Past

In this talk, we first review the de Rham complex and the finite element exterior calculus, a cohomological framework for structure-preserving discretisation of PDEs. From de Rham complexes, we derive other complexes with applications in elasticity, geometry and general relativity. The derivation, inspired by the Bernstein-Gelfand-Gelfand (BGG) construction, also provides a general machinery to establish results for tensor-valued problems (e.g., elasticity) from de Rham complexes (e.g., electromagnetism and fluid mechanics). We discuss some applications and progress in this direction, including mechanics models and the construction of bounded homotopy operators (Poincaré integrals) and finite elements.

In this presentation, I will discuss the construction and results of numerical simulations quantifying flows around several species of soft corals. In the first project, the flows near the tentacles of xeniid soft corals are quantified for the first time. Their active pulsations are thought to enhance their symbionts' photosynthetic rates by up to an order of magnitude. These polyps are approximately 1 cm in diameter and pulse at frequencies between approximately 0.5 and 1 Hz. As a result, the frequency-based Reynolds number calculated using the tentacle length and pulse frequency is on the order of 10 and rapidly decays as with distance from the polyp. This introduces the question of how these corals minimize the reversibility of the flow and bring in new volumes of fluid during each pulse. We estimate the Péclet number of the bulk flow generated by the coral as being on the order of 100–1000 whereas the flow between the bristles of the tentacles is on the order of 10. This illustrates the importance of advective transport in removing oxygen waste. In the second project, the flows through the elaborate branching structures of gorgonian colonies are considered.  As water moves through the elaborate branches, it is slowed, and recirculation zones can form downstream of the colony. At the smaller scale, individual polyps that emerge from the branches expand their tentacles, further slowing the flow. At the smallest scale, the tentacles are covered in tiny pinnules where exchange occurs. We quantified the gap to diameter ratios for various gorgonians at the scale of the branches, the polyp tentacles and the pinnules. We then used computational fluid dynamics to determine the flow patterns at all three levels of branching. We quantified the leakiness between the branches, tentacles and pinnules over the biologically relevant range of Reynolds numbers and gap-to-diameter ratios, and found that the branches and tentacles can act as either leaky rakes or solid plates depending upon these dimensionless parameters. The pinnules, in contrast, mostly impede the flow. Using an agent-based modeling framework, we quantified plankton capture as a function of the gap-to diameter ratio of the branches and the Reynolds number. We found that the capture rate depends critically on both morphology and Reynolds number. 

The Lott-Sturm-Villani curvature-dimension condition CD(K,N) provides a synthetic notion for a metric measure space to have curvature bounded from below by K and dimension bounded from above by N. It was proved by Juillet that the CD(K,N) condition is not satisfied in a large class of sub-Riemannian manifolds, for every choice of the parameters K and N. In a joint work with Tommaso Rossi, we extended this result to the setting of almost-Riemannian manifolds and finally it was proved in full generality by Rizzi and Stefani. In this talk I present the ideas behind the different strategies, discussing in particular their possible adaptation to the sub-Finsler setting. Lastly I show how studying the validity of the CD condition in sub-Finsler Carnot groups could help in proving rectifiability of CD spaces.

Mike will briefly describe the scope and shape of science within the Met Office and of his career there. He will also outline the coordination of the development of the NEMO ocean model, which he leads, and work to ensure the marine systems at the Met Office work efficiently on modern High Performance Computers (HPCs).  In the second half of the talk, Mike will focus on two of his current scientific interests: accurate calculation of horizontal pressure forces in models with steeply sloping coordinates; and dynamical interpretations of meridional overturning circulations and ocean heat uptake.

Mathematicians frequently present their own work in a diachronic fashion, e.g. by comparing their "modern" methods to those supposedly of the "Ancients," or by situating their latest theories as an "abstract" counterpart to more "classical" ones. The construction of such contrasts entangle mathematical labour and cultural life writ large. Indeed, it involves on the part of mathematicians the shaping up of correspondences between their technical achievements and intellectual discussions taking place on a much broader stage, such as those surrounding the concept of modernity, its relation to an imagined ancient past, or the characterisation of scientific progress as an increase in abstraction. This talk will explore the creation and use of such mathematical diachronies, the focus being on the works of Felix Klein, Hieronymus Zeuthen, and Hermann Schubert.

Stable commutator length (scl) is a measure of homological complexity in groups that has attracted attention for its various connections with geometric topology and group theory. In this talk, I will introduce scl and discuss the (hard) problem of computing scl in surface groups. I will present some results concerning isometric embeddings of free groups for scl, and how they generalise to surface groups for the relative Gromov seminorm.

