Past

If you are curious about using your maths outside academia, want to learn new skills, or just want a change of pace from your PhD, then consider a policy internship. During a three-month UKRI policy internship at the Joint Nature Conservation Committee, I worked on assessing the impact of human-made underwater noise on harbour porpoises. I got to see what it was like to work for a government advisory body, and how scientific modelling is used to inform policy and real-world decision making, all whilst occasionally spotting dolphins from my office window. In this talk, I will describe my project and use it as a starting point to discuss internships more broadly: what you can gain from them, how they differ from academic research, and how to apply.

The study of combinatorial designs includes some of the oldest questions at the heart of combinatorics. In a breakthrough result of 2014, Keevash proved the longstanding Existence Conjecture by showing the existence of (n,q,r)-Steiner systems (equivalently K_q^r-decompositions of K_n^r) for all large enough n satisfying the necessary divisibility conditions. Meanwhile, in recent decades, incremental progress has been made on the celebrated Nash-Williams' Conjecture of 1970, which posits that any large enough, triangle-divisible graph on n vertices with minimum degree at least 3n/4 admits a triangle decomposition. In 2021, Glock, Kühn, and Osthus proposed a generalization of these results by conjecturing a hypergraph version of the Nash-Williams' Conjecture, where their proposed minimum degree K_q^r-decomposition threshold is motivated by hypergraph Turán theory. By using the recently developed method of refined absorption and establishing a non-uniform Turán theory, we tie the K_q^r-decomposition threshold to its fractional relaxation. Combined with the best-known fractional decomposition threshold from Delcourt, Lesgourgues, and Postle, this dramatically closes the gap between what was known and the above conjecture. This talk is based on joint work with Luke Postle.

Recurrent neural networks (RNNs) provide a theoretical framework for understanding computation in biological neural circuits, yet classical results, such as Hopfield's model of associative memory, rely on symmetric connectivity that restricts network dynamics to gradient-like flows. In contrast, biological networks support rich time-dependent behaviour facilitated by their asymmetry. In this talk, I will introduce a general framework, known as ‘drift-diffusion matching’, for training continuous-time RNNs to represent arbitrary stochastic dynamical systems within a low-dimensional latent subspace. Allowing asymmetric connectivity, I will show that RNNs can embed the drift and diffusion of an arbitrary stochastic differential equation, including nonlinear and nonequilibrium dynamics such as chaotic attractors. As an application, we have constructed RNN realisations of stochastic systems that transiently explore various attractors through both input-driven switching and autonomous transitions driven by nonequilibrium currents, which we interpret as models of associative and sequential (episodic) memory. To elucidate how these dynamics are encoded in the network, I will introduce decompositions of the RNN based on its asymmetric connectivity and its time-irreversibility. These results extend attractor neural network theory beyond equilibrium, showing that asymmetric neural populations can implement a broad class of dynamical computations within low-dimensional manifolds, unifying ideas from associative memory, nonequilibrium statistical mechanics, and neural computation.

Quantization is notoriously difficult and even when we have found quantizations of, say, coordinate rings of varieties, these often appear ad-hoc.  But, using ideas that were just emerging when the speaker was embarking on their research journey in the Institute some twenty years ago, a remarkable alternative approach has been developed. We will describe the journey, culminating in a recent result with Pressland showing that quantization and categorification are intimately linked. Braidings, grading and mirror symmetry also feature in our story.

The Hitchin equations are an integrable system in two-dimensions that plays a variety of important roles across mathematics and physics and this talk will start with some of this motivation.  It will go on to discuss how the 4d Chern-Simons of Costello, Witten and Yamazaki fits into ideas from  30-40 years ago that sought to unify the study of integrable systems via the study of the self-duality equations and their twistor constructions.  In particular 4d Chern-Simons provides a uniform approach to 2d integrable systems and their canonical structures.  The Hitchin equations have been missing in this approach and this talk will explain I will explain how Hitchin equations are incorporated with reductions to Toda and Sine Gordon, and  gives new approaches to understanding canonical strucures associated with these equations.  This talk is based on joint work with Roland Bittleston and Faroogh Moosavian https://arxiv.org/abs/2601.05309.

Everyone is invited to celebrate International Women in Mathematics Day with a pizza lunch! We will be watching ‘Journeys of Women in Mathematics’, a powerful 20-minute film by the International Mathematical Union showcasing the experiences of women mathematicians worldwide. It follows three mathematicians from India, Cameroon, and Brazil from their home institutions to the (WM)² international meeting, showing their research and what it’s like to be part of the global maths community.

