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.

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.

An unpublished theorem of Clozel, proven with global techniques, says that the class of essentially discrete series representations of a connected reductive p-adic group is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.

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.

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.

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.

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. 

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.

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.

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.

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.

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.

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.

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.

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.

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.

Dr Steph Foulds will talk about; 'Lazy Quantum Walks with Native Multiqubit Gates'

Quantum walks, the quantum analogue to the classical random walk, have been shown to deliver the Dirac equation in the continuum limit. Recent work has shown that 'lazy', open quantum walks can be mapped to computational methods for fluid simulation such as lattice Boltzmann method, quantum fluid dynamics, and smoothed-particle hydrodynamics. This work concerns evaluating the ability of near-term hardware to perform small, proof-of-concept quantum walks - but crucially with the inclusion of a rest state to encompass 'lazy' quantum walks, providing an integral step towards quantum walks for fluid simulation.

Neutral atom hardware is a promising choice of platform for implementing quantum walks due to its ability to implement native multiqubit gates and to dynamically re-arrange qubits. Using detail realistic modelling for near-term multiqubit Rydberg gates via two-photon adiabatic rapid passage, SPAM, and passive error, we present the gate sequences and final state fidelities for quantum walks with and without a rest state on 4 to 16-node rings. This, along with results of an error model with improved two- and three-qubit gate fidelities, leads us to conclude that a native four-qubit gate is required for the near-term implementation of interesting quantum walks on neutral atom hardware.

Please note; this talk is hosted by Rutherford Appleton Laboratory, Harwell Campus, Didcot, OX11 0QX

I will review how the large N expansion can be used in the context of the renormalisation group to probe some strongly coupled regimes. In particular, I will discuss a work by Gawedzki and Kupiainen where the authors study the three-dimensional non-Gaussian infrared fixed point of Phi^4 in the case of a hierarchical model of rank-one covariance, and explain how their approach could generalise to more realistic models. 

This is a joint work with Ajay Chandra.  

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Speaker Prof Dr Maria Lukacova will talk about 'Numerical analysis of oscillatory solutions of compressible flows'

Oscillatory solutions of compressible flows arise in many practical situations.  An iconic example is the Kelvin-Helmholtz problem, where standard numerical methods yield oscillatory solutions. In such a situation,  standard tools of numerical analysis for partial differential equations are not applicable. 

We will show that structure-preserving numerical methods converge in general to generalised solutions, the so-called dissipative solutions. 
The latter describes the limits of oscillatory sequences. We will concentrate on the inviscid flows, the Euler equations of gas dynamics, and mention also the relevant results obtained for the viscous compressible flows, governed by the Navier-Stokes equations.

We discuss a concept of K-convergence that turns a weak convergence of numerical solutions into the strong convergence of
their empirical means to a dissipative solution. The latter satisfies a weak formulation of the Euler equations modulo the Reynolds turbulent stress.  We will also discuss suitable selection criteria to recover well-posedness of the Euler equations of gas dynamics. Theoretical results will be illustrated by a series of numerical simulations.  

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