The joint spectral radius of a finite family S of matrices measures the rate of exponential growth of the maximal norm of an element from the product set S^n as n grows. This notion was introduced by Rota and Strang in the 60s. It arises naturally in a number of contexts in pure and applied mathematics. I will discuss its basic properties and focus on a formula of Berger and Wang and results of J. Bochi that extend to several matrices the classical for formula of Gelfand that relates the growth rate of the powers of a single matrix to its spectral radius. I give new proofs and derive explicit estimates with polynomial dependence on the dimension, refining these results. If time permits I will also discuss connections with the Tits alternative, the notion of joint spectrum, and a geometric version of these results regarding groups acting on non-positively curved spaces.

In 1977, Milnor formulated the following conjecture: every discrete group of affine transformations acting properly on the affine space is virtually solvable. We now know that this statement is false; the current goal is to gain a better understanding of the counterexamples to this conjecture. Every group that violates this conjecture "lives" in a certain algebraic affine group, which can be specified by giving a linear group and a representation thereof. Representations that give rise to counterexamples are said to be non-Milnor. We will talk about the progress made so far towards classification of these non-Milnor representations.

At the heart of all quasi-Newton methods is an update rule that enables us to gradually improve the Hessian approximation using the already available gradient evaluations. Theoretical results show that the global performance of optimization algorithms can be improved with higher-order derivatives. This motivates an investigation of generalizations of quasi-Newton update rules to obtain for example third derivatives (which are tensors) from Hessian evaluations. Our generalization is based on the observation that quasi-Newton updates are least-change updates satisfying the secant equation, with different methods using different norms to measure the size of the change. We present a full characterization for least-change updates in weighted Frobenius norms (satisfying an analogue of the secant equation) for derivatives of arbitrary order. Moreover, we establish convergence of the approximations to the true derivative under standard assumptions and explore the quality of the generated approximations in numerical experiments.

Sketch-and-precondition techniques are popular for solving large overdetermined least squares (LS) problems. This is when a right preconditioner is constructed from a randomized 'sketch' of the matrix. In this talk, we will see that the sketch-and-precondition technique is not numerically stable for ill-conditioned LS problems. We propose a modifciation: using an unpreconditioned iterative LS solver on the preconditioned matrix. Provided the condition number of A is smaller than the reciprocal of the unit round-off, we show that this modification ensures that the computed solution has a comparable backward error to the iterative LS solver applied to a well-conditioned matrix. Using smoothed analysis, we model floating-point rounding errors to provide a convincing argument that our modification is expected to compute a backward stable solution even for arbitrarily ill-conditioned LS problems.

A regular bipartite tournament is an orientation of a complete balanced bipartite graph $K_{2n,2n}$ where every vertex has its in- and outdegree both equal to $n$. In 1981, Jackson conjectured that any regular bipartite tournament can be decomposed into Hamilton cycles. We prove this conjecture for sufficiently large $n$. Along the way, we also prove several further results, including a conjecture of Liebenau and Pehova on Hamilton decompositions of dense bipartite digraphs.

(jt work with David Cushing and George Stagg)

Prolog is a rather unusual programming language that was developed by Alain Colmerauer 50 years ago in one of the buildings on the way to the CIRM in Luminy. It is a declarative language that operates on a paradigm of first-order logic -- as distinct from imperative languages like C, GAP and Magma. Prolog operates by loading in a list of axioms as input, and then responds at the command line to queries that ask the language to achieve particular goals, given those axioms. It gained some notoriety through IBM’s implementation of ‘Watson’, which was a system designed to play the game show Jeopardy. Through a very efficiently implemented constraint logic programming module, it is also the worlds fastest sudoku solver. However, it has had barely any serious employment by pure mathematicians. So the aim of this talk is to advertise Prolog through an extended example: my co-authors and I used it to search for new simple Lie algebras over the field GF(2) and were able to classify a certain flavour of absolutely simple Lie algebra in dimensions 15 and 31, discovering a dozen or so new examples. With some further examples in dimension 63, we then extrapolated two previously undocumented infinite families of simple Lie algebras.

Shortwave radiation penetrating beneath an ice-sheet surface can cause internal melting and the formation of a near-surface porous layer known as the weathering crust, a dynamic hydrological system that provides home to impurities and microbial life. We develop a mathematical model, incorporating thermodynamics and population dynamics, for the evolution of such layers. The model accounts for conservation of mass and energy, for internal and surface-absorbed radiation, and for logistic growth of a microbial species mediated by nutrients that are sourced from the melting ice. I will discuss one-dimensional steadily melting solutions of the model, which suggest a range of changes in behaviour of the weathering crust and its microbial community in response to climate change. In addition, time-dependent solutions of the model give insight into the formation and removal of the weathering crust in frequently changing weather conditions.

