Past

We study the geometry of simplicial cochains on Vietoris–Rips complexes built from i.i.d. samples of a compact embedded manifold. By integrating differential forms over affine simplices, we define a cochain map from smooth forms to simplicial cochains and equip the latter with kernel-weighted inner products determined by ambient pairwise distances. At scales where the sampled complex recovers the homotopy type of the manifold, we prove quantitative high-probability convergence of these inner products and, under uniform sampling, of the associated codifferential energies to their continuum counterparts. As consequences, we obtain spectral upper-bounds for the discrete Hodge Laplacian and, for orientable manifolds, convergence of harmonic representatives of discretized harmonic one-forms and consistency of harmonic smoothing for circular coordinates.

This is joint work with Darrick Lee: https://arxiv.org/pdf/2608.11137

We'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?

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

We study model-free policy learning for discrete-time mean-field control (MFC) problems with finite state space and compact action space. In contrast to the extensive literature on value-based methods for MFC, policy-based approaches remain largely unexplored due to the intrinsic dependence of transition kernels and rewards on the evolving population state distribution, which prevents the direct use of likelihood-ratio estimators of policy gradients from classical single-agent reinforcement learning. We introduce a perturbation scheme on the state-distribution flow and prove that the gradient of the resulting perturbed value function converges to the true policy gradient as the perturbation magnitude vanishes. This construction yields a model-free estimator based solely on simulated trajectories and an auxiliary estimate of the sensitivity of the state distribution. Building on this framework, we develop MF-REINFORCE, a model-free policy gradient algorithm for MFC, and establish quantitative bounds on its bias and mean-squared error.

Manin and his collaborators initiated a program in 1989 with the goal of understanding the distribution of rational points of bounded height on Fano varieties, using methods from algebraic geometry and analytic number theory. Since then, there has been substantial progress on this program, including the formulation of a precise conjecture. I will discuss the achievements of this program since then, with a focus on surfaces, including some of my own work on quintic del Pezzo surfaces.

Professor Dr. Sebastian Pokutta will speak about; 'How Machines Explore, Conjecture, and Discover Mathematics'

Artificial Intelligence is increasingly becoming a genuine partner in mathematical research, not only as a computational tool, but as a driver of exploration, conjecture generation, and discovery. Under the umbrella of AI4Math, we develop methodologies that combine optimization, machine learning, and mathematical structure to navigate large, complex, and highly constrained search spaces that are inaccessible to traditional approaches.

In this talk, we illustrate this paradigm through a concrete case study: the Hadwiger–Nelson problem, a long-standing open problem in discrete geometry and extremal combinatorics concerning colorings of the plane without monochromatic unit-distance pairs. We show how neural networks can be used as expressive approximators to transform a mixed discrete–continuous geometric problem with hard constraints into a differentiable optimization problem with a probabilistic loss. This enables gradient-based exploration of admissible configurations and directly led to the discovery of two novel six-colorings, yielding the first improvement in thirty years for the off-diagonal variant of the problem.

Joseph Webb (Mathematical Institute, University of Oxford) is going to talk about "An optimisation framework for the well-conditioned training of physics-informed neural networks"

In mathematical models of opinion dynamics, individuals interact with each other and adjust their opinions based on their interactions. In opinion models, network structures determine which individuals can interact with each other and thereby affect how opinions change with time. In this talk, I will introduce opinion models and why scientists study them. I will also discuss several variants of bounded-confidence models (BCMs), in which opinions take continuous values in a region, and I will examine how network structure affects the formation of consensus, polarization, and fragmentation of populations.

Concordances between knots have provided much framework for the study and classification. The slice-ribbon conjecture posed by Fox in 1961 continues to be one of the largest open problems in the field. In this talk, I will provide background on concordances and the slice-ribbon conjecture, as well as explaining how concordances can induce maps on both Khovanov and Knot-Floer homology. I will then cover how this ties into my own current research and future directions.

Join us to discuss x+y: A Mathematician's Manifesto for Rethinking Gender by Eugenia Cheng.

We study the random motion of interfaces in stochastic Allen–Cahn equations. In one dimension, we show that, on a suitable long time scale, the solution stays close to the family of standing wave solutions, while the interface location converges to a diffusion. Our main novelty is the construction of functional correctors to cancel potential divergences. We also establish an analogous result for more singular noise leading to distribution-valued solutions. In this setting, we show that two divergent quantities arising from renormalization and Itô’s formula cancel each other.

