7 October 2022: Michela Ottobre (Heriot-Watt) Interacting Particle systems and (Stochastic) Partial Differential equations: modelling, analysis and computation
The study of Interacting Particle Systems (IPSs) and related kinetic equations has attracted the interest of the mathematics and physics communities for decades. Such interest is kept alive by the continuous successes of this framework in modelling a vast range of phenomena, in diverse fields such as biology, social sciences, control engineering, economics, game theory, statistical sampling and simulation, neural networks etc. While such a large body of research has undoubtedly produced significant progress over the years, many important questions in this field remain open. We will (partially) survey some of the main research directions in this field and discuss open problems.
In writing a biographical memoir of Zeeman for the Royal Society, I appreciated even more what a remarkable character he was, both in terms of his life, his leadership, his mathematics and his breadth of interests. I discovered a number of aspects that I don't think are very well known about his life and his contributions to topology, catastrophe theory and our department. In this colloquium I will try and give an overview of this.
Give a wave packet an initial energy and let it propagate in the whole space, then in the linear regime, the wave packet will scatter. But in non linear regimes, part of the energy may concentrate to form coherent non linear structures which propagate without deformation (solitons). And in more extreme cases singularities may form.
Whether or not singular structures arise is a delicate problem which has attracted a considerable amount of works in both mathematics and physics, in particular in the super critical regime which is the heart of the 6Th Clay problem on singularity formation for three dimensional viscous incompressible fluids. For another classical model like the defocusing Non Linear Schrodinger equation (NLS), Bourgain ruled out in a breakthrough work (1994) the existence of singularities in the critical case, and conjectured that this should extend to the super critical one.
I will explain how the recent series of joint works with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris 6) shades a new light on super critical singularities: in fact there exist super critical singularities for (NLS), and the new underlying mechanism is directly connected to the first description of singularities for three dimensional viscous compressible fluids.
Many relevant problems in Geometric Measure Theory can be ultimately understood in terms of the structure of certain classes of sets, which can be loosely described as "small" (in some sense or another). In this talk I will review a few of these problems and related results, and highlight the connections to other areas of Analysis.
11 November 2022: Yan Fyodorov (King's College London) "Escaping the crowds": extreme values and outliers in rank-1 non-normal deformations of GUE/CUE
Rank-1 non-normal deformations of GUE/CUE provide the simplest model for describing resonances in a quantum chaotic system decaying via a single open channel. In the case of GUE we provide a detailed description of an abrupt restructuring of the resonance density in the complex plane as the function of channel coupling, identify the critical scaling of typical extreme values, and finally describe how an atypically broad resonance (an outlier) emerges from the crowd. In the case of CUE we are further able to study the Extreme Value Statistics of the ''widest resonances'' and find that in the critical regime it is described by a distribution nontrivially interpolating between Gumbel and Frechet. The presentation will be based on the joint works with Boris Khoruzhenko and Mihail Poplavskyi.
The entropy rate of a stationary sequence of random symbols was introduced by Shannon in his foundational work on information theory in 1948. In the early 1950s, Kolmogorov and Sinai realized that they could turn this quantity into an isomorphism invariant for measure-preserving transformations on a probability space. Almost immediately, they used it to distinguish many examples called "Bernoulli shifts" up to isomorphism. This resolved a famous open question of the time, and ushered in a new era for ergodic theory.
In the decades since, entropy has become one of the central concerns of ergodic theory, having widespread consequences both for the abstract structure of measure-preserving transformations and for their behaviour in applications. In this talk, I will review some of the highlights of the structural story, and then discuss Bowen's more recent notion of `sofic entropy'. This generalizes Kolmogorov--Sinai entropy to measure-preserving actions of many `large' non-amenable groups including free groups. I will end with a recent result illustrating how the theory of sofic entropy has some striking differences from its older counterpart.
This talk will be aimed at a general mathematical audience. Most of it should be accessible given a basic knowledge of measure theory, probability, and a little abstract algebra.
2 December 2022: Tara Brendle (Glasgow) Twists and trivializations: encoding symmetries of manifolds
The classification of 2-manifolds in the first half of the 20th century was a landmark achievement in mathematics, as was the more recent (and more complicated) classification of 3-manifolds completed by Perelman. The story does not end with classification, however: there is a rich theory of symmetries of manifolds, encoded in their mapping class groups. In this talk we will explore some aspects of mapping class groups in dimensions 2 and 3, with a focus on illustrative examples.
Category theory provides a general framework for building models of dynamical systems. We explain this framework and illustrate it with the example of "stock and flow diagrams". These diagrams are widely used for simulations in epidemiology. Although tools already exist for drawing these diagrams and solving the systems of differential equations they describe, we have created a new software package called StockFlow which uses ideas from category theory to overcome some limitations of existing software. We illustrate this with code in StockFlow that implements a simplified version of a COVID-19 model used in Canada. This is joint work with Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson.