# 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

If you would like to give a talk, please contact one of the **Organisers**

Andrew RoutLink opens in a new window | Sotiris KotitsasLink opens in a new window | Ryan Acosta BabbLink opens in a new window |

## 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 ) relate toproperties of n-rectifiable sets in (completely general) metric spaces? We call a subset of a metric space n-rectifiable if it can be covered, up to a set of -null measure, by a countable number of Lipschitz images of pieces of . 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 Poisson**–**Dirichlet 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 Trotter**–**Kurtz 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.