Past

Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of sheaves of Y to the G-equivariant derived category of C^2. In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of positive Lefschetz stabilizations in symplectic geometry.

Fersztand, Jacquard, Nanda, and Tilmann ('24) introduced the Skyscraper Invariant, a filtration of the classical rank-invariant, for multiparameter persistence modules. It is defined by considering the Harder-Narasimhan (HN) filtration of the module along a special set of stability conditions.

This talk will begin with a post-hoc motivation for considering stability conditions on persistence modules. To compute an approximation of the Skyscraper Invariant we present a technique which, exploiting the geometry of low-dimensional bifiltrations, lets us perform a brute-force computation. We compare it against Cheng's algorithm [Cheng24] which can compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension.

To avoid unnecessary recomputation in our algorithm, we ask for which stability conditions the HN filtrations are equivalent. This partition of the space of stabililty conditions is called the wall-and-chamber structure. We show that for a finitely presented d-parameter module it is given by the lower envelopes of a set of multilinear polynomials of degree d-1. For d=2 it is then easy to compute this, enabling a faster algorithm to compute the Skyscraper Invariant up to arbitrary accuracy. As a proof of concept for data analysis, we use it to compute a filtered version of the Multiparameter Landscape for large modules from real world data.

The Springer correspondence parameterises the irreducible representations of the Weyl group of a complex semisimple Lie algebra by nilpotent orbits. A key ingredient in the construction is the convolution operation, which appears in various forms throughout geometric representation theory. In this talk, we'll introduce the geometry of the Springer resolution, describe the convolution operation, and illustrate how it gives rise to a geometric construction of Weyl group representations.

The characterisation of the physics of the M5-brane remains an important open problem in string theory. While the superconformal field theory that resides on a planar M5-brane in flat space is poorly understood, other configurations involving M5-branes wrapped on certain manifolds have well-known superconformal field theory descriptions, including N=2 class S field theories. In this talk, I will use new methods based on exceptional generalised geometry to describe the gravity duals of class S field theories, compute a universal sector of their light-operator spectrum, and provide, for the first time, a holographic match of their superconformal index.

Heterogeneity is a fundamental feature of biological systems. Oncology is one of the fields in which this feature is most evident, as its key players are characterised by mutability, plasticity, and often “uncontrolled” dynamics. Whether heterogeneity arises from spatial structure, environmental variability, or cellular traits, effective therapeutic strategies must explicitly account for it in order to eradicate or control tumours.

From a modern perspective, this requires balancing the hit-hard / keep-it-sensitive trade-off, while also considering not only medical but also broader patient-related side effects of treatments. Contemporary medicine is increasingly exploring ways to exploit the very characteristics that have historically made cancer so dangerous, turning them into potential advantages for therapy.

The multiscale nature of tumour systems, together with the need to predict the combined effects of multiple, non-parallelisable processes, makes the development of optimised mathematical tools particularly compelling. Such tools can address questions that are both scientifically challenging and highly relevant from a clinical and humanitarian perspective.

In this seminar, we will analyse tumour masses from a structured population perspective, focusing on the role of heterogeneity in shaping therapeutic strategies. We will first discuss how heterogeneity in phenotypic composition and nutrient distribution influences the eco-evolutionary dynamics of tumour growth. We will then consider more specifically its impact on radiotherapy.

In particular, we will highlight the advantages of mathematically rigorous modelling in bridging theory and biology. We will also adopt a more exploratory perspective, using these models to illustrate how mathematics can serve as a potential decision-support tool for the selection and optimisation of treatment protocols, within an image- and model-driven framework.

The final part of the seminar will focus on potential future developments, with the aim of fostering an open and collaborative discussion on novel perspectives to improve understanding, prediction, and therapeutic optimisation.

The Zilber—Pink conjecture describes the points on an algebraic variety which have 'special' properties. In this talk, I will discuss some new results which can be proved about this, focusing on the examples of subvarieties of a torus, an abelian variety, or a product of modular curves. The method of proof is a generalisation of the Buium—Coleman proof of the Manin—Mumford conjecture. Parts of this are joint work with Sudip Pandit (KCL) and with Arnab Saha (IIT Gandhinagar).

