Past

This will be a gentle introduction to the theory of pseudo-Anosov  flows on 3-manifolds, as seen from the perspective of a topologist and not a dynamicist.

I will start by considering geodesic flows on the unit tangent bundles of hyperbolic surfaces. This will lead to a definition of an Anosov and then a pseudo-Anosov flow on a 3-manifold. After discussing a couple of examples, I will outline some connections between pseudo-Anosov flows and other aspects of 3-manifold topology/ geometry/ group theory.

The notion of topological order was introduced by Xiao-Gang Wen in order to capture the features of the exotic phases of matter given by fractional quantum Hall phases. I will motivate why the corresponding mathematical structures are higher categories with additional properties. In 2+1-dimensions, I will explain in details how the definition of fusion category arises from physical and geometrical intuitions about topological orders. Finally, I will sketch how the notion of higher fusion category emerges in higher dimensions.

In this talk we will discuss the correlations of the Riemann Zeta in various ranges, and prove a new result for correlations of squares. This problem is closely related to correlations of the characteristic polynomial of CUE with a very subtle difference. We will explain where this difference comes from, and what it means for the moments of moments of the Riemann Zeta, and its maximum in short intervals.

We develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut, resulting in a compact cut space which is needed for localisation. We introduce the notion of a moment polyptych associated to a hypertoric variety and prove that the necessary isotropy data can be read off from it. Finally, the equivariant Hirzebruch-Riemann-Roch formula is applied to the cut spaces and expresses the dimension of the equivariant quantisation space as a finite sum over the fixed-points. This is joint work with Johan Martens.

We prove a positive characteristic analogue of the classical result that the centralizer of a nonconstant differential operator in one variable is commutative. This leads to a new, short proof of that classical characteristic zero result, by reduction modulo p. This is joint work with Justin Desrochers available at https://arxiv.org/abs/2201.04606.

Since their introduction in 1928 by Paul A. Dirac, Dirac operators have been playing essential roles in many areas of Physics and Mathematics. In particular, they provide powerful and efficient tools to clarify (and sometimes solve) important problems in representation theory of real Lie groups, p-adic groups or Hecke algebras, such as classification, unitarity and geometric realization. In this representation theoretic context, the Dirac index of a Harish-Chandra module is a virtual module induced by Vogan’s Dirac cohomology. Once we observe that Dirac index commutes with translation functors, we will associate a polynomial (on a Cartan subalgebra) with a coherent family of Harish-Chandra modules. Then we shall explain how this polynomial can be used to connect nilpotent orbits, associated cycles and the leading term of the Taylor expansion of the characters of Harish-Chandra modules. This is joint wok with P. Pandzic, D. Vogan and R. Zierau.
 

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue coloring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Trujić showed that if $H$ is a graph on $n$ vertices and maximum degree three, then $\hat{r}(H) = O(n^{8/5})$, improving upon the bound of $n^{5/3 + o(1)}$ by Kohayakawa, Rödl, Schacht and Szemerédi. In our paper, we show that $\hat{r}(H)\leq n^{3/2+o(1)}$. While the previously used host graphs were vanilla binomial random graphs, we prove our result by using a novel host graph construction.
We also discuss why our bound is a natural barrier for the existing methods.
This is joint work with Kalina Petrova.

Chimera states have attracted significant attention as symmetry-broken states exhibiting the coexistence of coherence and incoherence. Despite the valuable insights gained by analyzing specific systems, the understanding of the physical mechanism underlying the emergence of chimeras has been incomplete. In this presentation, I will argue that an important class of stable chimeras arise because coherence in part of the system is sustained by incoherence in the rest of the system. This mechanism may be regarded as a deterministic analog of noise-induced synchronization and is shown to underlie the emergence of so-called strong chimeras. These are chimera states whose coherent domain is formed by identically synchronized oscillators. The link between coherence and incoherence revealed by this mechanism also offers insights into the dynamics of a broader class of states, including switching chimera states and incoherence-mediated remote synchronization.

