We consider particles following a diffusion process in a multi-well potential and attracted by their barycenter (corresponding to the particle approximation of the Wasserstein flow of a suitable free energy). It is well-known that this process exhibits phase transitions: at high temperature, the mean-field limit has a single stationary solution, the N-particle system converges to equilibrium at a rate independent from N and propagation of chaos is uniform in time. At low temperature, there are several stationary solutions for the non-linear PDE, and the limit of the particle system as N and t go to infinity do not commute. We show that, in the presence of multiple stationary solutions, it is still possible to establish local convergence rates for initial conditions starting in some Wasserstein balls (this is a joint work with Julien Reygner). In terms of metastability for the particle system, we also show that for these initial conditions, the exit time of the empirical distribution from some neighborhood of a stationary solution is exponentially large with N and approximately follows an exponential distribution, and that propagation of chaos holds uniformly over times up to this expected exit time (hence, up to times which are exponentially large with N). Exactly at the critical temperature below which multiple equilibria appear, the situation is somewhat degenerate and we can get uniform in N convergence estimates, but polynomial instead of exponential.

The spectral gap (or bass note) of a closed hyperbolic surface is the smallest non-zero eigenvalue of its Laplacian. This invariant plays an important role in many parts of hyperbolic geometry. In this talk, I will speak about joint work with Will Hide on the question of which numbers can appear as spectral gaps of closed arithmetic hyperbolic surfaces.

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Adam Jones will explore the relationship between the category of smooth representations of a semisimple p-adic Lie group G and the module category over its associated pro-p Iwahori-Hecke algebra via the canonical invariance adjunction. This relationship is well understood in characteristic 0, in fact it yields a category equivalence equivalence, but in characteristic p it is very mysterious and largely defies understanding. We will explore methods of constructing an appropriate subcategory of Hecke modules which is well behaved under the adjunction, and which can be shown to contain all parabolic inductions. He will give examples of this yielding results when G has rank 1, and more recently when G = SL_3 in certain cases.

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Given a finite poset $P$ we ask how small a family of subsets of $[n]$ can be such that it does not contain an induced copy of the poset, but adding any other subset creates such a copy. This number is called the saturation number of $P$, denoted by $\operatorname{sat}^*(n,P)$. Despite the apparent similarity to the saturation for graphs, this notion is vastly different. For example, it has been shown that the saturation numbers exhibit a dichotomy: for any poset, the saturation number is either bounded, or at least $2 n^{1/2}$. In fact, it is believed that the saturation number is always bounded or exactly linear. In this talk we will be discussing the most recent advances in this field, with the focus on the diamond poset, whose saturation number was unknown until recently.

Joint with Sean Jaffe.

I'll discuss some of the history of the use of random matrices for studying the structure of operator algebras, starting with Voiculescu's notion free independence. We'll see that the original notions of convergence of random matrix models to certain infinite-dimensional operators is actually fairly weak, and discuss the more recent "strong convergence" phenomenon and its applications to C*-algebras. Finally, I'll touch upon some ongoing work, joint with A. Shiner and S. White, for continuing to use random matrix tools to prove structural properties of C*-algebras.

We present an adjoint method for optimization of the spatially inhomogeneous Boltzmann equation for rarefied gas dynamics. The adjoint method is derived using a "discretize then optimize" approach. Discretization (in time and velocity) is via the Direct Simulation Monte Carlo (DSMC) method, and adjoint equations are derived from an augmented Lagrangian.  The boundary conditions that are included in this analysis include spectral reflection, thermal reflection, and inflow boundary conditions. For thermal reflection, a "score function" is included as a statistical regularization. This is joint work with Yunan Yang (Cornell).

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Maximum likelihood estimation is a ubiquitous task in statistics and its applications. The task is: given some observations of a random variable, find the distribution(s) in your statistical model which best explains these observations. A modern perspective on this classical problem is to study the "likelihood geometry" of a statistical model. By focusing on models which have a polynomial parametrization, i.e., lie on an algebraic variety, this perspective brings in tools, algorithms and invariants from algebraic geometry and combinatorics.

In this talk, I will explain some of the key ideas in likelihood geometry and discuss its recent application to the study of likelihood asymptotics, i.e., understanding likelihood estimation for very large or very small observation counts. Agostini et al. showed that these asymptotics can be modeled and understood using tools from tropical geometry, and they used this to completely describe the asymptotics for linear models. In our work, we use the same approach to treat the class of log-linear models (also known as Gibbs distributions or maximum entropy models) and give a complete and combinatorial description of the likelihood asymptotics under some conditions.

This talk is based on joint work with Emma Boniface (UC Berkeley) and Serkan Hoşten (San Francisco SU), available at: https://epubs.siam.org/doi/full/10.1137/24M1656839

Linear discrete inverse problems arise in many areas of science and engineering, from medical imaging and geophysics to atmospheric modelling. Their numerical solution often relies on iterative algorithms, particularly Krylov subspace methods, that can efficiently handle large-scale, ill-posed systems. In many practical settings, however, exact computations of matrix–vector products, preconditioners, or right-hand sides are either infeasible or unnecessary, leading to inexact iterations. This talk explores the interplay between inexactness and the regularizing behaviour of Krylov subspace methods for inverse problems. We discuss how approximate computations influence the regularization effect inherent in early iterations, as well as  semiconvergence, and how controlled inexactness may be exploited to improve computational efficiency. The aim is to provide a broad perspective on recent insights and open questions at the interface of inverse problems, iterative solvers, and computational inexactness.