The path from classic Black& Scholes quant finance to AI-driven trading and hedging. We review a number of recent results and put them in context of a wider strategy.

Professor Colbrook is going to talk about: 'A Computational Framework for Infinite-Dimensional Nonlinear Spectral Problems' 

Nonlinear spectral problems -- where the spectral parameter enters operator families nonlinearly -- arise in many areas of analysis and applications, yet a systematic computational theory in infinite dimensions remains incomplete. In this talk, I present a unified framework based on a solve-then-discretise philosophy (familiar, for example, from Chebfun!), ensuring that truncation preserves convergence. The setting accommodates unbounded operators, including differential operators with spectral-parameter-dependent boundary conditions. 
In the first part, I introduce a provably convergent method for computing spectra and pseudospectra under the minimal assumption of gap-metric continuity of operator graphs -- the weakest natural setting in which the resolvent norm remains continuous. 
In the second part, I develop a contour-based framework for discrete spectra of holomorphic operator families, with a complete analysis of stability, convergence, and randomised sketching based on Gaussian probes. This perspective unifies and extends many existing contour integral methods. Examples throughout highlight practical effectiveness and subtle phenomena unique to infinite dimensions, including the perhaps unexpected sensitivity to probe selection when seeking to avoid spectral pollution.

It is in general nontrivial to construct a 2d worldsheet model whose correlators evaluate to the amplitudes of a target theory. In this talk I will go through a neat, self contained (and to my knowledge, isolated) example in which the matter content and vertex operators of the dual 2d theory can be straightforwardly read off from the action of a 4d theory. Specifically, we will see that a genus 0 worldsheet model whose correlators compute all the tree amplitudes for pure 4d N=4 super Yang-Mills can be essentially derived from the twistor action in elementary steps. We will then discuss the limitations of this approach. There are no twistorial prerequisites assumed.

Jing-Yuan Wang is going to talk about: 'A Runtime-Data-Driven Enhancement Preconditioner for PCG for a Sequence of SPD Linear Systems'
 

In this work, we propose a runtime-data-driven enhancement preconditioner for improving the convergence of a preconditioned conjugate gradient method for solving a sequence of symmetric positive definite linear systems of equations. The methodology is designed for the situation where a subset of the systems has been solved and the convergence is considered too slow. In such a situation, data generated from the solved problems (residual vectors, intermediate solution vectors, approximate error vectors) are first analyzed by an unsupervised learning algorithm as a 3-step process: (1) dimension reduction; (2) classification of the slow features; (3) construction of projections to each of the feature subspaces. Based on the results of the analysis, one or more enhancement preconditioners are constructed using projection matrices corresponding to the features extracted from the slow convergence subspaces. The enhancement preconditioners are additively incorporated into the existing preconditioners and are employed to solve other systems in the sequence. The enhancement preconditioner can be further enhanced when necessary by repeating this process. Numerical experiments for time-dependent problems, including parabolic and hyperbolic equations, and stochastic elliptic equations demonstrate that the proposed approach improves the convergence considerably for other systems in the sequence when classical preconditioners are insufficient.

For more than a hundred years, scientists have carefully analysed the apparently random fluctuations in Brownian trajectories to learn about soft systems. In a more general sense, however, the information hidden within experimental fluctuations is typically underexploited, due to challenges in unambiguously linking fluctuation signatures to underlying physical mechanisms. In this talk, I will discuss our recent work developing new approaches to interpreting fluctuations in experimental data from a variety of soft systems, and thereby turn ‘noise’ into signal. In particular, I will share some recent results taking a fresh look at fluctuations in equilibrium colloidal monolayers. Here, we have combined experiment, simulation and theory to explore how simply counting colloids can reveal details of self and collective dynamics in interacting systems [1,2,3]. I will then discuss ongoing work to extend this understanding to confined driven systems [4], with the long-term goal of elucidating characteristic fluctuations in our synthetic nanopore experiments [5].