We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider on the discretized circle the rate equation for the process where pairs of particles sitting on the same side jump simultaneously to the two neighboring sites, and the reverse jump where a pair of particles sitting on a common site jump simultaneously to the side in the middle. Depending on the rates, in the vanishing mesh size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. We identify the limiting gradient structure to be driven by entropy with respect to a generalization of the diffusive transport type with nonlinear mobility via EDP convergence. Furthermore, the DLSS equation with nonlinear mobility of the power type shares qualitative similarities with the fast diffusion and porous medium equations, since we find traveling wave solutions with algebraic tails and polynomial compact support, respectively.    
       

Joint work with Alexander Mielke and Artur Stephan arXiv:2510.07149. The DLSS part is based on joints works with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré.

We will begin with an overview of Stark's conjectures before discussing the case of imaginary quadratic fields, covering both the limit formula and the existence of elliptic units. The classical expositions of these are at times lacking in intuition, but thanks to Kato's deep insights 20 years ago, we can present more geometric and illuminating proofs of both results.

One can learn a lot about a compact manifold if one can show that it fibres over the circle - in essence, this allows us to view a static n-dimensional manifold as a manifold of dimension n-1 that evolves in time.Being fibred (over the circle) is a relatively rare property. It is much more common to be virtually fibred, that is, to admit a finite cover that is fibred. For example, it was the content of a conjecture of William Thurston, now two theorems by Ian Agol and Dani Wise, that all finite-volume hyperbolic 3-manifolds are virtually fibred; in fact, this property is extremely common among irreducible 3-manifolds.The situation is less clear in higher dimensions. On the obstruction side, we know that virtually fibred manifolds must have vanishing Euler characteristic. This immediately shows that compact hyperbolic manifolds in even dimensions will not be virtually fibred. A more involved obstruction comes from L2-homology: virtually fibred manifolds must be L2-acyclic. The motivation behind the research I will present lies in trying to find situations in which the vanishing of L2-homology is is not only necessary, but also sufficient for virtual fibring. It turns our that a lot more can be said if we replace aspherical manifolds by their homological cousins: Poincare duality groups. Concretely, if G is an n-dimensional Poincare-duality group over the rationals, and if G satisfies the RFRS property, then G is L2-acyclic if and only if there is a finite-index subgroup G0 of G and an epimorphism from G0 onto the integers such that its kernel is a Poincare-duality group over the rationals of dimension n-1. (This last theorem is joint with Sam Fisher and Giovanni Italiano.)The RFRS property was introduced in Agol's work on Thurston's conjecture. A countable group is RFRS if and only if it is residually {virtually abelian and poly-Z}. All compact special groups in the sense of Haglund-Wise satisfy this property, so there is a ready supply of RFRS groups, also among fundamental groups of hyperbolic manifolds in high dimensions.

We study the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

Given a Fano variety X with smooth anticanonical divisor D, one may consider the enumerative geometry of X, of the pair (X,D) or of D. A-model Quantum Lefschetz, Residue and Serre relate counts of genus 0 curves in X,  (X,D) and D. While the A-model statements are fairly involved, they become standard integral transforms when formulated as B-model correspondences within the Intrinsic Mirror Construction of Gross-Siebert. I will explain how this works. Time permitting, I will explain how for K-polystable del Pezzo surfaces, genus 0 log BPS instanton expansions transform into modular forms.

Associate Professor Nicolas Boumal will talk about: 'Smooth, globally Polyak-Łojasiewicz functions are nonlinear least-squares'

Polyak-Łojasiewicz (PŁ) functions abound in the literature, especially in nonconvex optimization. When they are also smooth, they become surprisingly simple---with an exotic twist. The plan is for us to discover the structure of those functions and of their sets of minimizers via gradient flow and fiber bundles.

Joint work with Christopher Criscitiello and Quentin Rebjock.

In a joint work with Yu Deng (University of Chicago) and Xiao Ma (University of Michigan), we extended Lanford’s theorem to long times—specifically, for as long as the solution of the Boltzmann equation exists. This allowed us to fully carry out Hilbert’s program and derive the fluid equations in the Boltzmann–Grad limit. The underlying strategy builds on earlier joint work with Yu Deng that resolved a parallel problem in which colliding particles are replaced by nonlinear waves, thereby establishing the mathematical foundations of wave turbulence theory. In this talk, we will review this progress and discuss some related problems and future directions. 

This week’s Fridays@2 session is intended to provide advice on exam preparation and how to approach the Part A, B, and C exams.  A panel consisting of past examiners and current students will answer any questions you might have as you approach exam season.

