Past

A common challenge in persistent homology is choosing "good" representative cycles for homology classes in a way compatible with persistence. In this talk, we discuss a geometric framework for codimension-1 persistent homology that addresses this issue using Alexander duality.

For an embedded filtered simplicial complex, connected components of the complement induce cycle representatives for a homology basis. The evolution of these cycles along the filtration can be tracked via the merge tree of the complement and the elder rule. This leads to the notion of cycle progression barcodes, associating to each persistence interval a sequence of representative cycles evolving through the filtration.

Applying geometric functionals to these progressions produces generalized persistence landscapes, which extend classical persistence landscapes and allow geometric information about cycle representatives to be captured without fixing a single filtration value. This provides a way to distinguish data sets with similar persistent homology but different geometric structure.

Everything you have been taught about Turing patterns is wrong! (Well, not everything, but qualifying statements tend to weaken a punchy first sentence). Turing patterns are universally used to generate and understand patterns across a wide range of biological phenomena. They are wonderful to work with from a theoretical, simulation and application point of view. However, they have a paradoxical problem of being too easy to produce generally, whilst simultaneously being heavily dependent on the details. In this talk I demonstrate how to fix known problems such as small parameter regions and sensitivity, but then highlight a new set of issues that arise from usually overlooked issues, such as boundary conditions, initial conditions, and domain shape. Although we’ve been exploring Turing’s theory for longer than I’ve been alive, there’s still life in the old (spotty) dog yet.

Dr Anna Lisa Vari will talk about: 'The orbital structure of the Hill's problem'

Hill’s problem is a limiting case of the circular restricted gravitational three-body problem in which the mass ratio between the two massive bodies tends to zero, leaving a small region surrounding the secondary in which it remains gravitationally dominant. Originally formulated in terms of point masses, Hill’s problem may be modified to include a secondary of finite extent, thus providing a more realistic description of the dynamics internal to a stellar cluster orbiting within a host galaxy. By considering stellar energies above the cluster escape energy, we may investigate the dynamics that underpin the process of stellar escape from star clusters -- a topical issue in contemporary astrophysics. Specifically, we construct a self-consistent formulation of Hill’s problem using a tidally perturbed cluster model for the secondary body. The behaviour of energetically unbound stellar orbits within such a self-consistent problem, as characterised using Poincaré surfaces of section, is then numerically explored via a structure-preserving integrator, revealing a previously unknown bifurcation in the orbital structure.

Speaker Francesco Hrobat will talk about; 'Lanczos with compression for symmetric matrix Lyapunov equations'

Large-scale symmetric matrix Lyapunov equations arise in control theory, model order reduction, and the discretization of PDEs. State-of-the-art algorithms, such as standard and rational Krylov methods, aim to approximate the solution with a low-rank matrix. However, the standard polynomial Krylov method (also referred to as the Lanczos method) often converges slowly and faces a memory bottleneck as the dimension of the Lanczos basis grows. Conversely, rational Krylov alternatives, while effective for low-rank approximations, require the solution of expensive shifted linear systems involving a large coefficient matrix.

In this talk, I will present a low-memory variant of the Lanczos algorithm for solving symmetric Lyapunov equations. Our approach leverages a polynomial Krylov subspace while employing rational subspaces associated with small matrices to compress the Lanczos basis. This method accesses the large coefficient matrix exclusively through matrix-vector products and maintains fixed storage requirements. The resulting low-rank solution has a rank that is independent of the dimension of the underlying polynomial Krylov subspace.

Rare and extreme events are notoriously hard to handle in any complex stochastic system: They are simultaneously too rare to be reliably observable in experiments or numerics, but at the same time often too impactful to be ignored. Large deviation theory provides a classical way of dealing with events of extremely small probability, but generally only yields the exponential tail scaling of rare event probabilities. In this talk, I will discuss theory, and algorithms based upon it, that improve on this limitation, yielding sharp quantitative estimates of rare event probabilities from a single computation and without fitting parameters. Notably, these estimates require the computation of determinants of differential operators, which in relevant cases are not traceclass and require appropriate renormalization. We demonstrate that the Carleman--Fredholm operator determinant is the correct choice. Throughout, I will demonstrate the applicability of these methods to high-dimensional real-world systems, for example coming from atmosphere and ocean dynamics.