[1] E. K. R. Mackay, B. Sprinkle, S. Marbach, A. L. Thorneywork, Phys. Rev X. (2024)

[2] A. Carter, ALT et al., Soft Matter, 21, 3991, (2025)

[3] E. K. R. Mackay, ALT et al., arXiv:2512.17476, (2025)

[4] S. F. Knowles, E. K. R. Mackay, A. L. Thorneywork, J. Chem. Phys., (2024)

[5] S. F. Knowles, A. L. Thorneywork et al., Phys. Rev. Lett, 127, 137801, (2021)

We reformulate the statement that the theory of the free group is stable in terms of simplicial diagram chasing and profinite sets, without any terminology from logic. This includes three characterisations of stability (via indiscernible sequences, counting types, and definable types), and the notions of a first order theory and a model.

We do so by generalising slightly and allowing the universe of a first order structure/model to be an arbitrary (symmetric) simplicial set: formulas and basic predicates now may denote sets of simplices of an arbitrary (symmetric) simplicial set rather than sets of tuples of elements of a set. In this generalised sense the type space functor of a theory is its universal model classifying its usual models: taking the type of a tuple gives a map from a usual model of a theory to its type space functor. We define a property of simplicial maps weaker then being a fibration, and find it appears in the conditions characterising which maps correspond to models, when the generalised semantics is well-behaved, and which symmetric simplicial profinite sets correspond to first order theories.

In 1982, Conway introduced the angel-devil game, which is played on an infinite chess board.  For fixed k, the angel moves at most distance k from its current position on its turn.  The devil then blocks a square permanently.  The devil wins if the angel eventually has no legal moves left.  Berlekamp showed the devil wins against the 1-angel.  Conway asked whether there exists k such that the k-angel has a winning strategy against the devil.  This was resolved independently by Kloster, Máthé, and Bowditch in 2006.  Bowditch proposed playing the game on Cayley graphs of finitely generated groups.  A group for which the devil beats the k-angel for every k is called diabolical.  We will explore the ends of these diabolical groups.

 I will talk about recent progress on (a) a conjecture of Fyodorov and Keating on supercritical asymptotics of moments of moments of characteristic polynomials of the circular beta ensemble and (b) on the correlation functions of the sine beta point process. This is joint work with Joseph Najnudel.

Let G be a finite Chevalley group, i.e., the group of F_q points of a reductive group over F_q. Virtual representations of G in defining characteristic can be constructed in two ways, either by Brauer-Nesbitt reduction of complex representations, or by restricting an algebraic representation. G. Lusztig conjectured the shape of formulas connecting the two procedures; I will discuss a realization of his proposal related to decomposition of the class of diagonal for G/B coming from summands in the push-forward of the structure sheaf under Frobenius. 

Time permitting I will discuss a different, unrelated at present, way to describe such virtual representations linking it to homology of an affine Springer fiber. This found application in the work of Tony Feng and Viet Bao Le Hung on Breuil-Mezard conjectures. 

Based on joint works with Finkelberg, Kazhdan and Morton-Ferguson and with Boixeda Alvarez, McBreen and Yun respectively.

Let G be a finite Chevalley group, i.e., the group of F_q points of a reductive group over F_q. Virtual representations of G in defining characteristic can be constructed in two ways, either by Brauer-Nesbitt reduction of complex representations, or by restricting an algebraic representation. G. Lusztig conjectured the shape of formulas connecting the two procedures; I will discuss a realization of his proposal related to decomposition of the class of diagonal for G/B coming from summands in the push-forward of the structure sheaf under Frobenius. 

Time permitting I will discuss a different, unrelated at present, way to describe such virtual representations linking it to homology of an affine Springer fiber. This found application in the work of Tony Feng and Viet Bao Le Hung on Breuil-Mezard conjectures. 

Based on joint works with Finkelberg, Kazhdan and Morton-Ferguson and with Boixeda Alvarez, McBreen and Yun respectively.

It is known for a long time, due to a celebrated theorem of Ornstein and Weiss, that (classical/plain) orbit equivalence offers no information about ergodic probability measure preserving actions of amenable groups. On the other hand, conjugacy is too intractable, and effectively hopeless to study in full generality. Quantitative orbit equivalence aims to bridge this gap by adding intermediate layers of rigidity— a strategy that has borne fruit already in the late 1960s but was used as a general framework only semi-recently. In this talk, Spyridon Petrakos will introduce aspects of quantitative orbit equivalence and present a complete picture of it for integer odometers. This is joint work with Petr Naryshkin.