Drug discovery is slow and expensive, and a growing body of AI work tackles this by training generative models that propose new candidate molecules directly, searching chemical space far faster than a human chemist could. Most of this work has focused on standard small molecules, leaving more specialized but valuable classes underexplored.

Macrocycles are ring-shaped molecules that offer a promising alternative to small-molecule drugs due to their enhanced selectivity and binding affinity against difficult targets. Despite their chemical value, they remain underexplored in generative modeling, likely owing to their scarcity in public datasets and the challenges of enforcing topological constraints in standard deep generative models.

We introduce MacroGuide: Topological Guidance for Macrocycle Generation, a diffusion guidance mechanism that uses Persistent Homology to steer the sampling of pretrained molecular generative models toward the generation of macrocycles, in both unconditional and conditional (protein pocket) settings. At each denoising step, MacroGuide constructs a Vietoris-Rips complex from atomic positions and promotes ring formation by optimizing persistent homology features. Empirically, applying MacroGuide to pretrained diffusion models increases macrocycle generation rates from 1% to 99%, while matching or exceeding state-of-the-art performance on key quality metrics such as chemical validity, diversity, and PoseBusters checks.

Accepted to ICML 2026. Paper: https://arxiv.org/abs/2602.14977

Prostate cancer progression and therapeutic resistance present significant clinical challenges, particularly in the transition to castration-resistant disease. Although androgen deprivation therapy and second-generation drugs have improved patient outcomes, resistance frequently develops, reflecting tumour heterogeneity and the influence of its microenvironment. This talk presents two interdisciplinary studies that address these issues through data-driven mathematical approaches. We show how integrating experimental data with mathematical and statistical modelling can improve our understanding of prostate cancer dynamics and inform more effective, context-specific therapeutic strategies. The first study examines drug resistance and tumour evolution under treatment. We develop a multi-scale hybrid modelling framework to capture processes occurring across different temporal scales. Partial differential equations describe the behaviour of drugs and other chemicals in the tumour microenvironment (over the ‘fast’ timescale), while a cellular automaton captures the dynamics of tumour cells (over the ‘slow’ timescale). Through computational analysis of the model solutions, we examine the spatial dynamics of tumour cells, assess the efficacy of different drug therapies in inhibiting prostate cancer growth, and identify effective drug combinations and treatment schedules to limit tumour progression and prevent metastasis. The second study focuses on the role of host–microbiome interactions in obesity-associated prostate cancer. Using data from experiments with the TRAMP mouse model, we apply statistical and machine learning methods, including generalised linear models, Granger causality, and support vector regression, to characterise microbial dynamics and their responses to treatment. These findings are incorporated into a dynamical systems framework that captures microbiome–tumour co-evolution under therapeutic and dietary perturbations, providing insight into how dietary fat and combination therapies involving enzalutamide and phytocannabinoids influence tumour progression and gut microbiota composition.

Professor Po-Ling Loh will talk about; 'Private estimation in stochastic block models'

We study the problem of private estimation for stochastic block models, where the observation comes in the form of an undirected graph, and the goal is to partition the nodes into unknown, underlying communities. We consider a notion of differential privacy known as node differential privacy, meaning that two graphs are treated as neighbors if one can be transformed into the other by changing the edges connected to exactly one node. The goal is to develop algorithms with optimal misclassification error rates, subject to a certain level of differential privacy.

We present several algorithms based on private eigenvector extraction, private low-rank matrix estimation, and private SDP optimization. A key contribution of our work is a method for converting a procedure which is differentially private and has low statistical error on degree-bounded graphs to one that is differentially private on arbitrary graph inputs, while maintaining good accuracy (with high probability) on typical inputs. This is achieved by considering a certain smooth version of a map from the space of all undirected graphs to the space of bounded-degree graphs, which can be appropriately leveraged for privacy. We discuss the relative advantages of the algorithms we introduce and also provide some lower-bounds for the performance of any private community estimation algorithm.

This is joint work with Laurentiu Marchis, Ethan D'souza, and Tomas Flidr.

The integration of AI into computational fluid dynamics (CFD) represents a transformative frontier for engineering, yet realizing this potential requires navigating the complexities inherent to fluid mechanics. Bridging the methodological gap between deep learning and traditional CFD simulation, this talk presents work (outlined in the recent preprint: Fluids Intelligence: A forward look on AI foundation models in computational fluid dynamics) to produce a novel scaling law tailored specifically for a fluids foundation model. We explore the theoretical and practical opportunities, analyzing the critical inflection points where model training compute begins to eclipse the high costs of traditional data generation. We conclude by discussing the technical challenges and opportunities the fluids and machine learning communities must collaboratively address to operationalize autonomous computational engineering.