Arham Farid (MI EDI Officer) will join us to chat about current EDI initiatives and to hear our thoughts about ways EDI can improve in the Maths Institute.

Holographic renormalization provides a framework that makes the AdS/CFT correspondence computationally precise. It systematically resolves the divergences and ambiguities that arise when relating bulk gravitational actions to boundary correlation functions. In this seminar, I will review how correlation functions of a conformal field theory can be extracted from gravitational dynamics in asymptotically AdS spacetimes using this method. I will explain how divergences of the on-shell bulk action near the AdS boundary reflect ultraviolet divergences in the dual field theory, and how these are removed by introducing covariant boundary counterterms. The resulting renormalized action generates well-defined one- and two-point functions, while bulk interactions are encoded in Witten diagrams that compute higher-point correlators.

I will present a construction and characterization of the (massive) sine-Gordon EQFT up to 6π in the full space.  The construction relies on a systematic study of the renormalization flow equation and a forward backward stochastic differential equation (FBSDE) which give good control of the EQFT and allows to derive various additional properties.

This is based on joint work with Massimiliano Gubinelli.

We introduce equivariant bivariant K-theory for bornological algebras by taking a presentable refinement of the bivariant K-theory of Lafforgue and Paravicini. An upshot of this refinement is that we may purely formally define a Bost-Connes assembly map via localisation in the sense of Meyer-Nest. Another feature built into the refinement is a large UCT-class; on this UCT-class, we show that the rationalised Chern-Connes character from KK-theory to local cyclic homology is an equivalence. This is joint work with Anupam Datta.

The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we outline our program to prove the conjecture that symplectic 4-manifolds with $b^+>1$ obey the Bogomolov-Miyaoka-Yau inequality. Our method uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles, where the Morse function is a Hamiltonian for a natural circle action and natural two-form.  We shall describe generalizations of Donaldson’s symplectic subspace criterion (1996) from finite to infinite dimensions. These generalized symplectic subspace criteria can be used to show that the natural two-form is non-degenerate and thus an almost symplectic form on the moduli space of non-Abelian monopoles. This talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789  (to appear in AMS Mathematical Surveys and Monographs), https://arxiv.org/abs/2206.14710 and https://arxiv.org/abs/2410.13809. 

In 1984 Cannon showed that cocompact discrete hyperbolic groups have finitely many cone types. In this talk, I will demonstrate how this result can be extended to non-positively curved k-fold triangle groups. I will further show how this implies that such groups have an automatic structure and how we can use this information to construct top dimensional l^2 cycles.

Colour Refinement is a combinatorial method that distinguishes vertices in graphs based on their local neighborhood structure. By encoding these local properties into vertex colours that are refined iteratively, the process eventually stabilises into a final colouring which serves as an isomorphism test on a large class of graphs.

The central complexity parameter of the algorithm is the number of iterations required to reach stabilisation. For $n$-vertex graphs, the upper bound is $n−1$. We call graphs that attain this maximum long-refinement graphs. Their final colourings are discrete, meaning every vertex is uniquely identified by its colour.  For a long time, it was not clear whether such graphs actually exist. My talk provides an overview of the history of this graph class and reports on recent work towards a full characterisation of it.

By restricting our scope to graphs with small degrees, we have constructed infinite families of long-refinement graphs. Furthermore, by reverse-engineering connections between colour classes, we obtained a complete classification of long-refinement graphs with small (or, equivalently, large) degrees. This analysis offers deep insights into the dynamics of the refinement process, revealing that all long-refinement graphs with maximum degree 3 can be described by compact strings over a remarkably small alphabet.

The talk is based on collaborations with Brendan D. McKay and T. Devini de Mel.

Physical networks are spatially embedded complex networks composed of nodes and links that are tangible objects which cannot overlap. Examples of physical networks range from neural networks and networks of bio-molecules to computer chips and disordered meta-materials. It is hypothesized that the unique features of physical networks, such as the non-trivial shape of nodes and links and volume exclusion affect their network structure and function. However, the traditional tool set of network science cannot capture these properties, calling for a suitable generalization of network theory. Here, I present recent efforts to understand the impact of physicality through tractable models of network formation.