Let G be a group acting on a tree. I will discuss necessary conditions for G to have a finitely generated infinite normal subgroup of infinite index. When the edge stabilisers are virtually cyclic this naturally leads to considering (virtual) fibering of G. I will give an “if and only if” criterion for (virtual) fibering in the special case of amalgamated free products over virtually cyclic subgroups. The talk will be based on joint work with Jon Merladet.

We introduce a new spanning tree model which we call Random Spanning Trees in Random Environment (RSTRE), which was introduced independently by A. Kúsz. As the inverse temperature beta varies in the underlying Gibbs measure, it interpolates between the uniform spanning tree and the minimum spanning tree. On the complete graph with n vertices, we show that with high probability, the diameter of the random spanning tree is of order n^1/2 when β=o(n/log n), and is of order n^1/3 when β > n^4/3 log n. We conjecture that the diameter exponent linearly interpolates between these two regimes as the power exponent of beta varies. Based on joint work with L. Makowiec and M. Salvi.

Robustness to perturbation is a key topic in the study of complex systems occurring across a wide variety of applications from epidemiology to biochemistry. In this talk I will examine the eigenspectrum of the Jacobian matrices associated to a general class of networked dynamical systems, which contains information on how perturbations to a stationary state develop over time. I will show that stability is always determined by a spectral outlier, but with pronounced differences to the corresponding eigenvector in different regimes. Depending on model details, instability may originate in nodes of anomalously low or high degrees, or may occur everywhere in the network at once. Our results have potentially useful applications in network monitoring to predict or prevent catastrophic failures.

This talk discusses various applications of the relative entropy method in the context of fluid mechanics, focusing on weak-strong uniqueness results and asymptotic limits. Particular attention is given to Euler-type equations involving nonlocal interactions. Furthermore, I will present recent results regarding a novel approach to pressureless Euler equations.

Another application of the relative entropy method to be discussed is the unconditional stability of certain radially symmetric steady states for compressible viscous fluids in domains with inflow/outflow boundary conditions. Specifically, we demonstrate that any solution to the associated evolutionary problem, not necessarily radially symmetric, converges to a unique radially symmetric steady state.

Theta operators are weight-shifting differential operators on  automorphic forms. They play an important role in studying congruences between Hecke eigenforms and their p-adic variation. For instance, the classical theta operator, which acts on q-expansions of modular forms as q·(d/dq), is used crucially in Edixhoven’s proof of the weight part of Serre’s conjecture, Katz’s construction of p-adic L-functions over CM fields, and Coleman’s classicality theorem.

Recent years have witnessed extensive works on understanding theta operators over general Shimura varieties, from both geometric and representation-theoretic perspectives. In this talk, I will hint at some aspects of this fascinating area of research. If time permits, I will discuss my ongoing work on overconvergent theta operators over Siegel Shimura varieties.

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noice. This talk will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability.

(Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich.)

The stable Andrews-Curtis conjecture remains one of the most notorious unsolved problems in group theory. It proposes that every balanced presentation of the trivial group can reduced to the standard presentation (with one generator and one relation) using a sequence of simple moves. In my talk, I will focus on group presentations that are ‘thickenable’, which means that their associated 2-complex embeds in a 3-manifold. For such presentations, the stable Andrews-Curtis conjecture is known to hold. In my talk, I will explain how one can also get an explicit exponential-type upper bound on the number of stable Andrews-Curtis moves that are required. This is in sharp contrast to what is known about non-thickenable presentations.

This talk will present a new approach to the geometry of moduli spaces of hyperbolic monopoles.  It is well-known that the L^2 metric on the moduli space of hyperbolic monopoles, defined using a Coulomb gauge fixing condition, diverges. Recently we have shown that a supersymmetry-inspired gauge-fixing condition cures this divergence, resulting in a pluricomplex geometry that generalises the hyperkaehler geometry of euclidean monopole moduli spaces.  We will compare this with metrics introduced by Nash and Bielawski—Schwachhofer, and present explicit calculations of both metrics for charge 2 monopoles.