Speaker Jung Eun Huh will talk about: 'Adaptive preconditioning for linear least-squares problems via iterative CUR'

Large-scale linear least-squares problems arise in many areas of computational science and data analysis, where efficiency and scalability are crucial. In this talk, we introduce a randomized preconditioning framework for iterative solvers based on low-rank approximations of small sketches of the original problem. The key idea is to iteratively construct low-rank preconditioners that reshape the singular value distribution in a favourable way. By tightly coupling the preconditioning and Krylov solving phases within an iterative CUR decomposition -- a low-rank approximation built from selected of columns and rows of the original matrix -- the proposed algorithm achieves faster and earlier convergence than existing methods. The algorithm performs particularly well on problems that are large in both dimensions, as well as on sparse and ill-conditioned systems. 

This is a joint work with Coralia Cartis and Yuji Nakatsukasa.

Gromov-hyperbolic groups are classically defined geometrically, by the negative curvature of their Cayley graphs. Interestingly, an algorithmic characterization of hyperbolicity is possible in terms of properties of the formal languages of quasigeodesics (geodesics up to bounded error) in their Cayley graphs. Holt and Rees proved, roughly speaking, that these formal languages are regular in the case of hyperbolic groups. More recently Hughes, Nairne, and Spriano established the converse. In this talk, I will discuss progress towards a conjectured strengthening of the result, where we consider context-free quasigeodesic languages. This is based on my summer project, supervised by Joseph MacManus and Davide Sprianoc

Compactifying topological actions using only de Rham forms fails to capture torsion sectors encoded in integral cohomology. Differential cohomology remedies this by combining integral characteristic classes, differential-form curvatures, and holonomy data into a single framework. In the context of deriving SymTFTs from M-theory, such a refinement is crucial for capturing background gauge fields for discrete 1-form global symmetries in the physical theory. In this talk, we will review the construction of differential cohomology and, time permitting, show how a refined Kaluza-Klein compactification leads to background gauge fields that encode these higher-form symmetries.

We discuss the asymptotic local behavior of the second correlation functions of the characteristic polynomials of a certain class of Gaussian N X N non-Hermitian random band matrices with a bandwidth W. Given W,N → ∞, we show that this behavior near the point in the bulk of the spectrum exhibits the crossover at W ∼√N: it coincides with those for Ginibre ensemble for W ≫√N, and factorized as 1 ≪ W ≪√N. The behavior of the correlation function near the threshold (W/√N →C) will be also discussed.

Tropicalization is a process by which we replace algebraic geometry with the geometry of piecewise linear (tropical) objects. One of the central questions in the field is when this process can be reversed: that is, when can we realize a tropical object with an honest algebraic one. In this talk, I'll discuss some recent work on the tropical to Lagrangian correspondence, and state under what conditions homological mirror symmetry allows us to transfer Lagrangian realizations into algebraic ones.

Real complex systems exhibit rich collective behavior, yet identifying which components of an interaction network drive such dynamics remains a central challenge. Here, we show that complexity itself can resolve this problem. In large random and empirical networks, structural disorder and heterogeneity induce spectral localization, causing Laplacian modes to concentrate on small subsets of nodes. This converts global modes into identifiable dynamical units tied to specific structural components. Exploiting this principle, we develop a node-resolved stability framework that predicts instability onsets, identifies the nodes responsible for collective transitions, and restores interpretability in systems where classical modal theories fail. In heterogeneous reaction networks, the same mechanism enables collective states beyond those usually associated with homogeneous assumptions. More broadly, our results show that complexity can be revealed, rather than obscure, the microscopic drivers of macroscopic dynamics.

The well-known "Dunkl operators" associated to a finite real reflection group constitute a commutative parameter family of deformations of the directional derivatives in Euclidean space. These operators are "differential-reflection" operators. Heckman and Cherednik have defined trigonometric versions of Dunkl's operators. The interest for these operators lies in their deep ties to Macdonald polynomials and hypergeometric functions, to the Calogero-Moser quantum integrable system, and to the representation theory of Hecke algebras. 

"Hypergeometric shift operators" are powerful tools to study Weyl group symmetric structures and functions in these contexts. In this talk, Eric Opdam presents a theorem of existence and uniqueness of ''nonsymmetric shift operators'' for the Dunkl operators. These are themselves differential reflection operators which "shift" the parameters of the Dunkl operators by integers by means of a "transmutation relation".