An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially discrete series representations of a connected reductive p-adic group is stable under twist by automorphisms of the complex numbers, and hence this class is defined over $\bar{\mathbb{Q}_\ell}$. Recent work of Solleveld, building on work of Kazhdan-Varshavsky-Solleveld, says that the same is true of the class of standard representations. Stefan Dawydiak will give a geometric proof of this result for the principal block, and use this to deduce a local proof of Clozel's theorem for the general linear group. Time permitting, Stefan will also give geometric formulas for certain Harish-Chandra Schwartz functions that help illustrate these results.

Gukov and Vafa have proposed that a conformal field theory describing a string compactification on a manifold is rational (an RCFT) if and only if the manifold admits complex multiplication (CM). We investigate and extend the Gukov-Vafa proposal by constructing Hodge structures of CM type using only RCFT data, without reference to a geometric interpretation. 

We use the chiral and boundary states of the RCFT to construct the complex and rational vector spaces underlying the Hodge structure. Using the known notion of Galois symmetry of RCFTs and some elementary Galois theory, we are able to show that these Hodge structures are of CM-type, subject to some technical assumptions that can be verified explicitly for large classes of theories, including those without known geometric interpretation. We also discuss briefly the relation of complex multiplication to arithmetic geometry.

This talk is based on arXiv:2510.25708 with H. Jockers and M. Sarve.

Topological data analysis (TDA) deals with quantifying the "shape of data" using tools from algebraic topology and computational geometry. In many contexts, data comes equipped with a labelling (for example, cell type annotations in spatial biology), and one is interested in quantifying not just the global structure of the data but the spatial relationships between labelled subsets of the data. I will give a brief introduction to TDA and then talk about chromatic Delaunay filtrations, a recently developed family of computational methods in TDA that can address the problem of quantifying spatial relationships in labelled point cloud datasets.

A quasihomomorphism is a map that satisfies the homomorphism relation up to bounded error. Fujiwara and Kapovich proved a rigidity result for quasihomomorphisms taking values in discrete groups, showing that all quasihomomorphisms can be built from homomorphisms and sections of bounded central extensions. We study quasihomomorphisms with values in real linear algebraic groups, and prove an analogous rigidity theorem.  Based on joint work with Sami Douba, Francesco Fournier Facio, and Simon Machado.

Gaussian fields arise in a variety of contexts in both pure and applied mathematics. While their geometric properties are well understood, their topological features pose deeper mathematical challenges. In this talk, I will begin by highlighting some motivating examples from different domains. I will then outline the classical theory that describes the geometric behaviour of Gaussian fields, before turning to more recent developments aimed at understanding their topology using the Wiener chaos expansion.

Alday & Maldacena conjectured an equivalence between string amplitudes in AdS5 ×S5 and null polygonal Wilson loops together with a duality with amplitudes for planar N = 4 super-Yang-Mills (SYM).  At strong coupling this identifies SYM amplitudes with (regularized) areas of minimal surfaces in AdS.  They reformulated the minimal surface problem as a Hitchin system and in collaboration with Gaiotto, Sever & Vieira they introduced a Y-system and a thermodynamic Bethe ansatze (TBA) expressing the complete integrability that could in principle be used to solve for the amplitude at strong coupling. This lecture will review the parts of this material that we need and use them to identify new geometric structures on the spaces of kinematics for super Yang-Mills amplitudes/null polygonal Wilson loops.   In AdS3, the kinematic space is the cluster variety  M_{0.n} X M_{0,n}, where M_{0,n} is the moduli space of n points on the Riemann sphere moduli Mobius transformations.   The nontrivial part of these amplitudes at strong coupling, the remainder function,  turns out to be the (pseudo-)K ̈ahler scalar for a (pseudo-)hyper-Kaher geometry. It satisfies an integrable system and we give its its Lax form. The result follows from a new perspective on Y-systems more generally as defining the natural twistor space associated to the hyperkahler geometry of the Hitchin moduli space for these minimal surfaces. These connections in particular allows us to prove that  the amplitude at strong coupling satisfies the Plebanski equations for a hyperKahler scalar for  these pseudo-hyperk ̈ahler and related geometries. These hyperkahler geometries are nontrivial, (not semiflat) with a nontrivial TBA that encodes the mutations of the cluster structure.  These new structures underpinning the N=4 SYM amplitudes  will be important beyond strong coupling.  This is based on joint work with Hadleight Frost and Omer Gurdogan, https://arxiv.org/abs/2306.17044.