Part 1: In the first part of this talk, we develop a convergence analysis for training neural PDEs in the overparameterized limit. Many engineering and scientific fields have recently become interested in modelling terms in PDEs with neural networks (NNs), which requires solving the inverse problem of learning NN terms from observed data in order to approximate missing or unresolved physics in the PDE model. The resulting neural PDE model, being a function of the NN parameters, can be calibrated to the available ground truth data by optimizing over the PDE using gradient descent, where the gradient is evaluated in a computationally efficient manner by solving an adjoint PDE. We study the convergence of the adjoint gradient descent optimization method for training neural PDE models in the limit where both the number of hidden units and the training time tend to infinity, proving convergence of the trained neural PDE solution to the target data.

Part 2: For the second part, we turn towards developing a convergence analysis of the regularized Newton method for training NNs in the overparameterized limit. As the number of hidden units tends to infinity, the NN training dynamics converge in probability to the solution of a deterministic limit equation involving a „Newton neural tangent kernel“ (NNTK). Explicit rates characterizing this convergence are provided and, in the infinite-width limit, we prove that the NN converges exponentially fast to the target data. We show that this convergence is uniform across the frequency spectrum, addressing the spectral bias inherent in gradient descent. Mathematical challenges that need to be addressed in our analysis include the implicit parameter update of the Newton method with a potentially indefinite Hessian matrix and the fact that the dimension of this linear system of equations tends to infinity as the NN width grows.

One of the problems posed by Kadison in 1967 asks whether every separably acting von Neumann algebra is generated by a single element. The problem remains open in its full generality but significant progress has been made since. One can of course ask the same question in the C*-algebraic setting where, however, counterexamples are abundant even among commutative C*-algebras. I will give an overview of the history of the problem and then discuss some recent results on single generation of C*-algebras associated to graphs and C*-algebras with Cartan subalgebras.

I will discuss stochastic systems of interacting particles with non-overlapping constraints, which give rise to so-called excluded-volume interactions. The aim is to derive effective macroscopic equations governing the evolution of particle densities from the underlying microscopic dynamics. When particles possess nontrivial size or shape, geometric constraints become essential: they complicate the coarse-graining process and strongly influence the emergent behaviour of the system. I will present two representative examples, hard spheres and infinitely thin needles, highlighting how geometry enters the macroscopic description

This session is aimed at first-year undergraduates preparing for Prelims exams. A panel of lecturers and current students will share key advice on exam technique and revision strategies, offering practical tips from their own experience.

Cells make decisions across developmental biology, immunology, and synthetic biology. These processes are typically described using systems of ordinary differential equations, where mathematical analysis focuses on steady-state solutions. However, understanding how the timing of cell decisions is controlled requires moving beyond this paradigm. In this talk, I will discuss two complementary molecular mechanisms for controlling dynamical speed. First, I will show how timing can be regulated through critical slowing down, and how combining different bifurcations can generate emergent temporal behaviours even in small gene regulatory networks. Secondly, I will address developmental tempo, where embryos from different species execute remarkably similar genetic programmes at different speeds. I will present a mathematical framework based on orbit invariance that allows us to explore potential molecular mechanisms underlying species-specific differences in developmental timing.

7 years ago, also in Oxford, Sylvy Anscombe and I asked this question, which is part of the general effort to try and understand the model theory of henselian valued fields through dividing lines. In 2024, Sylvy Anscombe and Franziska Jahnke completely classified NIP henselian valued fields. Their methods can be extended, with the help of works of Chernikov, Kaplan and Simon and of Kuhlmann and Rzepka, to NTP2 henselian valued fields, obtaining the following:

• if a henselian valued field is NTP2, then it is semitame and its residue field is NTP2;
• if a henselian valued field is separably algebraically maximal Kaplansky and its residue field is NTP2, then it is NTP2.

This covers a large class of fields, but there is still a gap. Notably, Fp((Q)) is in the middle: it is semitame but not Kaplansky.

To answer this question, we studied so called tame henselian fields with finite residue field, and derived quantifier elimination results, namely, we prove that any formula in the language of valued fields reduces to a formula of the form (∃y f(x,y)=0) ∧ φ(v(x)) ∧ ψ(res(x)), where φ and ψ are formulas in the language of ordered groups and of rings, respectively.

In Fp((Q)) specifically, the valuation ring itself is definable with a diophantine formula (ie of the form ∃y f(x,y)=0), reducing further our quantifier elimination result.

