Past

The synthetic theory of Ricci curvature lower bounds introduced more than 15 years ago by Lott-Sturm-Villani has been largely succesful in describing the geometry of metric measure spaces. However, this theory fails to include sub-Riemannian manifolds (an important class of metric spaces, the simplest example being the so-called Heisenberg group). Motivated by Villani's ``great unification'' program, in this talk we propose an extension of Lott-Sturm-Villani's theory, which includes sub-Riemannian geometry. This is a joint work with Barilari (Padua) and Mondino (Oxford). The talk is intended for a general audience, no previous knowledge of optimal transport or sub-Riemannian geometry is required.

The Balog-Szemerédi-Gowers theorem is a powerful tool in additive combinatorics, that allows one to roughly convert any “large energy” estimate into a “small sumset” estimate. This has found applications in a lot of results in additive combinatorics and other areas. In this talk, we will provide a friendly introduction and overview of this result, and then discuss some proof ideas. No hardcore additive combinatorics pre-requisites will be assumed.

In this talk, we consider differential equations perturbed by multiplicative fractional Brownian noise. Depending on the value of the Hurst parameter $H$, the resulting equation is pathwise viewed as an ordinary ($H>1$), Young  ($H \in (1/2, 1)$) or rough  ($H \in (1/3, 1/2)$) differential equation. In all three regimes we show regularisation by noise phenomena by proving the strongest kind of well-posedness  for equations with irregular drifts: strong existence and path-by-path uniqueness. In the Young and smooth regime $H>1/2$ the condition on the drift coefficient is optimal in the sense that it agrees with the one known for the additive case.

In the rough regime $H\in(1/3,1/2)$ we assume positive but arbitrarily small drift regularity for strong 
well-posedness, while for distributional drift we obtain weak existence. 

This is a joint work with Máté Gerencsér.

I will present classification results for 4-manifolds with boundary and infinite cyclic fundamental group, obtained in joint work with Anthony Conway and with Conway and Lisa Piccirillo.  Time permitting, I will describe applications to knotted surfaces in simply connected 4-manifolds, and to investigating the difference between the relations of homotopy equivalence and stable homeomorphism. These will also draw on work with Patrick Orson and with Conway,  Diarmuid Crowley, and Joerg Sixt.

We study the problem of determining when a compact group can be realized as the group of isometries of a norm on a finite dimensional real vector space.  This problem turns out to be difficult to solve in full generality, but we manage to understand the connected groups that arise as connected components of isometry groups. The classification we obtain is related to transitive actions on spheres (Borel, Montgomery-Samelson) on the one hand and to prehomogeneous spaces (Vinberg, Sato-Kimura) on the other. (joint work with Martin Liebeck, Assaf Naor and Aluna Rizzoli)

According to the correspondence principle, classical physics emerges in the limit of large quantum numbers. We examine three examples of the semiclassical description of conformal field theory data: large charge boundary operators in the O(2) model, large spin impurities in the free triplet scalar field theory and large charge Wilson lines in QED. By simultaneously taking the coupling to zero and quantum numbers to infinity, we can connect the microscopic to the emergent classical description smoothly.

Abstract: 

Welcome (back) to Fridays@4! To start the new academic year in this session we’ll introduce what Fridays@4 is for our new students and colleagues. This session will be a chance to meet current students and ECRs from across Maths and Stats who will share their hints and tips on conducting successful research in Oxford. There will be lots of time for questions, discussions and generally meeting more people across the two departments – everyone is welcome!

Discrete structures have a long history of use in applied mathematics. Graphs and hypergraphs provide models of social networks, biological systems, academic collaborations, and much more. Network science, and more recently hypernetwork science, have been used to great effect in analyzing these types of discrete structures. Separately, the field of applied topology has gathered many successes through the development of persistent homology, mapper, sheaves, and other concepts. Recent work by our group has focused on the convergence of these two areas, developing and applying topological concepts to study discrete structures that model real data.

This talk will survey our body of work in this area showing our work in both the theoretical and applied spaces. Theory topics will include an introduction to hypernetwork science and its relation to traditional network science, topological interpretations of graphs and hypergraphs, and dynamics of topology and network structures. I will show examples of how we are applying each of these concepts to real data sets.

Cell fate decision-making is responsible for development and homeostasis, and is dysregulated in disease. Despite great promise, we are yet to harness the high-resolution cell state information that is offered by single-cell genomics data to understand cell fate decision-making as it is controlled by gene regulatory networks. We describe how we leveraged joint dynamics + genomics measurements in single cells to develop a new framework for single-cell-informed Bayesian parameter inference of Ca2+ pathway dynamics in single cells. This work reveals a mapping from transcriptional state to dynamic cell fate. But no cell is an island: cell-internal gene regulatory dynamics act in concert with external signals to control cell fate. We developed a multiscale model to study the effects of cell-cell communication on gene regulatory network dynamics controlling cell fates in hematopoiesis. Specifically, we couple cell-internal ODE models with a cell signaling model defined by a Poisson process. We discovered a profound role for cell-cell communication in controlling the fates of single cells, and show how our results resolve a controversy in the literature regarding hematopoietic stem cell differentiation. Overall, we argue for the need to consider single-cell-resolved models to understand and predict the fates of cells.

