# Junior Analysis and Probability Seminar

Welcome to the Junior Analysis and Probability Seminar, a space for PG Students and Young Researchers to present and share their research in areas related to Mathematical Analysis and Probability.

We welcome talks in

• Harmonic Analysis
• Ordinary, Partial and Stochastic Differential Equations
• Probability Theory
• Measure Theory and Geometry

## Term 1 (Oct – Dec 2022)

Talks will take place on Thursdays 1pm-2pm in B1.01.

Date Speaker Affiliation Title
Nov 10 (W6) Simon Gabriel Warwick
On the Poisson–Dirichlet diffusion and Trotter–Kurtz approximations
Nov 17 (W7) Jakub Takáč Warwick Norms in finite dimensions and rectifiability in metric spaces
Nov 24 (W8) Giacomo Del Nin Warwick Isoperimetric shapes in Penrose tilings
Dec 1 (W9) Milos Tasic Warwick Rigorous derivation and the propagation of chaos

#### Abstracts

##### Milos Tasic: Rigorous derivation and propagation of chaos

Deriving continuum models from atomistic descriptions is a long lasting and fundamental problem in mathematical physics. We will discuss what we mean by derivation of a continuum model and give a few examples. Finally, we will discuss a result by Matthies and Theil which gives a long time derivation for the gainless Boltzmann equation and explain their approach via marked trees. For simplicity we shall restrict ourselves to 3 dimensions and impose spatial homogeneity.

##### Giacomo Del Nin: Isoperimetric shapes in Penrose tilings

Given a tiling of the plane, for any fixed integer N we consider the configuration of N tiles whose union has minimal perimeter. When N goes to infinity, is the best configuration (once suitably translated and rescaled down) approaching some specific shape? In this talk I will explain how to answer this question for a large class of quasi-periodic tilings, such as the Penrose tiling. Before that, I will describe the connection with the closely related problem of interacting particle systems, that will help us obtain the solution. Based on a joint work with Mircea Petrache (Pontificia Universidad Católica de Chile).

##### Jakub Takáč: Norms in finite dimensions and rectifiability in metric spaces

How does fitting a linearly squished convex set into another convex set (in $Rn$) relate toproperties of n-rectifiable sets in (completely general) metric spaces? We call a subset of a metric space $X$ n-rectifiable if it can be covered, up to a set of $HnX$-null measure, by a countable number of Lipschitz images of pieces of $Rn$. We will first discuss a recent result of Bate on characterising n-rectifiable subsets of metric spaces, and then we describe the connection of these to finite-dimensional norms (due to Kirchheim). In the bulk of the talk, we shall study certain geometric (“squishing”) properties of unit balls of these norms which can be transferred into general n-rectifiable metric spaces, strengthening thereupon some of the results of Bate. Particular attention will be paid to the case of 2-dimensional norms in which case the arising problems are a bit easier to understand.

##### Simon Gabriel: On the Poisson–Dirichlet diffusion and Trotter–Kurtz approximations

The aim of the talk is to shed light on the PoissonDirichlet diffusion, which is a stochastic process arising in population genetics. After introducing briefly the connection between stochastic processes and generators of semigroups, I will discuss how approximating the process is helpful in understanding the underlying dynamics. We use stochastic particle systems and the TrotterKurtz theorem to prove convergence to the aforementioned model.
The talk is based on joint work with Paul Chleboun and Stefan Grosskinsky, but does not assume knowledge on stochastic processes.