In higher dimensions, we construct stochastic mean curvature flow driven by noise that is white in time and smooth in space, and identify it as the sharp interface limit of the stochastic Allen–Cahn equation with the same noise slightly smoothed in time. To construct this flow, we use a transformation developed by Buckdahn, Keller, Ma and Zhang (2020), which reduces the stochastic equation to a classical PDE with random coefficients.

This talk is based on joint works with Weijun Xu and Wenhao Zhao.

Let $W=W^*$ be a large random matrix. For a broad range of ensembles it is known that the resolvent $G(z):=(W-z)^{-1}$ concentrates around a deterministic quantity as long as $|\Im z|$ is larger than the local eigenvalue spacing of $W$. This is the so-called single-resolvent local law. Multi-resolvent local laws extend this framework by showing that products of several resolvents also concentrate. In this talk, I will discuss two phenomena that provide additional information beyond the standard local laws: decorrelation in energy space and gains from taking expectations. I will also present several recent applications of these results, which would not be accessible via standard local laws. These applications include non-Gaussian corrections to fluctuations in local laws, law of fractional logarithm for Wigner minor process and hyperuniformity of eigenvalues of non-Hermitian random matrices.  The talk is based on several recent joint works with Z. Bao, G. Cipolloni, L. Erd{\H o}s and J. Henheik.

In this talk I will discuss different types of coproducts on cohomological Hall algebras associated to 3-Calabi-Yau categories, as recently defined by Kinjo-Park-Safronov and Descombes. In general, these are actually a more exotic structure we call a vertex coproduct. I will explain the important special case of a quiver with potential. In this case, these coproducts are related to deformed Drinfeld coproducts on Yangians. I will also discuss how vertex coproducts are related to factorization type coproducts, recently defined in work of Davison-Hennecart-Kinjo-Schiffmann-Vasserot. This is joint work with Jindal and Latyntsev and work in progress with Latyntsev and Descombes.

Diffusion models are central to generative modeling and have been adapted to graphs by diffusing adjacency matrix representations. The challenge of having up to $n!$ such representations for graphs with $n$ nodes is only partially mitigated by using permutation-equivariant learning architectures. Despite their computational efficiency, existing graph diffusion models struggle to distinguish certain graph families and their spectra, unless graph data are augmented with ad hoc features.
This shortcoming stems from enforcing the inductive bias within the learning architecture.
In this work, we leverage random matrix theory to analytically extract the spectral properties of the diffusion process, allowing us to push most of the inductive bias from the architecture into the dynamics. Building on this, we introduce the Dyson Diffusion Model, which employs Dyson's Brownian motion to capture the spectral dynamics of an Ornstein-Uhlenbeck process on the adjacency matrix.
Furthermore, conditioned on the spectral dynamics, we formulate a Lie group diffusion, appropriately modeling the remaining degrees of freedom. Strikingly, the resulting learning problem becomes permutation invariant at the Lie algebra level.
We demonstrate that the Dyson Diffusion Model learns graph spectra accurately and outperforms existing graph diffusion models.
This is joint work with Cai Dieball, Constantin Kogler, Renaud Lambiotte, Arnaud Doucet, Aljaz Godec, and George Deligiannidis.

Paper: Schwarz, T., Dieball, C., Kogler, C., Lambiotte, R., Doucet, A., Godec, A., Deligiannidis, G., 2026, Permutation-Invariant Spectral Learning via Dyson Diffusion, Accepted at NeurIPS 2026

