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.
13 January 2023: Tom Hudson (Warwick) Recent developments in the modelling and theory of crystalline defects
Crystalline solids are found all around us, and are made up of atoms arranged in a translation-invariant structure. Although this is a significant part of the story for such materials, crystals also contain many defects which break this symmetry, and it is these defects which turn out to be crucial in determining many of the physical properties of a crystalline material. I will begin by motivating the study of such defects, and provide an overview of the range of physical theories used to model and simulate their behaviour. I will then discuss some of the important recent developments in this area (many of which have a Warwick connection), including both novel machine-learning-based approaches and rigorous mathematical developments.
Kolmogorov's K41 theory of fully developed turbulence advances quantitative predictions on anomalous dissipation in incompressible fluids: although smooth solutions of the Euler equations conserve the energy, in a turbulent regime information is transferred to small scales and dissipation can happen even without the effect of viscosity, and it is rather due to the limited regularity of the solutions. In rigorous mathematical terms, however, very little is known. In a recent work in collaboration with M. Colombo and M. Sorella we consider the case of passive-scalar advection, where anomalous dissipation is predicted by the Obukhov-Corrsin theory of scalar turbulence. In my talk, I will present the general context and illustrate the main ideas behind our construction of a velocity field and a passive scalar exhibiting anomalous dissipation in the supercritical Obukhov-Corrsin regularity regime. I will also describe how the same techniques provide an example of lack of selection for passive-scalar advection under vanishing diffusivity, and an example of anomalous dissipation for the forced Euler equations in the supercritical Onsager regularity regime (this last result has been obtained in collaboration with E. Brue', M. Colombo, C. De Lellis, and M. Sorella).
Forcing is one of the key techniques in modern set theory. It is one of the main tools with which we study provability and independence. In this talk we will talk about what is set theory, how it came to be a foundation of mathematics (and what does that even mean?), as well as the basic ideas behind forcing, what it is, and what do people even research about it these days?
10 February 2023: Thomasina Ball (Warwick) —cancelled—
The idea of comparing the complexity of two equivalence relations on the reals by searching for a _Borel reduction_ from one to the other has transformed the last three decades of descriptive set theory. In addition to making precise the intuitive notion that some classification problems are "harder" than others, it has inspired fruitful connections with other areas of math such as dynamical systems and combinatorics.
We give an introduction to this theory, working through some specific examples and sampling some recent developments. Particular attention will be paid to how group- and graph-theoretic notions manifest in the Borel reducibility hierarchy.
24 February 2023: Steven Tobias (Leeds) "So Many DynamoS": Some interesting mathematical problems in dynamo theory
The generation of magnetic field in the Earth's interior and the origin of the eleven year solar cycle are both thought to lie in hydromagnetic dynamo action. In both cases fluid motions interact with rotation to sustain electrical currents and hence magnetic field. In this talk I will give an introduction to the mathematical structure of the equations for the generation of magnetic field and highlight some interesting unsolved mathematical problems. I'll conclude by drawing an analogy with transition to turbulence in flow down a pipe.
3 March 2023: Olivia Caramello (University of Insubria, Como) Grothendieck toposes as unifying 'bridges' in Mathematics
I will explain the sense in which Grothendieck toposes can act as unifying 'bridges' for relating different mathematical theories to each other and studying them from a multiplicity of points of view. I shall first present the general techniques underpinning this theory and then discuss a number of selected applications in different mathematical fields.
10 March 2023: Matej Balog (DeepMind London) Discovering faster matrix multiplication algorithms with reinforcement learning
Improving the efficiency of algorithms for fundamental computational tasks such as matrix multiplication can have widespread impact, as it affects the overall speed of a large amount of computations. Automatic discovery of algorithms using machine learning offers the prospect of reaching beyond human intuition and outperforming the current best human-designed algorithms. In this talk I’ll present AlphaTensor, our learned agent for discovering efficient and provably correct algorithms for the multiplication of arbitrary matrices. AlphaTensor discovered algorithms that outperform the state-of-the-art complexity for many matrix sizes. Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor’s algorithm improves on Strassen’s two-level algorithm for the first time since its discovery 50 years ago. In this talk I’ll present the mathematical problem underlying algorithm discovery for matrix multiplication, and our formulation of this problem as a single-player game. Then I'll describe general ingredients for tackling mathematical problems using machine learning techniques, and show how these ingredients enable AlphaTensor to learn to play this challenging game well, and thereby to discover faster algorithms for matrix multiplication.
In engineering applications (heat transfer, fluid flow, elasticity etc), we often encounter physics-based models consisting of partial differential equations (PDEs) with uncertain inputs, which are reformulated as so-called parametric PDEs. Given a probability distribution for the inputs, the forward UQ problem consists of trying to estimate statistical quantities of interest related to the model solution. Conversely, given (usually, noisy) data relating to the model solution, the inverse UQ problem consists of trying to infer the uncertain inputs themselves.