The representation theory of Lie algebras over fields of positive characteristic behaves quite differently to the characteristic zero case. For example, in positive characteristic, the dimension of all simple modules is finite and bounded. In this talk, we’ll begin by recalling the classification of finite simple representations of sl_2, and then explore how this changes when we move to the positive characteristic setting. Along the way, we’ll discuss the additional structures that appear in positive characteristic, such as restricted Lie algebras, the p-centre, and reduced enveloping algebras.

ACFA is the model companion of the theory of a field endowed with a distinguished endomorphism. This theory has been extensively studied by Chatzidakis and Hrushovski. Notably, it was shown that any non-principal ultraproduct of algebraically closed fields with powers of the Frobenius map gives rise to a model of ACFA.

In this talk, I will discuss the model theory of pairs of ACFA. In particular, we will give an axiomatization of those pairs in which the smaller one is transformally algebraically closed in the larger one. These are precisely the ultraproducts of pairs of algebraically closed fields equipped with powers of the Frobenius map. This theory also provides an example of beautiful pairs in the sense of Cubides Kovacsics, Hils, and Ye.

This is joint work with Martin Hils, Udi Hrushovski, and Jinhe Ye.

Graph causal optimal transport is a recent generalisation of causal optimal transport in which the allowed couplings satisfy causal restrictions given by a directed graph. Inspired by applications to structural causal models, it was originally introduced in Eckstein and Cheridito (2023). We study fundamental properties of graph causal optimal transport, with a particular focus on its induced Wasserstein distance. Our main result is a full characterisation of the directed graphs for which this associated Wasserstein distance is indeed a metric, an open problem in the original paper. We fully characterise the gluing properties of graph causal couplings, prove denseness of Monge maps, and provide a dynamic programming principle. Finally, we present an application to continuity of stochastic team problems. Based on joint work with Jan Obloj.

Over 30 years has passed since the original proof of Fermat's Last Theorem by Wiles and Taylor—Wiles. There are now several proofs known to humanity, and I'm currently teaching one of them to a computer. This made me try to find out what the most ergonomic route was nowadays, and I found it by asking Richard Taylor what it was. In the talk I will summarise how to prove Fermat's Last Theorem in 2026, highlighting the differences between the modern method and the original route discovered by Wiles (we do use p=3, but in a different way). I won't talk much at all about Lean and essentially none of the work I will present is my own; this will just be a standard number theory seminar, and probably everything in it will already be known to the experts, but hopefully younger people will learn something.

Dr Jindong Wang will talk about; 'Stabilised Finite Element Methods for General Convection–Diffusion Equations'

This talk presents several stabilised finite element methods for general convection–diffusion equations, with particular emphasis on recent extensions to vector-valued problems arising in magnetohydrodynamics (MHD). Owing to the non-self-adjoint structure of the operator and the potentially large disparity between convective and diffusive scales, standard Galerkin discretisations may exhibit non-physical oscillations. We design a class of upwind-type schemes and exponentially fitted methods for vector-valued problems that mitigate these effects, highlighting both their shared stabilisation mechanisms and the distinctive features that arise in the vector-valued setting. These developments illustrate concrete strategies for the design and analysis of finite element discretisations for general convection–diffusion problems.

Roy Makhlouf will talk about: 'Random Embeddings for Global Optimization: Convergence Results Beyond Low Effective Dimension'
 

Timely optimization problems are high-dimensional, calling for dimensionality reduction techniques to solve them efficiently. The random embedding strategy, which optimizes the objective along a low-dimensional subspace of the search space, is arguably the simplest possible dimensionality reduction method. Recent works quantify the probability of success of this strategy to solve the original problem by lower bounding the probability of a random subspace to intersect the set of approximate global minimizers. These works showed that, when the objective has low effective dimension (i.e., is only varying along a low-dimensional subspace of the search space), random embeddings of sufficiently large dimension solve the original high-dimensional problem with probability one. In this work, we relax the low effective dimension assumption by considering objectives with anisotropic variability, namely, Lipschitz continuous functions whose Lipschitz constant is small (though nonzero) when the function is restricted to a high-dimensional subspace. Exploiting tools from stochastic geometry, we lower bound the probability for a random subspace to intersect the set of approximate global minimizers of these objectives, hence, the probability of random embeddings to succeed in solving (approximately) the original global optimization problem. Our findings offer deeper insights into the role of the dimension of the optimization problem in this probability of success.