In this talk I will present and explain some recent results motivated by the fundamental conjecture of Erdős and Hajnal (1969) on the existence of high-girth high-chromatic subgraphs. Formally, their conjecture was that for every positive integer g there is a (smallest) function f_g such that every graph of chromatic number at least f_g(k) has a subgraph of girth at least g and chromatic number at least k. This has only been proved for the smallest case g=4 by Rödl, whose proof yields a bound on f_4(k) which is a tower of height Θ(k^2\log k). In this talk I will present a surprising connection between the problem of finding subgraphs of large odd girth and multicolor Ramsey numbers of odd cycles. Using this connection and at the same time extending the recent first superexponential lower bound on the multicolor Ramey numbers of triangles due to OpenAI to fixed odd cycles, we prove that for every odd g\ge 5 there exists a function h_g growing as a power-tower of height (g-3)/2 such that every graph of chromatic number at least h_g(k) contains a subgraph of odd girth at least g and chromatic number at least k. This proves a conjecture of Mohar and Wu (2018), addresses a problem of Erdős and Hajnal (1975) and in the g=5 case reduces the tower-type bound for f_4(k) to a single exponential. 
We then present a much more general meta theorem, which establishes that for a rich family of graph parameters f, including the fractional chromatic number, the Hall ratio and the strict vector chromatic number (a.k.a. Lovász Theta function of the complement), every graph G with f(G) sufficiently large has a subgraph G' of large odd girth with f(G') still large. 
Along the way, I will also explain OpenAI's construction for the multicolor Ramsey lower bound for triangles. 
Comment: The conjecture of Erdős and Hajnal has recently been disproved using AI, but the first paper with this result was not put on arXiv and appeared on a problem-bounty website. Recently, Nguyen and Walczak wrote the first readable exposition of this disproof. I will not present the disproof in my talk and just briefly mention it along the way.

The sl_3 web category is a combinatorial model for the representation theory of the quantum group U_q(sl_3). We study the category from the cellular point of view, exploiting diagrammatic methods and light-ladder bases to compute branching rules, homomorphisms, and composition factors of cell modules when q is a root of unity. We then apply these results to the study of clasps. In the semisimple case, these higher rank analogues of the Jones—Wenzl idempotents can be constructed using Elias’s recursive triple-clasp expansion. When q is a root of unity, however, their construction is considerably more subtle, and we use the aforementioned results from cellularity to obtain formulas for some of the non-semisimple clasps.

Protected sectors of four-dimensional N=2 superconformal field theories connect their operator spectra to algebraic geometry and vertex operator algebras. I will introduce a proposal to encode the Higgs branch and the Schur and Macdonald indices in a single (filtered graded) affine scheme. Its reduced structure recovers the Higgs branch, while its nonreduced structure and filtration captures additional protected information through its arc space. I will explain how these schemes can be constructed from OPE decoupling relations, with their defining relations constrained by "syzygy maximization", and discuss the proposed relation of their coordinate rings to Zhu’s C2-algebras. Argyres–Douglas theories provide concrete examples. Furthermore, Higgs branch RG flows admit an algebraic description through restriction of a defining potential to Slodowy slices. I will conclude with recent progress towards a uniform construction for broad families of Argyres--Douglas theories, Dp(G, O), via Jacobian ideals related to so-called polynomial twisted periods.

We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space R^3 . When the small rigid body shrinks to a point in the sense that its density is constant, we prove that the solution of the fluid-rigid body system converges to a solution of the Navier–Stokes equations in the full space. Based on some L^p − L^q estimates of the fluid–structure semigroup and a fixed point argument, we obtain a uniform estimate of velocity of the rigid body. This allows us to construct admissible test functions which plays a key role in the procedure of passing to the limit. This is a joint work with Pei Su (Nantes).

Selberg’s central limit theorem shows that the modulus of the Riemann zeta function is asymptotically log-normal. Recent work has extended the theorem to the large-value regime. In this talk, I will survey these results and discuss ongoing work on the small-value regime, where new behavior emerges.

Z/2 harmonic 1-forms are generalizations of harmonic 1-forms that allow topological twisting around a subspace of codimension 2. These objects were introduced by Taubes to compactify the moduli spaces of solutions to generalized Seiberg-Witten equations, and they show up in many other gauge theoretical problems.

Donaldson showed there is a deformation theory for so-called non-degenerate Z/2-harmonic 1-forms. In this presentation we study the deformations of the remaining degenerate solutions. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form.

