Seminars in Term 1. Talks are held 16-17 in B3.03.
Oct 4- Tom Klose, University of Warwick
Title: Large deviations for the Φ^4_3 measure via Stochastic Quantisation
Oct 11- Ellen Powell, University of Durham
Title: Characterising the Gaussian free field
Oct 18- Anna Maltsev, Queen Mary University
Title: Bulk Universality for Complex Non-Hermitian Gauss-divisible Matrices
In this talk I will discuss universality of the k-point correlation function for Gauss divisible non-Hermitian matrices. We consider NxN matrices with centred, independent and identically distributed complex entries that have a small Gaussian component. We prove that the bulk correlation functions are universal in the large N limit using Householder transformations, supersymmetry, and Laplace method. Assuming the entries have finite moments and are supported on at least three points, the Gaussian component is removed by the four moment theorem. This is based on joint work with Mohammed Osman.
Oct 25- Julien Sabin, University of Rennes
Title: Nonlinear Hartree dynamics for density matrices
Abstract: In this talk I will review results concerning the mean-field dynamics of fermionic quantum particles governed by the nonlinear Hartree equation. The particularity of this equation is that its unknown is a bounded operator on a Hilbert space, rather than a (wave)function as is the case for most PDEs. I will explain how to deal with setting, with a focus on the large time behaviour of solutions.
Nov 1- Erlend Grong, University of Bergen
Title: Sub-Riemannian geometry, most probable paths and transformations.Abstract: Doing statistics on a Riemannian manifold becomes very complicated for the reason that we lack tools to define such things as mean and variance. Using the Riemannian distance, we can define a mean know as the Fréchet mean, but this gives no concept of asymmetry, also known as anisotropy. We introduce an alternative definition of mean called the diffusion mean, which is able to both give a mean and the analogue of a covariance matrix for a dataset on a Riemannian manifolds. Surprisingly, computing this mean and covariance is related to sub-Riemannian geometry. We describe how sub-Riemannian geometry can be applied in this setting, and mention some finite dimensional and infinite-dimensional applications. The results are part of joint work with Stefan Sommer (Copenhagen, Denmark)
Nov 8- Balint Toth, University of Bristol
Nov 15- Luisa Andreis, Politecnico di Milano
Title: Spatial coagulation processes: large deviations and phase transitions.
Abstract: We consider a spatial Markovian particle system with pairwise coagulation: after independent exponential random times, particle pairs merge into a single particle, and their masses are summed. We derive an explicit formula for the joint distribution of the particle configuration at a given fixed time, which involves the binary trees describing the history of how each of the particles has been formed via coagulations. While usually these processes are studied with the help of PDE (generalisation of the well-known Smoluchowski equation), our approach comes from statistical mechanics. The description is indeed in terms of a reference process, a Poisson point process of point group distributions, where each of the histories is an independent tree, and the non-coagulation between any two of them induces an exponential pair-interaction. Based on this formula, we can give a (conditional) large-deviation principle for the joint distribution of the particle histories in the limit of many particles with explicit identification of the rate function. We characterise its minimizer(s) and give criteria for the occurrence of a gelation phase transition, i.e., a loss of mass in the limiting configuration. This talk is based on an ongoing joint work with W. König, H. Langhammer and R.I.A. Patterson (WIAS Berlin).
Nov 22- Cristoforos Panagiotis, University of Bath
Nov 29- Marielle Simon, University of Lyon
Dec 6- Ilya Chevyrev, University of Edinburgh
Title: Decorated path spaces with applications to fast-slow systems
Abstract: In this talk, I will present a space of decorated paths that allows one to keep track of oscillations of paths that happens in infinitesimal time. Despite its simple definition as a naive completion of the Skorokhod space, this notion is fruitful in the study of ordinary differential equations with jumps, generalising the framework of Marcus, and applies in situations where classical Skorokhod topologies are too restrictive. As an application, I will show how homogenisation theorems of superdiffusive fast-slow systems, including billiards with flat cusps, can be stated and proved in this framework. Based on a joint work with Alexey Korepanov and Ian Melbourne.