Neurons interact via spikes, which is a pulse-like, discontinuous mechanism. Their mean-field PDE description gives Fokker-Planck equations with novel nonlinearities. From a probability point of view, these give rise to Mckean-Vlasov equations involving hitting times. Similar mechanisms also arise in models for systemic risk in mathematical finance, and the supercooled Stefan problem. In this talk, we will first present models for spiking neurons: both microscopic particle models and macroscopic PDE models, with an emphasis on the general mathematical structure. A central question for these equations is the finite-time blow-up of the firing rate, which scientifically corresponds to the synchronization of a neuronal network. We will discuss how to continue the solution physically after the blow-up, by introducing a new timescale. The new timescale also helps us to understand the long term behavior of the equation, as it reveals a hidden contraction structure in the hyperbolic case. Finally, we will present a recently developed numerical solver based on this framework. Numerical tests show that during the synchronization the standard microscopic solver suffers from a rather demanding time step requirement, while our macro-mesoscopic solver does not.

In this talk I will present a few topics of recent interest that centre around the theme of “driven interfacial hydrodynamics”: fluid mechanical systems in which droplets and particles are self-propelled through interaction with the environment. I will also present some very recent work on using differentiable physics (a branch of physics-informed machine learning) to determine constitutive relations for highly plasticised metals.

This talk will contain elements of fluid dynamics, experimental mechanics, dynamical systems, statistical physics, and machine learning.

This talk will provide an overview of the landscape of bicommutant categories, these are tensor categories with a strong functional-analytic flavour. I will discuss the evolution of the definition (and give the current version of the definition) and explain precisely how they categorify von Neumann algebras, in the same way a tensor category can be viewed as a categorification of an algebra. We will also introduce the string-calculus that renders the coherences in the definition transparent and workable. 

The necessary background from functional analysis (in particular, operator theory) will be reviewed, and I will conclude with open questions (if waiting for the end of talk is not your style, there are 75 Open problems on André’s website). 

A central goal in the study of quantum chaos is being able to make universal statements about the dynamics of generic Hamiltonian systems. Under time evolution, an initially local operator progressively explores the Hilbert space of a system becoming increasingly non-local in the process. We will see that this idea lends itself to a natural notion of operator complexity measured (in the Hilbert space of operators) by the overlap of a time-evolving operator with a basis naturally adapted to time evolution and stratified by the growth in the operator's support. The information contained in this so-called Krylov basis is encoded in a sequence called the Lanczos coefficients which quantify the rate at which an operator is "pushed" along the Krylov basis to successively more complex elements. The universal operator growth hypothesis is then the conjecture that the Lanczos coefficients grow asymptotically linearly in any quantum chaotic system. In this talk, I will present an overview of these ideas and see how they manifest in the example of the well-studied SYK model. This talk is primarily based on 1812.08657.

I will discuss a 2-dimensional model of random walk in random environment known as line model. The environment is described by two independent families of i.i.d. random variables dictating rates of jumps in vertical, respectively horizontal directions, and whose values are constant along vertical, respect. horizontal lines. When jump rates are heavy-tailed in one of the directions, the random walk becomes superdiffusive in that direction, with an explicit scaling limit written as a two-dimensional Brownian motion time-changed (in one of the components) by a process introduced by Kesten and Spitzer in 1979. I will present ideas of the proof of this result, which relies on appropriate time-change arguments.  In the case of a fully degenerate environment, I will present a sufficient condition for non-explosion of the process (which is also believed to be sharp), as well as conjectures on the associated scaling limit.

This is based on joint work with J.-D. Deuschel (TU Berlin). 

(Joint seminar with Random Matrix Theory)

Graph products of groups were introduced by Green as a construction that encompasses both direct products and free products. Likewise, the notion of graph product of von Neumann algebras, introduced by Caspers and Fima, recovers both tensor products and free products. Camille Horbez will present rigidity theorems for graph products of tracial von Neumann algebras, and discuss the computation of their symmetries, drawing parallels with the case of groups. This is a joint work with Adrian Ioana. 

Random walks on graphs can mix slowly. To speed it up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon > 0$ that we can choose it. We show that in this case, at least for graphs of bounded degree, there is a way to steer the walk so that we visit every vertex in $n^{1+o(1)}$ many steps. The key to this result is a way to decompose arbitrary graphs into small-diameter pieces.