Professor Zhanzing Zhu is going to talk about; 'Understanding Transformers: Token Distinguishability, Learnability, and Safety Alignment'

The underlying mechanism of Large Language Models’ (LLMs) reasoning capabilities remains a big puzzle. This talk will introduce several theoretical works from our group, which attempt to answer some fundamental questions about the Transformer in LLMs:

1. Does softmax attention still work for a very long context (e.g. millions) to distinguish each token for learning? How can we scale the logit to produce a sharp attention distribution?
2. Why can Transformers acquire reasoning ability through Chain-of-Thought (CoT) with fine-tuning? What is the difference between supervised fine-tuning (SFT) and reinforcement learning (RL) for fine-tuning LLMs?
3. Safety alignment in LLMs is fragile in part because it is often shallow: fine-tuning mainly reshapes the model’s behavior near the first few output tokens. Why does the fine-tuning lead to such fragile safety alignment? How can we robustify it?

What should you expect in intercollegiate classes?  What can you do to get the most out of them?  In this session, experienced class tutors will share their thoughts, and a current student will offer tips and advice based on their experience.

All undergraduate and masters students welcome, especially Part B and MSc students attending intercollegiate classes. (Students who attended the Part C/OMMS induction event will find significant overlap between the advice offered there and this session!)

A natural energy arising in the modelling of thin ferromagnetic films and liquid crystals is the Dirichlet energy, along with a penalisation term which favours a particular configuration. In this talk, I present recent work which establishes a quantitative analysis for sequences of minimisers to both the Baby-Skyrme model and two regularisation arising from geometry and physics respectively.

Hyperquiver representations are  generalisations of quiver representations. They are directed hypergraphs with vector spaces assigned to vertices, and for every directed hyperedge, an assignment of a linear map from the tensor product of the source vector spaces to the tensor product of the target vector spaces. We study the moduli of hyperquiver representations, where we use affine GIT to describe the (in)stability of such objects under the action of a product of general linear groups. Our main result introduces the Harder-Narasimhan filtration of a hyperquiver representation. We also derive a formula for the weights of this unique filtration, which specialise to the Harder-Narasimhan slopes of a quiver representation.

Representation theory of $p$-adic groups in characteristic $p$ is something which is still not understood too well. One issue, which we already see for modular representations of finite groups, is that things are not semisimple anymore, and so understanding irreducible representations is not quite enough to have full information. During the talk I will present some of my attempts at creating homological methods to understand extensions of representations of $\operatorname{SL}_2(\mathbb{Q}_p)$, and what those results imply about injective smooth representations (that are my main focus of research).

Reaction-diffusion processes underpin countless biological phenomena. In these systems, fluctuations play an important role, particularly when the number of interacting units is small. In chemical reactions, for example, mean-field approaches such as the law of mass action neglect fluctuations and thus fail to capture absorbing states. In this talk, I will introduce Doi-Peliti field theory (DPFT), a microscopic framework that uses ladder operators to represent particle dynamics in second-quantized form. Its path-integral formulation is exact at the level of the action and allows observables to be calculated systematically in a perturbative expansion.

I will show how DPFT can be used to model biological systems, including oscillating neuronal avalanches [1] and disease spreading on general graphs [2]. I will also present recent progress in microscopic theories of active matter, describing the emergence of effective attraction between interacting run-and-tumble particles [3,4] and the role of delay in reconfigurable chiral microswimmers [5].

[1] J. Pausch, R. Garcia-Millan, and G. Pruessner, Time-dependent branching processes: a model of oscillating neuronal avalanches, Sci. Rep. 10, 13678 (2020). 
[2] I. Bordeu, S. Amarteifio, R. Garcia-Millan, B. Walter, N. Wei, and G. Pruessner, Volume explored by a branching random walk on general graphs. Sci. Rep. 9, 15590 (2019).
[3] R. Garcia-Millan, L. Cocconi, Z. Zhang, M. Bothe, L. Chen, Z. Zhen, G. Pruessner, Effective attraction by repulsion, arXiv:2605.01421 (2026). 
[4] R. Garcia-Millan, Z. Zhang, L. Cocconi, M. Bothe, L. Chen, Z. Zhen, G. Pruessner, Microscopic theory of soft run- and-tumble particles, arXiv:2605.01422 (2026).
[5] J. Peltier, C. Britton, L. Alvarez, R. Garcia-Millan, in preparation (2026).

The model-theoretic construction of the binding group in the presence of internality has proven to be one of the most important group existence theorems. It saw numerous applications, including those in differential Galois theory, envisaged by Poizat, and pursued by Pillay in his generalisation of the strongly normal theory of Kolchin.