Finally, a large chunk of these formulas are known to be NTP2: when f(x,y) is additive in y, the formula ∃y f(x,y)=z is NTP2 (with respect to x and z). Unfortunately, that does not cover all formulas, so the answer to the titular question is still unknown.

A question of Ben Green asks whether every finite set $A$ of integers with doubling constant $K$ must contain a subset $A'$ of comparable size whose doubling is at most $K + o(1)$ due to some explicit algebraic structure on $A'$. This was previously understood in the regime $K < 4 - o(1)$ by work of Eberhard, Green, and Manners, who showed that one can find such a subset $A'$ with density at least $1/2 + o(1)$ inside a long arithmetic progression. In this talk, I will provide a brief survey of this question as well as mention some new progress towards this. This is joint work with Yifan Jing.

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.  

The period matrix of a smooth complex projective variety encodes the isomorphism between its singular homology and its algebraic De Rham cohomology. Numerical approximations with high precision of the entries of the period matrix allow to recover some algebraic invariants of the variety, such as the Néron-Severi group in the case of surfaces. In this talk, we will see a method relying on the computation of an effective description of the homology for obtaining such numerical approximations of periods of algebraic varieties, and showcase implementations and applications, in particular to computation of the Picard rank of certain K3 surfaces related to Feynman diagrams.

In this seminar, we consider an isoperimetric problem for planar tilings with possibly unequal repeating cells. We present general existence and regularity results, and we study the classification of planar isoperimetric double tilings, namely tilings with two repeating cells of minimal perimeter. In this case, we explicitly determine the associated energy profile and provide a complete description of the phase transitions. We also comment on possible extensions and discuss some open problems. This is based on joint work with M. Novaga and E. Paolini.

Daniel Cortild is going to talk about: 'Regularization Methods for Hierarchical Programming'

We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and equilibrium selection problems. In a real Hilbert space setting, we combine a Tikhonov regularization and a proximal penalization to develop a flexible double-loop method for which we prove asymptotic convergence and provide rate statements in terms of gap functions. Our method is flexible, and effectively accommodates a large class of structured operator splitting formulations for which fixed-point encodings are available. 

Joint work with Meggie Marschner, and Mathias Staudigl (University of Mannheim)

Insects use flight as far more than a means of getting from A to B. Flight creates an aeiral theatre for interaction, whether between species or among members of the same species. For example, a male dragonfly must hunt for food, fend off rival males, and pursue evasive females in order to reproduce, tasks that all revolve around chasing fast-moving targets. Despite the remarkable diversity of insect species and their aerial behaviours, common patterns emerge in how they exploit speed and manoeuvrability to achieve these goals. Simple geometric guidance laws can describe these flight trajectories with surprising accuracy, revealing shared strategies that underpin insect aerial combat.

Pseudo-Anosov flows are a rich class of dynamical systems on 3-manifolds which are studied for their deep connections to the geometry and topology of the underlying space. A modern tool for studying these flows is to capture them with combinatorial objects called veering triangulations. This correspondence lets us study the flows from a computational perspective. In this talk, I will first give an introduction to pseudo-Anosov flows and how they are captured by these ‘old’ triangulations. I will then give a ‘new’ triangulation which captures these flows in greater generality, giving us many new explicit examples. I will finish by discussing how to algorithmically pass between the old and the new.

In this talk, we give a construction of the Abelian Yang--Mills--Higgs measure on the two-dimensional torus via the variational approach initiated by Barashkov--Gubinelli. The construction is carried out through a disintegration of measures: we first construct the conditional Higgs measure given a rough gauge field, and then construct the gauge field marginal. This leads to iterated variational problems, one for the Higgs field and one for the gauge field. At the technical level, the starting point is the construction of the renormalised covariant Laplacian associated to a rough gauge field, together with the study of its resolvent. This allows us to define the covariant Gaussian free field, which serves as the reference Gaussian field for the conditional Higgs measure. Finally, we analyse the ratios of determinants that arise from the change-of-measure formula for Gaussian measures. This is joint work with Nikolay Barashkov, Ajay Chandra, Ilya Chevyrev, and Andreas Koller.