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

I will review the series of recent results with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris Sorbonne) concerning the description of implosion mechanisms for viscous three dimensional compressible fluids. I will explain how the problem is connected to the description of blow up mechanisms for classical super critical defocusing models. 

A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely understood notion of uniform exponential growth. We will see how considering acylindrical actions on hyperbolic spaces can help us, and give a particular application to mapping class groups.

 "In this talk, we will discuss invariant measures techniques to establish probabilistic global well-posedness for PDEs. We will go over the limitations that the Gibbs measures and the so-called fluctuation-dissipation measures encounter in the context of energy-supercritical PDEs. Then, we will present a new approach combining the two aforementioned methods and apply it to the energy supercritical Schrödinger equations. We will point out other applications as well."

For almost calibrated Lagrangian mean curvature flow it is known that all singularities are of Type II. To understand the finer structure of the singularities forming, it is thus necessary to understand the structure of general ancient solutions arising as potential limit flows at such singularities. We will discuss recent progress showing that ancient solutions with a blow-down a pair of static planes meeting along a 1-dimensional line are translators. This is joint work with J. Lotay and G. Szekelyhidi.

We study global 1- and (d−2)-form symmetries for gauge theories based on disconnected gauge groups which include charge conjugation. For pure gauge theories, the 1-form symmetries are shown to be non-invertible. In addition, being the gauge groups disconnected, the theories automatically have a Z2
global (d−2)-form symmetry. We propose String Theory embeddings for gauge theories based on these groups. Remarkably, they all automatically come with twist vortices which break the (d−2)-form global symmetry. 

Title: Topics on regularity theory for fully non-linear integro-differential equations

Abstract: We will focus on local and non-local Monge Ampere type equations, equations with deforming kernels and convex envelopes of functions with optimal special conditions. We discuss global solutions and their regularity properties.

Title: Quasilinear Conservative Collisional Transport in Kinetic Mean Field models

Abstract: We shall focus the on the interplay of nonlinear analysis  and numerical approximations to mean field models in particle physics where kinetic transport flows in momentum are strongly nonlinearly  modified by macroscopic quantities in classical or spectral density spaces. Two noteworthy models arise: the classical Fokker-Plank Landau dynamics as a low magnetized plasma regimes in the modeling of perturbative non-local high order terms. The other one corresponds to perturbation under strongly magnetized dynamics for fast electrons  in momentum space  give raise to a coupled system of classical kinetic diffusion processes described by the balance equations for electron probability density functions (electron pdf) coupled to the time dynamics on spectral energy waves  (quasi-particles) in a quantum process of their resonant interaction. Both models are rather different, yet there are derived form the Liouville-Maxwell system under different scaling. Analytical tools and some numerical  simulations show a presence of  strong hot tail anisotropy  formation taking the stationary states away from Classical equilibrium solutions stabilization for the iteration in a three dimensional cylindrical model. The semi-discrete schemes preserves the total system mass, momentum and energy, which are enforced by the numerical scheme. Error estimates can be obtained as well.

Work in collaboration with Clark Pennie and Kun Huang

Persistent homology is an important methodology from topological data analysis which adapts theory from algebraic topology to data settings and has been successfully implemented in many applications. It produces a statistical summary in the form of a persistence diagram, which captures the shape and size of the data. Despite its widespread use, persistent homology is simply impossible to implement when a dataset is very large. In this talk, I will address the problem of finding a representative persistence diagram for prohibitively large datasets. We adapt the classical statistical method of bootstrapping, namely, drawing and studying smaller multiple subsamples from the large dataset. We show that the mean of the persistence diagrams of subsamples—taken as a mean persistence measure computed from the subsamples—is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. Given the complex algebraic and geometric nature of persistent homology, we adapt the convexity and stability properties in the space of persistence diagrams together with random set theory to achieve our theoretical results for the general setting of point cloud data. We demonstrate our approach on simulated and real data, including an application of shape clustering on complex large-scale point cloud data.