Factorization homology is an arguably abstract formalism which produces
well-behaved topological invariants out of certain "higher algebraic"
structures. In this talk, I'll explain how this formalism can be made
fairly concrete in the case where this input algebraic structure is a
braided tensor category. If the category at hand is semi-simple, this in
fact essentially recovers skein categories and skein algebras. I'll
present various applications of this formalism to quantum topology and
representation theory.
 

Given a finite graph, the arboreal gas is the measure on forests (subgraphs without cycles) in which each edge is weighted by a parameter β greater than 0. Equivalently this model is bond percolation conditioned to be a forest, the independent sets of the graphic matroid, or the q→0 limit of the random cluster representation of the q-state Potts model. Our results rely on the fact that this model is also the graphical representation of the nonlinear sigma model with target space the fermionic hyperbolic plane H^{0|2}, whose symmetry group is the supergroup OSp(1|2).

The main question we are interested in is whether the arboreal gas percolates, i.e., whether for a given β the forest has a connected component that includes a positive fraction of the total edges of the graph. We show that in two dimensions a Mermin-Wagner theorem associated with the OSp(1|2) symmetry of the nonlinear sigma model implies that the arboreal gas does not percolate for any β greater than 0. On the other hand, in three and higher dimensions, we show that percolation occurs for large β by proving that the OSp(1|2) symmetry of the non-linear sigma model is spontaneously broken. We also show that the broken symmetry is accompanied by massless fluctuations (Goldstone mode). This result is achieved by a renormalisation group analysis combined with Ward identities from the internal symmetry of the sigma model.

In this talk I'll discuss some applications of the mean curvature flow to self shrinkers in R^3 and R^4.

Sparse inversion and classification problems are ubiquitous in modern data science and imaging. They are often formulated as non-smooth minimisation problems. In sparse inversion, we minimise, e.g., the sum of a data fidelity term and an L1/LASSO regulariser. In classification, we consider, e.g., the sum of a data fidelity term and a non-smooth Ginzburg--Landau energy. Standard (sub)gradient descent methods have shown to be inefficient when approaching such problems. Splitting techniques are much more useful: here, the target function is partitioned into a sum of two subtarget functions -- each of which can be efficiently optimised. Splitting proceeds by performing optimisation steps alternately with respect to each of the two subtarget functions.

In this work, we study splitting from a stochastic continuous-time perspective. Indeed, we define a differential inclusion that follows one of the two subtarget function's negative subdifferential at each point in time. The choice of the subtarget function is controlled by a binary continuous-time Markov process. The resulting dynamical system is a stochastic approximation of the underlying subgradient flow. We investigate this stochastic approximation for an L1-regularised sparse inversion flow and for a discrete Allen-Cahn equation minimising a Ginzburg--Landau energy. In both cases, we study the longtime behaviour of the stochastic dynamical system and its ability to approximate the underlying subgradient flow at any accuracy. We illustrate our theoretical findings in a simple sparse estimation problem and also in low- and high-dimensional classification problems.

5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled UV fixed points, whose realization hinges on string theoretic methods: they can be constructed by compactifying M-theory on local Calabi-Yau threefold singularities or alternatively from the world-volume of 5-brane-webs in type IIB string theory. There is a correspondence between 5-brane-webs and toric Calabi-Yau threefolds, however this breaks down when multiple 5-branes are allowed to end on a single 7-brane. In this talk, we extend this connection and provide a geometric realization of brane configurations including 7-branes. Along the way, we also review techniques developed in the past few years to describe the Higgs branch of these 5d SCFTs, including magnetic quivers and Hasse diagram for symplectic singularities. 

This session will introduce the opportunities for entrepreneurship and generating commercial impact available to researchers and students across MPLS. Representatives from the Maths Institute and across the university will discuss training and resources to help you begin enterprising and develop your ideas. We will hear from Paul Gass and Dawn Gordon about the support that can be provided by Oxford University Innovation, discussing commercialisation of research findings, consultancy, utilising your expertise and the protection and licensing of Intellectual Property. 

Please see below slides from the talk:

20230217 Short Seminar - Maths Fri@4_FINAL- Dept (1)_0.pdf

Introductory talk Maths 2023.pdf

Leadership and Innovation Presentation.pdf