Over the last two decades a myriad of numerical schemes have been developed to tackle the forward and inverse UQ problems for PDE models. The vast majority of these are sampling schemes and are non-intrusive in the sense that users do not have to modify existing solvers and codes for the associated deterministic PDEs. This is very attractive in industrial settings. Stochastic Galerkin methods, also known as intrusive polynomial chaos methods, standard apart and are much less widely used in practise. In this talk, I will outline the intrusive approach, its advantages and limitations, and explain why, if properly implemented, it sometimes offers advantages over sampling methods.
In the study of a discrete dynamical system defined by polynomials, we hope as a starting point to understand the growth of the degrees of the iterates of the map. This growth is measured by the dynamical degree, an invariant which controls the topological, arithmetic, and algebraic complexity of the system. I will discuss the history of this question and the recent surprising construction, joint with Bell, Diller, and Jonsson, of a transcendental dynamical degree for an invertible map of this type, and how our work fits into the general phenomenon of power series taking transcendental values at algebraic inputs.
Constructive mathematicians and computer scientists have long been interested in logical theories in which all mathematical statements have computational content. In such systems, any proof of the existence of some natural number automatically gives an algorithm for computing the number. Most modern computer "proof assistants", that is, programs aimed at helping the user construct and verify the correctness of mathematical statements, are based on a class of such systems call *type theories*.
Around 15 years ago, however, it was discovered that the way type theories represent equality meant that, rather than describing constructive *sets*, these systems should more properly be thought of as describing constructive *homotopy types*. This has led to a number of new connections between homotopy theory, higher category theory, computer science and logic. In this talk, I will describe some of these ideas and the results that they have led to.
The theory of quiver representations plays an important role in algebraic geometry and in geometric representation theory. The aim of this talk is to describe some of this theory and to give a glimpse of an extension to 'higher quivers' (which are, very roughly speaking, to quivers as higher categories are to categories), concentrating on 2-quivers.
Starting from the humble set of 2x2 matrices over the complex numbers we will build up to the world of Lie algebras, showing that 2x2 matrices are still running the show. We will then discuss analogous results as we move away from the complex numbers. The final stop is ongoing work (joint with David Stewart, Manchester) in the characteristic 2 case where we will see a link to the game 'Lights Out'.
In this talk I will discuss two instabilities present in thin films that depend strongly on the rheology of the fluid. I will focus on viscoplastic fluids which have both fluid- and solid-like properties (e.g. toothpaste). The first instability is the viscoplastic version of the well-studied Saffman-Taylor instability where the fluid film deformation is dominated by shear. The second is a recently proposed two-dimensional extensional instability. I will present some theoretical and experimental results showing some surprising results when trying to observe these instabilities in the lab.
2 June 2023, Room B3.03: James Maynard (Oxford) Primes and patterns of zeros of the Riemann zeta function
The Riemann Hypothesis is one of the most famous open problems in mathematics, and if proven it would have amazing consequences for our understanding of prime numbers. It turns out many of these applications to primes would still be true even if the Riemann Hypothesis was false, provided counterexamples to the Riemann Hypothesis are suitably ‘rare’. I’ll give an accessible talk about this and some recent work showing how ‘patterns’ of zeros are related to our understanding of primes.
We will come at this question from two different angles: first, from the viewpoint of model theory, a subject in which for nearly half a century the notion of stability has played a central role in describing tame behaviour; secondly, from the perspective of combinatorics, where so-called regularity decompositions have enjoyed a similar level of prominence in a range of finitary settings, with remarkable applications including to patterns in the primes. In recent years, these two fundamental notions have been shown to interact in interesting ways. In particular, it has been shown that mathematical objects that are stable in the model-theoretic sense admit particularly well-behaved regularity decompositions. In this talk we will illustrate this fruitful interplay in the context of both finite graphs and subsets of abelian groups.
Counting the number of subgroups in a finite group has numerous applications, ranging from enumerating certain classes of finite graphs (up to isomorphism), to counting how many isomorphism classes of finite groups there are of a given order. In this talk, I will discuss the history behind the question; why it is important; and what we currently know.
Ramsey theory is a branch of combinatorics which seeks to find patterns in disorganized situations. One of its main achievements, Szemeredi’s theorem on arithmetic progressions, got an ergodic theoretic proof in 1977 when Furstenberg created a Correspondence Principle to encode combinatorial information about sets of integers into a dynamical system. Since then ergodic methods have been very successful in obtaining new Ramsey theoretic results, some of which still have no purely combinatorial proof.
I will survey some of the history of how ergodic theory and Ramsey theory are interconnected, leading to a recent result involving infinite sumsets.