Theory of mean-field games (MFGs) has recently experienced an exponential growth. Existing analytical approaches to find Nash equilibrium (NE) solutions for MFGs are, however, by and large restricted to contractive or monotone settings, or rely on the uniqueness of the NE. We propose a new mathematical paradigm to analyze discrete-time MFGs without any of these restrictions. The key idea is to reformulate the problem of finding NE solutions in MFGs as solving an equivalent optimization problem, called MF-OMO (Mean-Field Occupation Measure Optimization), with bounded variables and trivial convex constraints. It is built on the classical work of reformulating a Markov decision process as a linear program, and by adding the consistency constraint for MFGs in terms of occupation measures, and by exploiting the complementarity structure of the linear program. This equivalence framework enables finding multiple (and possibly all) NE solutions of MFGs by standard algorithms such as projected gradient descent, and with convergence guarantees under appropriate conditions. In particular, analyzing MFGs with linear rewards and with mean-field independent dynamics is reduced to solving a finite number of linear programs, hence solvable in finite time. This optimization reformulation of MFGs can be extended to variants of MFGs such as personalized MFGs.

For positive integers $k$ and $n$ let $\sigma_k(n)$ denote the sum of the $k$th powers of the divisors of $n$. Erd\H{o}s and Kac conjectured that, for every $k$, the number $\alpha_k = \sum_{n\geq 1} \frac{\sigma_k(n)}{n!}$ is irrational. This is known conditionally for all $k$ assuming difficult conjectures like the Hardy-Littlewood prime $k$-tuples conjecture. Before our work it was known unconditionally that $\alpha_k$ is irrational if $k\leq 3$. We discuss some of the ideas in our recent proof that $\alpha_4$ is irrational. The proof involves sieve methods and exponential sum estimates.

I will discuss the following statement, a definable version of the (p,q) theorem of Jiří Matoušek from combinatorics, conjectured by Chernikov and Simon:

Suppose that T is NIP and that phi(x,b) does not fork over a model M. Then there is some formula psi(y) in tp(b/M) such that the partial type {phi(x,b’) : psi(b’)} is consistent.

It has long been appreciated that in QCD-like theories without fundamental matter, confinement can be given a sharp characterization in terms of symmetry. More recently, such symmetries have been identified as 1-form symmetries, which fit into the broader category of generalized global symmetries.  In this talk I will discuss obstructions to the existence of a 1-form symmetry in large N QCD, where confinement is a sharp notion. I give general arguments for this disconnect between 1-form symmetries and confinement, and use 2d scalar QCD on the lattice as an explicit example.  

Hilbert complexes are chains of spaces linked by operators, with properties that are crucial to establishing the well-posedness of certain systems of partial differential equations. Designing stable numerical schemes for such systems, without resorting to nonphysical stabilisation processes, requires reproducing the complex properties at the discrete level. Finite-element complexes have been extensively developed since the late 2000's, in particular by Arnold, Falk, Winther and collaborators. These are however limited to certain types of meshes (mostly, tetrahedral and hexahedral meshes), which limits options for, e.g., local mesh refinement.

In this talk we will introduce the Discrete De Rham complex, a discrete version of one of the most popular complexes of differential operators (involving the gradient, curl and divergence), that can be applied on meshes consisting of generic polytopes. We will use a simple magnetostatic model to motivate the need for (continuous and discrete) complexes, then give a presentation of the lowest-order version of the complex and sketch its links with the CW cochain complex on the mesh. We will then briefly explain how this lowest-order version is naturally extended to an arbitrary-order version, and briefly present the associated properties (Poincaré inequalities, primal and adjoint consistency, commutation properties, etc.) that enable the analysis of schemes based on this complex.

Come along for free pizza and to hear about the Mathematrix events this term.

We present a framework to generate and evaluate thematic recommendations based on multilayer network representations of knowledge graphs (KGs).  We represent the relative importance of different types of connections (e.g., Directing/acting) with an intuitive salience matrix that can be learnt from data, tuned to incorporate domain knowledge, address different use cases, or respect business logic. We apply an adaptation of the personalised PageRank algorithm to multilayer network models of KGs to generate item-item recommendations. These recommendations reflect the knowledge we hold about the content, and are suitable for thematic or cold-start settings.