This is joint work with Yueqi Cao (Imperial College London).

We rely on soil to support the crops on which we depend. Less obviously we also rely on soil for a host of 'free services' from which we benefit. For example, soil buffers the hydrological system greatly reducing the risk of flooding after heavy rain; soil contains very large quantities of carbon, which would otherwise be released into the atmosphere where it would contribute to climate change. Given its importance it is not surprising that soil, especially its interaction with plant roots, has been a focus of many researchers. However the complex and opaque nature of soil has always made it a difficult medium to study. 

In this talk I will show how we can build a state of the art image based model of the physical and chemical properties of soil and soil-root interactions, i.e., a quantitative, model of the rhizosphere based on fundamental scientific laws.
This will be realised by a combination of innovative, data rich fusion of structural and chemical imaging methods, integration of experimental efforts to both support and challenge modelling capabilities at the scale of underpinning bio-physical processes, and application of mathematically sound homogenisation/scale-up techniques to translate knowledge from rhizosphere to field scale. The specific science questions I will address with these techniques are: (1) how does the soil around the root, the rhizosphere, function and influence the soil ecosystems at multiple scales, (2) what is the role of root- soil interface micro morphology on plant nutrient uptake, (3) what is the effect of plant exuded mucilage on the soil morphology, mechanics and resulting field and ecosystem scale soil function and (4) how to translate this knowledge from the single root scale to root system, field and ecosystem scale in order to predict how the climate change, different soil management strategies and plant breeding will influence the soil fertility. 

Effectiveness of COVID-19 vaccines was first demonstrated in randomised trials, but many questions of vital importance to vaccination policies could only be addressed in subsequent observational studies. The pandemic led to a step change in the availability of population-level linked electronic health record data, analysed in privacy-protecting Trusted Research Environments, across the UK. I will discuss methodological approaches to estimating causal effects of COVID-19 vaccines, and their application in estimating vaccine effectiveness and quantifying waning vaccine effectiveness. I will present results from recent analyses using detailed linked data on up to 24 million people in the OpenSAFELY Trusted Research Environment, which was developed by the University of Oxford's Bennett Institute for Applied Data Science.

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

MHD equilibrium is an important topic for fusion (and other MHD applications). A tokamak, in principle, is a toroidally symmetric fusion device and so MHD equilibrium can be reduced to solving the time independent MHD equations in axisymmetry. This produces the Grad-Shafranov equation (a two dimensional, nonlinear PDE) which has been solved using various techniques in the fusion community including finite difference, finite elements and spectral methods. A similar PDE exists if there is a plasma column with helical symmetry. Non-axisymmetric plasmas do occur in tokamaks as a result of instabilities and applied fields. However, if there is no symmetry angle there is no PDE to be solved. The current workhorse for finding non-axisymmetric equilibria uses energy minimization to find the equilibrium. New approaches to this problem that can use state of the art techniques are desirable. The speaker has formulated a coupled set of PDEs for the non-axisymmetric MHD equilibrium problem assuming that flux surfaces are nested (i.e. there are no magnetic islands) and has written this in weak form to use finite element method to solve the equations. The questions are around whether there is an optimal way to try to formulate the problem for FEM and to couple the equations, what sort of elements to use, if other solution techniques would be better suited and so on.

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

The possibility of `tacit collusion', in which interactions across market-making algorithms lead to an outcome similar to collusion among market makers, has increasingly received regulatory scrutiny. 
    We model the interaction of market makers in a dealer market as a stochastic differential game of intensity control with partial information and study the resulting dynamics of bid-ask spreads. Competition among dealers is modeled as a Nash equilibrium, which we characterise in terms of a system of coupled Hamilton-Jacobi-Bellman (HJB) equations, while Pareto optima correspond to collusion. 
    Using a decentralized multi-agent deep reinforcement learning algorithm to model how competing market makers learn to adjust their quotes, we show how the interaction of market-making algorithms may lead to tacit collusion with spread levels strictly above the competitive equilibrium level, without any explicit sharing of information.