Pfaffian functions, and by extension Pfaffian and semi-Pfaffian sets, play a crucial role in various areas of mathematics, including o-minimal theory. Incidence combinatorics has recently experienced a surge of activity, fuelled by the introduction of the polynomial partitioning method of Guth and Katz. While traditionally restricted to simple geometric objects such as points and lines, focus has shifted towards incidence questions involving higher dimensional algebraic or semi-algebraic sets. We present a generalization of the polynomial partitioning method to semi-Pfaffian sets and illustrate how this leads to Pfaffian generalizations of classic results in incidence geometry, such as the Szemerédi-Trotter Theorem. Finally, we outline an application of semi-Pfaffian geometry and Khovanskii's bound to the robustness of neural networks.

This session will explore practical ways to manage your time effectively as a student. We’ll discuss how to find the right balance between revising and working on problem sheets, tools and strategies to help you plan your workload, and how to set realistic priorities. We’ll also talk about what kind of study balance makes sense over the Christmas break. Come along to pick up useful tips for staying organised, focused, and on top of your studies.

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

Microbial communities contain many evolving and interacting bacteria, which makes them complex systems that are difficult to understand and predict. We use theory – including game theory, agent-based modelling, ecological network theory and metabolic modelling - and combine this with experimental work to understand what it takes for bacteria to succeed in diverse communities. One way is to actively kill and inhibit competitors and we study the strategies that bacteria use in toxin-mediated warfare. We are now also using our approaches to understand the human gut microbiome and its key properties including ecological stability and the ability to resist invasion by pathogens (colonization resistance). Our ultimate goal is to both stabilise microbiome communities and remove problem species without the use of antibiotics.

As discovered by Perelman, the study of ancient Ricci flows which are $\kappa$-noncollapsed is a crucial prerequisite to understanding the singularity behaviour of more general Ricci flows. In dimension three, these so-called "$\kappa$-solutions" have been fully classified through the groundbreaking work of Brendle, Daskalopoulos, and Šešum. Their classification result can be extended to higher dimensions, but only for those Ricci flows that have uniformly positive isotropic curvature (PIC), as well as weakly-positive isotropic curvature of the second type (PIC2); it appears the classification result fails with only minor modifications to the curvature assumption. Indeed, with the alternative assumption of non-negative curvature operator, a rich variety of new examples emerge, as recently constructed by Buttsworth, Lai, and Haslhofer; Haslhofer himself has conjectured that this list of non-negatively curved $\kappa$-solutions is now exhaustive in dimension four. In this talk, we will discuss some recent progress towards resolving Haslhofer's conjecture, including a compactness result for non-negatively curved $\kappa$-solutions in dimension four, and a symmetry improvement result for bubble-sheet regions. This is joint work with Anusha Krishnan and Timothy Buttsworth. 

I will discuss a uniform waist inequality in codimension 2 for the family of finite covers of a Riemannian manifold whose fundamental group has Kazhdan‘s property T. I will describe a general strategy to prove waist inequalities based on a higher property T for Banach spaces. The general strategy can be implemented in codimension 2 but is conjectural in higher codimension. We speculate about the situation for lattices in semisimple Lie groups. Based on joint work with Uri Bader

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Your chance to ask Mathematrix DPhil students about the process of applying to PhD programs, including written stages and interviews! 

Wave energy has the theoretical potential to meet global electricity demand, but it remains less mature and less cost-competitive than wind or solar power. A key barrier is the absence of engineering convergence on an optimal wave energy converter (WEC) design. In this work, I demonstrate how geometry optimisation can deliver step-change improvements in WEC performance. I present methodology and results from optimisations of two types of WECs: an axisymmetric point-absorber WEC and a top-hinged WEC. I show how the two types need different optimisation frameworks due to the differing physics of how they make waves. For axisymmetric WECs, optimisation achieves a 69% reduction in surface area (a cost proxy) while preserving power capture and motion constraints. For top-hinged WECs, optimisation reduces the reaction moment (another cost proxy) by 35% with only a 12% decrease in power. These result show that geometry optimisation can substantially improve performance and reduce costs of WECs.

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Blood vessels are among the most vital structures in the human body, forming intricate networks that connect and support various organ systems. Remarkably, during early embryonic development—before any blood vessels are visible—their precursor cells are arranged in stereotypical patterns throughout the embryo. We hypothesize that these patterns guide the directional growth and fusion of precursor cells into hollow tubes formed from initially solid clusters. Further analysis of cells within these clusters reveals unique organization that may influence their differentiation into endothelial and blood cells. In this work, I revisit the problem of pattern formation through the lens of active matter physics, using both developing embryonic systems and in vitro cell culture models where similar patterns are observed during tissue budding. These different systems exhibit similar patterning behavior, driven by changes in cellular activity, adhesion and motility.

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WCMB, University of Oxford and University of Bristol

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