(Joint work with Valerio Toledano Laredo) 

Recently a new inhabitant entered the zoo of convexity notions for vectorial variational problems: functional convexity. I would like to report of progress in understanding the corresponding integrands, but also new insight into fine properties of most general class of related integrands: It turns out that rank-one convex functions share surprisingly many pointwise differentiablity properties with ordinary convex functions.

Fix an Artin representation rho. Work in progress by Emmanuel Lecouturier and Loïc Merel claims that the special values L(f,rho,1) for certain modular forms f see some global data related to the L-function attached to rho. We first give a brief exposition on Mazur’s Eisenstein ideal, which lies at the heart of their work. We then describe this conjectural phenomenon in a few simple cases, the last being related to a conjecture of Harris and Venkatesh.

We consider the question of whether almost-homomorphisms are close to honest homomorphisms. I’ll survey a few historical results, with different source/target collections of algebras, and also consider what to take as the definition of “almost-homomorphisms”. If we end up having time, I will sketch an elementary proof that almost-characters from commutative C*-algebras are close to honest characters.

The Euler characteristic transform (ECT) is an emerging and powerful framework within topological data analysis for quantifying the geometry of shape. The applicability of ECT has been limited due to its sensitivity to noisy data. Here, we introduce SampEuler, a novel ECT-based shape descriptor designed to achieve enhanced robustness to perturbations. We provide a theoretical analysis establishing the stability of SampEuler and validate these properties empirically through pairwise similarity analyses on a benchmark dataset and showcase it on a thymus dataset. The thymus is a primary lymphoid organ that is essential for the maturation and selection of self-tolerant T cells, and within the thymus, thymic epithelial cells are organized in complex three-dimensional architectures, yet the principles governing their formation, functional organization, and remodeling during age-related involution remain poorly understood. Addressing these questions requires robust and informative shape descriptors capable of capturing subtle architectural changes across developmental stages. We develop and apply SampEuler to a newly generated two-dimensional imaging dataset of mouse thymi spanning multiple age groups, where SampEuler outperforms both persistent homology-based methods and deep learning models in detecting subtle, localized morphological differences associated with aging. To facilitate interpretation, we develop a vectorization and visualization framework for SampEuler, which preserves rich morphological information and enables identification of structural features that distinguish thymi across age groups. Collectively, our results demonstrate that SampEuler provides a robust and interpretable approach for quantifying thymic architecture and reveals age-dependent structural changes that offer new insights into thymic organization and involution.

Gauge/gravity duality is more than AdS/CFT.  In this talk I will discuss how the holographic dictionary generalises to non-conformal settings, focusing on maximally supersymmetric Yang-Mills theories in diverse dimensions and their Dp-brane supergravity duals. Scaling covariance replaces conformal invariance as the unifying principle on both sides of the duality. On the gravity side, I will show how to systematically organise effective actions and Witten diagram rules for arbitrary correlators of scalar and spin-1 Kaluza-Klein modes. On the field theory side, scale covariance fixes the kinematic structure of 2- and 3-point functions at strong coupling, with the latter admitting closed-form expressions in terms of Appell functions. I will illustrate these results with explicit examples, focussing on 3d MSYM.

Reaction systems are continuos-time dynamical systems with polynomial right-hand side, and are very common in biochemistry, cell signaling, population dynamics, and many other biological applications. We discuss global stability (i.e., the existence of a globally attracting point) and persistence (i.e., robust absence of extinction) for large classes of reaction systems. In particular, we describe recent progress on the proof of the Global Attractor Conjecture (which says that vertex-balanced reaction systems are globally stable) and the Persistence Conjecture (which says that weakly-reversible reaction systems are persistent), and how these results can be extended outside their classical setting using the notion of “disguised reaction systems". We will also discuss analogous results for the case where reaction systems are replaced by generalized Lotka-Volterra systems of arbitrary degree. 

A test case for the Langlands functoriality principle is the tensor product lifting of automorphic representations of $\mathrm{GL}(m) \times \mathrm{GL}(n)$ to automorphic representations of $\mathrm{GL}(mn)$. This has been established in several key instances: for $m=n=2$ by Ramakrishnan (2000), for $m=2$ and $n=3$ by Kim-Shahidi (2002), and more recently for $m=2$ and arbitrary $n$ over $\mathbb{Q}$ by Arias-de-Reyna-Dieulefait-Pérez (2025) under certain assumptions, including that the $\mathrm{GL}(2)$ factor has level 1. I will discuss some new results in the case of $\mathrm{GL}(2) \times \mathrm{GL}(n)$, as well as ideas for how to go further, when $m>2$, using a p-adic propagation technique introduced by Newton-Thorne (2021).