will discuss how approximate subrings mirror approximate subgroups and how model-theoretic tools, such as a stabilizer lemma for approximate subrings due to Krupiński, lead to structural results implying a general, non-effective sum-product phenomenon in arbitrary rings: either sets grow rapidly under sum and product, or nilpotent ideals govern their structure. I will also outline related results for infinite approximate subrings and conjectures unifying known (effective) sum-product phenomena.

Based on joint work with Krzysztof Krupiński.

We generalize the seminal polynomial partitioning theorems of Guth and Katz [1, 2] to a set of semi-Pfaffian sets. Specifically, given a set $\Gamma \subseteq \mathbb{R}^n$ of $k$-dimensional semi-Pfaffian sets, where each $\gamma \in \Gamma$ is defined by a fixed number of Pfaffian functions, and each Pfaffian function is in turn defined with respect to a Pfaffian chain $\vec{q}$ of length $r$, for any $D \ge 1$, we prove the existence of a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$ such that each connected component of $\mathbb{R}^n \setminus Z(P)$ intersects at most $\sim \frac{|\Gamma|}{D^{n - k - r}}$ elements of $\Gamma$. Also, under some mild conditions on $\vec{q}$, for any $D \ge 1$, we prove the existence of a Pfaffian function $P'$ of degree at most $D$ defined with respect to $\vec{q}$, such that each connected component of $\mathbb{R}^n \setminus Z(P')$ intersects at most $\sim \frac{|\Gamma|}{D^{n-k}}$ elements of $\Gamma$. To do so, given a $k$-dimensional semi-Pfaffian set $\gamma \subseteq \mathbb{R}^n$, and a polynomial $P \in \mathbb{R}[X_1, \ldots, X_n]$ of degree at most $D$, we establish a uniform bound on the number of connected components of $\mathbb{R}^n \setminus Z(P)$ that $\gamma$ intersects; that is, we prove that the number of connected components of $(\mathbb{R}^n \setminus Z(P)) \cap \gamma$ is at most $\sim D^{k+r}$. Finally, as applications, we derive Pfaffian versions of Szemeredi-Trotter-type theorems and also prove bounds on the number of joints between Pfaffian curves.

These results, together with some of my other recent work (e.g., bounding the number of distinct distances on plane Pfaffian curves), are steps in a larger program - pushing discrete geometry into settings where the underlying sets need not be algebraic. I will also discuss this broader viewpoint in the talk.

This talk is based on multiple joint works with Saugata Basu, Antonio Lerario, Martin Lotz, Adam Sheffer, and Nicolai Vorobjov.

[1] Larry Guth, Polynomial partitioning for a set of varieties, Mathematical Proceedings of the Cambridge
Philosophical Society, vol. 159, Cambridge University Press, 2015, pp. 459–469.

[2] Larry Guth and Nets Hawk Katz, On the Erdős distinct distances problem in the plane, Annals of
mathematics (2015), 155–190.