Past

Rivers choose their size and shape, and spontaneously organize into ramified networks. Yet, they are essentially a channelized flow of water that carries sediment. Based on laboratory experiments, field measurements and simple theory, we will investigate the basic mechanisms by which rivers form themselves, and carve the landscapes that surround us.

The overall goal of the Sherwood lab is to advance genomic and precision medicine applications through high-throughput, multi-disciplinary science. In this talk, I will review a suite of high-throughput genomic and cellular perturbation platforms using CRISPR-based genome editing that the lab has developed to improve our understanding of genetic disease, gene regulation, and genome editing outcomes.

This talk will focus on recent efforts using combined analysis of rare coding variants from the UK Biobank and genome-scale CRISPR-Cas9 knockout and activation screening to improve the identification of genes, coding variants, and non-coding variants whose alteration impacts serum LDL cholesterol (LDL-C) levels. Through these efforts, we show that dysfunction of the RAB10 vesicle transport pathway leads to hypercholesterolemia in humans and mice by impairing surface LDL receptor levels. Further, we demonstrate that loss of function of OTX2 leads to robust reduction in serum LDL-C levels in mice and humans by increasing cellular LDL-C uptake. Finally, we unveil an activity-normalized base editing screening framework to better understand the impacts of coding and non-coding variation on serum LDL-C levels, altogether providing a roadmap for further efforts to dissect complex human disease genetics.

Classical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow non-discrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.

Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups.

Zarankiewicz's problem for hypergraphs asks for upper bounds on the number of edges of a hypergraph that has no complete sub-hypergraphs of a given size. Let M be an o-minimal structure. Basit-Chernikov-Starchenko-Tao-Tran (2021) proved that the following are equivalent:

(1) "linear Zarankiewicz's bounds" hold for hypergraphs whose edge relation is induced by a fixed relation definable in M

(2) M does not define an infinite field.

We prove that the following are equivalent:

(1') linear Zarankiewicz bounds hold for sufficiently "distant" hypergraphs whose edge relation is induced by a fixed relation definable in M

(2') M does not define a full field (that is, one whose domain is the whole universe of M).

This is joint work (in progress) with Aris Papadopoulos.

After a brief introduction, I elucidate a technique, dubbed "bootstrap'', which generates an infinite family of D_p(G) theories, where for a given arbitrary group G and a parameter b, each theory in the same family has the same number of mass parameters, same number of marginal deformations, same 1-form symmetry, and same 2-group structure. This technique is utilized to establish the presence or absence of the 2-group symmetries in several classes of D_p(G) theories. I, then, argue that we found the presence of 2-group symmetries in a class of Argyres-Douglas theories, called D_p(USp(2N)), which can be realized by Z_2-twisted compactification of the 6d N=(2,0) of the D-type on a sphere with an irregular twisted puncture and a regular twisted full puncture. I will also discuss the 3d mirror theories of general D_p(USp(2N)) theories that serve as an important tool to study their flavor symmetry and Higgs branch.

For the first time, a method has become available for fast computation of near-best rational approximations on arbitrary sets in the real line or complex plane: the AAA algorithm (Nakatsukasa-Sète-T. 2018).  After a brief presentation of the algorithm this talk will focus on twenty demonstrations of the kinds of things we can do, all across applied mathematics, with a black-box rational approximation tool.
 

In this talk I will discuss our recent work on two problems. The first problem concerns with capillary rise between rough structures, a fundamental wetting phenomenon that is functionalised in biological organisms and prevalent in geological or man-made materials. Predicting the liquid rise height is more complex than currently considered in the literature because it is necessary to couple two wetting phenomena: capillary rise and hemiwicking. Experiments, simulations and analytic theory demonstrate how this coupling challenges our conventional understanding and intuitions of wetting and roughness. For example, the critical contact angle for hemiwicking becomes separation-dependent so that hemiwicking can vanish for even highly wetting liquids. The rise heights for perfectly wetting liquids can also be different in smooth and rough systems. The second problem concerns with droplets (or condensates) formed via a liquid-liquid phase separation process in biological cells. Despite the widespread importance of surface tension for the interactions between these droplets and other cellular components, there is currently no reliable technique for their measurement in live cells. To address this, we develop a high-throughput flicker spectroscopy technique. Applying it to a class of cellular droplets known as stress granules, we find their interface fluctuations cannot be described by surface tension alone. It is necessary to consider elastic bending deformation and a non-spherical base shape, suggesting that stress granules are viscoelastic droplets with a structured interface, rather than simple Newtonian liquids. Moreover, given the broad distributions of surface tension and bending rigidity observed, different types of stress granules can only be differentiated via large-scale surveys, which was not possible previously and our technique now enables.