Evaluating thematic recommendations from user data presents unique challenges. Our method only recommends items that are 'thematically' related; that is, easily reachable following connections in the KG. We develop a variant of the widely-used Normalised Discounted Cumulative Gain (NDCG) to evaluate recommendations based on user-item ratings, respecting their thematic nature.

We apply our methods to a KG of the movie industry and MovieLens ratings and in an internal AB test. We learn the salience matrix and demonstrate that our approach outperforms existing thematic recommendation methods and is competitive with collaborative filtering approaches.

Profinite rigidity is essentially the study of which groups can be distinguished from each other by their finite quotients. This talk is meant to give a gentle introduction to the area - I will explain which questions are the right ones to ask and give an overview of some of the main results in the field. I will assume knowledge of what a group presentation is.

In this talk, I will discuss the notion of quantum limits from different viewpoints: Cordes' work on the Gelfand theory for pseudo-differential operators dating from the 70’s as well as the micro-local defect measures and semi-classical measures of the 90’s. I will also explain my motivation and strategy to obtain similar notions in subRiemannian or subelliptic settings. 

I will describe the interaction between a single soliton and a gas of solitons, providing for the first time a mathematical justification for the kinetic theory as posited by Zakharov in the 1970s.  Then I will explain how to use random matrix theory to introduce randomness into a large collection of solitons.

We formulate and sketch the proof of the K-theoretic Farrell-Jones Conjecture for
for the Hecke algebras of reductive p-adic groups. This is the first time that
a version of the farrell-Jones Conjecture for topological groups is formulated. It implies that
the reductive projective class group of the Hecke algebra of a reductive p-adic group
is the colimit of these for all compact open subgroups. This has been proved rationally by
Bernstein and Dat using representation theory. The main applications of our result
will concern the theory of smooth representations
In particular we will prove a conjecture of Dat.

The proof is much more involved than the one for instance for discrete CAT(0)-groups.
We will only give a very brief sketch of it and the new problems occurring in the setting of
totally disconnected groups. Most of the talk will be devoted
an introduction to the Farrell-Jones Conjecture and the theory of
smooth representations of reductive p-adic groups, and
discussion of  applications.

This is a joint project with Arthur Bartels.

An occupational hazard of mathematicians is the investigation of objects that are "optimal" in a mathematically precise sense, yet may be far from optimal in practice. This talk will discuss an extreme example of this effect: Gauss-Hermite quadrature on the real line. For large numbers of quadrature points, Gauss-Hermite quadrature is a very poor method of integration, much less efficient than simply truncating the interval and applying Gauss-Legendre quadrature or the periodic trapezoidal rule. We will present a theorem quantifying this difference and explain where the standard notion of optimality has failed.

Let F be a p-adic field, and k an algebraically closed field of characteristic l different from p. In this talk, we will first give a category decomposition of Rep_k(SL_n(F)), the category of smooth k-representations of SL_n(F), with respect to the GL_n(F)-equivalent supercuspidal classes of SL_n(F), which is not always a block decomposition in general. We then give a block decomposition of the supercuspidal subcategory, by introducing a partition on each GL_n(F)-equivalent supercuspidal class through type theory, and we interpret this partition by the sense of l-blocks of finite groups. We give an example where a block of Rep_k(SL_2(F)) is defined with several SL_2(F)-equivalent supercuspidal classes, which is different from the case where l is zero. We end this talk by giving a prediction on the block decomposition of Rep_k(A) for a general p-adic group A.

In this talk we explore structural results about sets with small doubling in k dimensions. We start in the continuous world with a sharp stability result for the Brunn-Minkowski inequality conjectured by Figalli and Jerison and work our way to the discrete world, where we discuss the natural extension: we show that non-degenerate sets in Z^k with doubling close to 2^k are close to convex progressions i.e. convex sets intersected with a sub-lattice. This talk is based on joint work with Peter van Hintum and Hunter Spink.

Starting from the principle of locality in quantum field theory, which
states that an object is influenced directly only by its immediate

surroundings, I will first briefly review some features of the notion of
locality arising in physics and mathematics. These are then encoded
in  locality relations, given by symmetric binary relations whose graph
consists of pairs of "mutually independent elements".

I will mention challenging questions that arise from  enhancing algebraic
structures to their locality counterparts, such as i) when  is the quotient
of a locality vector space by a linear subspace, a locality vector space, if
equipped with the quotient locality relation,  ii) when does  the locality
tensor product of two locality vector spaces  define a locality vector
space. These are discussed in recent joint work  with Pierre Clavier, Loïc
Foissy and Diego López.

Locality morphisms, namely maps that factorise on   products of  pairs of
"mutually independent" elements, play a key role in the context of
renormalisation in
multiple variables. They include "locality evaluators", which we use to

consistently evaluate meromorphic germs in several variables at
their poles. I will  also report on recent joint work with Li Guo and Bin
Zhang. which gives a classification of locality evaluators on certain
classes of algebras of meromorphic germs.