We show how to view the binding group construction as an instance of Janelidze's categorical Galois theory. This allows us to apply it in areas beyond model theory, for example to the scheme-theoretic treatment of Pillay's differential Galois theory.

We will discuss connections with Hrushovski’s work on groupoids.

Recently, data-driven methods based on path signatures have gained prominence in mathematical finance. They rely on universal approximation theorems stating that continuous functionals on path space can be approximated uniformly on compact sets by linear functionals of the signature. In financial applications, this has led to the use of Stratonovich-signatures, although Itô integration is often the natural modeling framework.

In this talk, we establish a universality result for signatures of non-geometric rough paths. By augmenting the path with its rough path bracket, we obtain a quasi-shuffle structure that provides the algebraic basis for universality. For continuous semimartingales, this yields a universal approximation property for Itô signatures.

This talk is based on joint work with A. P. Kwossek and D. J. Prömel.

Professor Liza Rebrova will talk about: 'On designing efficient randomized solvers for structured linear systems and optimization problems'

Randomized iterative solvers approach large linear systems through a sequence of small random subproblems, usually with the aim of reducing overall computational cost. While inexpensive iterations alone do not guarantee an efficient solver, I will discuss how these methods can be very effective in multiple contexts, and why their effectiveness depends on local updates exploiting the global structure of the problem well.

Specifically, based on recent work on sketch-and-project methods, I plan to discuss both the design of randomized linear solvers and their role within optimization methods. I will first discuss subspace-constrained randomized coordinate descent, which uses a fixed Nyström core together with fresh block updates to provide implicit preconditioning. This allows a low-rank approximation to accelerate the solution of the full system, with convergence gains quantified by a tail condition number that can be much smaller than the standard condition number. I will then discuss accelerated sketch-and-project as an inner solver for online Newton methods and its effect on the statistical efficiency of averaged iterates.

Luisa Plato (WIAS Berlin) is going to talk about: "Electrically driven mixing in anisotropic electrolytes: an optimal control approach"

Mixing fluids on lab-on-a-chip devices is challenging: flows at this scale are highly viscous, and mechanical stirring is impractical. An alternative is to drive the flow electrically. Applying an electric field to an electrolyte moves the ions, and they in turn set the fluid in motion. We model this by an anisotropic, time-dependent Stokes–Nernst–Planck–Poisson system. The control is the electric potential applied on the boundary, subject to upper and lower bounds, and the aim is to choose the potential so as to maximize the mixing over a given time horizon. In this talk, we prove the existence of an optimal control, present a finite element discretization of the system, and show first numerical results for the discrete optimal control problem, computed with automatically derived adjoints.

The study of cortical dynamics during different states such as decision making, sleep and movement, is an important topic in Neuroscience. Modelling efforts aim to relate the neural rhythms present in cortical recordings to the underlying dynamics responsible for their emergence. We present an effort to characterise the neural activity from the cortex of a mouse during natural sleep, captured through local field potential measurements. Our approach relies on using a discretised Wilson–Cowan Amari neural field model for neural activity, along with a data assimilation method that allows the Bayesian joint estimation of the state and parameters. We demonstrate the feasibility of our approach on synthetic measurements before applying it to a dataset available in literature. Our findings suggest the potential of our approach to characterize the stimulus received by the cortex from other brain regions, while simultaneously inferring a state that aligns with the observed signal.

Using the techniques of arc presentations and open book foliations, we prove new lower bounds on the crossing number and arc index of a satellite knot in terms of it's companion, pattern, and wrapping number.

How to mathematically describe the structure of D-brane charges in string theory turns out to be a surprisingly rich question. Stacking D-branes has a natural mathematical interpretation as summing vector bundles, but the mathematical analogue of brane-antibrane annihilation requires the sophisticated machinery of topological K-theory. In this talk I will present a physics-motivated introduction to the subject using the language of non-abelian gauge theory. 

In recent joint work with Felipe Flores, it was shown that the C*-algebra generated by the quasi-regular representation of Thompson's F < T is purely infinite. The proof involves an ultrapower technique developed by Robert and later considered by Ozawa. In this talk, we will discuss why ultrapowers are relevant in this context and how one works with them, focusing on a few proof techniques.