The formulation of Mean Field Games (MFG) via partial differential equations typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we will present results on the analysis and numerical approximation of stationary second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we will propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We present results that guarantee the existence of unique weak solutions to the stationary MFG PDI under a monotonicity condition similar to one that has been considered previously by Lasry and Lions. Moreover, we will propose a monotone finite element discretization of the weak formulation of the MFG PDI, and present results that confirm the strong H^1-norm convergence of the approximations to the value function and strong L^q-norm convergence of the approximations to the density function. The performance of the numerical method will be illustrated in experiments featuring nonsmooth solutions. This talk is based on joint work with my supervisor Iain Smears.

We survey the theory of $\ell^2$-invariants, their applications in group theory and topology, and introduce a positive characteristic version of $\ell^2$-theory. We also discuss the Atiyah and Lück approximation conjectures, two of the central problems in this area.

Equivariant Jiang-Su stability is an important regularity property for group actions on C*-algebras.  In this talk, I will explain this property and how it arises naturally in the context of the classification of C*-algebras and their actions. Depending on the time, I will then explain a bit more about the nature of equivariant Jiang- Su stability and the kind of techniques that are used to study it, including a recent result of Gábor Szabó and myself establishing an equivalence with equivariant property Gamma under certain conditions.
 

Neural networks are the most practically successful class of models in modern machine learning, but there are considerable gaps in the current theoretical understanding of their properties and success. Several authors have applied models and tools from random matrix theory to shed light on a variety of aspects of neural network theory, however the genuine applicability and relevance of these results is in question. Most works rely on modelling assumptions to reduce large, complex matrices (such as the Hessians of neural networks) to something close to a well-understood canonical RMT ensemble to which all the sophisticated machinery of RMT can be applied to yield insights and results. There is experimental work, however, that appears to contradict these assumptions. In this talk, we will explore what can be derived about neural networks starting from RMT assumptions that are much more general than considered by prior work. Our main results start from justifiable assumptions on the local statistics of neural network Hessians and make predictions about their spectra than we can test experimentally on real-world neural networks. Overall, we will argue that familiar ideas from RMT universality are at work in the background, producing practical consequences for modern deep neural networks.

Consider a random group $\Gamma$ with $k$ generators and $r$ random relators of large length $N$. We study the geometry of the character variety of $\Gamma$ with values in $\SL(2,\C)$ or more generally any semisimple Lie group $G$. This is the moduli space of group homomorphisms from $\Gamma$ to $G$ up to conjugation. We are in particular able to determine its dimension, number of components and Galois group, with an excellent control on the probability of exceptions. The proofs use effective Chebotarev type theorems as well as new spectral gap bounds  for Cayley graphs of finite simple groups. They are also conditional on GRH. Joint work with Peter Varju and Oren Becker.

Harish-Chandra's Lefschetz principle suggests that representations of real and p-adic split reductive groups are closely related, even though the methods used to study these groups are quite different. The local Langlands correspondence (as formulated by Vogan) indicates that these representation theoretic relationships stem from geometric relationships between real and p-adic Langlands parameters. In this talk we will discuss how the geometric structure of real and p-adic Langlands parameters lead to functorial relationships between representations of real and p-adic groups. I will describe work in progress which applies this functoriality to the study of unitary representations and signatures of invariant hermitian forms for GL(n). The main result expresses signatures of invariant hermitian forms on graded affine Hecke algebra modules in terms of signature characters of Harish-Chandra modules, which are computable via the unitary algorithm for real reductive groups by Adams-van Leeuwen-Trapa-Vogan.

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states (i.e. present or not present) of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such models typically arise in renormalisation arguments applied to independent percolation models, or percolation models with finite range dependencies. A 1-independent model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 Balister and Bollobás defined $p_{\textrm{max}}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine, or give good bounds on, the value of $p_{\textrm{max}}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\textrm{max}}(\mathbb{Z}^n)\leq p_{\textrm{max}}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty}p_{\textrm{max}}(\mathbb{Z}^n)$.

In this talk we will present a significantly improved upper bound for this limit as well as improved upper and lower bounds for $p_{\textrm{max}}(\mathbb{Z}^2)$. We will also show that with high confidence we have $p_{\textrm{max}}(\mathbb{Z}^n)<p_{\textrm{max}}(\mathbb{Z}^2)$ for large $n$ and discuss some open problems concerning 1-independent models on other graphs.

This is joint work with Tom Johnston and Alex Scott.

Molecules can be broken apart with a high-powered laser or an electron beam. The position of charged fragments can then be detected on a screen. From the mass to charge ratio, the identity of the fragments can be determined. The covariance of two fragments then gives us the projection of a distribution related to the initial scattering distribution. We formulate the mathematical transformation from the scattering distribution to the covariance distribution obtained from experiments. We expand the scattering distribution in terms of basis functions to obtain a linear system for the coefficients, which we use to solve the inverse problem. Finally, we show the result of our method on three examples of test data, and also with experimental data.

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.

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.

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.

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.