Past

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.

Wireless communication systems with many antennas and users naturally give rise to large random channel matrices. In one-bit precoding, the transmitted signal is restricted to binary values and is designed using the channel matrix. It is then multiplied by the same channel matrix during transmission. This dependence, together with the entrywise sign nonlinearity, makes the performance difficult to characterize. Focusing on the standard i.i.d. Gaussian channel model, this talk studies the large-system behaviors of two classes of one-bit precoding schemes. For linear-quantized precoding, we use a recursive representation of Haar matrices, known as Householder Dice, to construct an asymptotic scalar signal-plus-independent-Gaussian-noise model.  We next consider nonlinear symbol-level precoding, focusing on a scheme based on convex relaxation followed by one-bit quantization. Using approximate message passing and state evolution, we characterize the limiting empirical laws of the relaxed solution and of the received signal–symbol pairs. These results provide explicit performance predictions and guidance for precoder design.

Any countable group G comes equipped with a canonical dynamical system, namely its conjugation action on its space of subgroups Sub(G). This 0-dimensional compact space is a central object in measured and geometric group theory, especially because it supports invariant and stationary random subgroups.

In general, describing this space is hard. In 2024, Carderi, Gaboriau, Le Maître, and Stalder initiated the study of the space of subgroups of non-amenable Baumslag-Solitar groups BS(m,n). They provided an explicit description of the perfect kernel of Sub(BS(m,n)). This is the largest closed subspace without isolated points, i.e. the space that remains after performing successive derivations that remove the isolated points.

In this talk, I will provide a complete classification of the spaces Sub(BS(m,n)) up to homeomorphism. I will prove that there exist exactly four homeomorphism types among the non-amenable ones. This relies on a detailed study of the Cantor-Bendixson erasing process, which depends on the arithmetical properties of the parameters m,n. This is based on a joint work with Damien Gaboriau, François Le Maître, and Yves Stalder.

Infinite-dimensional hyperbolic spaces serve as universal objects containing all finite-dimensional hyperbolic spaces at once and carry group actions with surprising properties. For example, the Mostow–Karpelevich theorem, which says that a semisimple Lie group acting on a symmetric space preserves a copy of its own symmetric space, fails there. Using kernels of hyperbolic type, Monod–Py and Monod obtain exotic actions on infinite-dimensional real and complex hyperbolic spaces. We explain these constructions and classify all such actions for any combination of real, complex and quaternionic hyperbolic spaces. The classification is partially based on joint work in progress with Bruno Duchesne and Gonzalo Ruiz Stolowicz.

The study of the maximum diameter of $d$-dimensional pseudomanifolds on $n$ vertices was initiated by Criado and Santos as an abstraction of the analogous problem for polytopes, in relation to the Polynomial Hirsch Conjecture.
A series of works by Santos, Criado, Newman and Bohman establish the asymptotics. Here we use a mixture of deterministic and random tools to 
determine the exact value for every large enough $n$ when $d=2$, and for a positive fraction of $n$ when $d\geq 3$.  

Our pseudomanifolds with maximum diameter crucially depend on a surprising connection to the cube of Euler trails in uniform hypergraphs. To this end, for every fixed uniformity $d$ and power $r \geq 2$, we show that satisfying the natural divisibility conditions implies the existence of the $r$th power of an Euler tour/trail in any large enough $d$-uniform hypergraph with large enough codegree.  The talk represents joint work with Stefan Glock, Olaf Parczyk, and Silas Rathke.

The local Langlands correspondence is a conjectural relation between the representations of a p-adic group and its L-parameters (which one can think of as representations of a `dual' group). Given a representation, it is natural to construct the hermitian dual representation. In this talk we present an explicit conjectural local Langlands correspondence, beyond the tempered case and including enhancements. We then show, assuming this conjecture, that constructing the hermitian dual corresponds to twisting its enhanced L-parameter by the compact real form on the dual group, generalising a result of Adams and Vogan. If time permits we will also show that standard desiderata of the local Langlands correspondence are in fact equivalent to this compatibility with hermitian duals.

The study of closed geodesics on hyperbolic surfaces is a subject that shares deep connections with many areas of mathematics. We survey classic and recent results in the subject, emphasizing the relations between three different perspectives: topology, geometry, and arithmetic. In particular, we discuss recently discovered connections with mapping class groups and Teichmüller dynamics. The talk assumes no previous knowledge on the subject.

In this talk, we study the numerical approximation of the stochastic heat equation on the sphere driven by an additive Lévy process. For this, we first prove new regularity results for the solution of the stochastic heat equation under different regularity assumptions on the initial value and driving Lévy process. In these settings, we perform a spectral approximation based on the truncation of the series expansion with respect to the real-valued spherical harmonic functions. Further, we apply a forward resp. backward Euler-Maruyama scheme for the temporal approximation. We prove strong and weak convergence rates for the introduced approximation scheme and present numerical simulations that confirm our theoretical results.

This is joint work with Annika Lang and Andrea Papini.

Hypertoric varieties are conical symplectic singularities equipped with a hamiltonian action of a torus of maximal possible dimension. They behave analogously to toric varieties in many ways, with the hamiltonian torus replacing the dense torus. Examples include: (crepant partial resolutions of) rational surface singularities of type A, the cotangent bundle of projective space, and Nakajima quiver varieties with the `all 1s' dimension vector.

Arbo and Proudfoot conjectured a combinatorial classification of hypertoric varieties by zonotopal tilings. In this talk I will discuss a proof of the conjecture.

Ian Osband from DeepMind is going to talk about: 'The Price of Learning'

Policy gradient sits at the heart of modern RL, including LLM post-training. Yet on image classification, where the gradient of accuracy is exact, it loses to cross-entropy, even on accuracy. Our explanation is that learning is a sequential resource allocation problem. Data, compute and capacity are scarce, and each update sets where the next one starts. Policy gradient ignores this and spends greedily. This talk presents simple methods that price learning, and shows how they improve robustness, compute efficiency and accuracy.

 

TBA

Multiparameter persistent homology (MPH) associates to a vector-valued filter function on a simplicial complex a multiparameter persistence module. In this talk, I will present recent joint work with Heather Harrington and Ulrike Tillmann on the inverse problem for MPH: given a persistence module, what can we say about the data that generated it? To study the geometry of the fibers of MPH, we equip both the space of filters and the moduli space of essentially finite persistence modules with natural stratifications that are preserved by the map. Over each stratum in its image, MPH restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We also bound the dimension of MPH fibers in terms of multigraded Betti numbers, recovering the one-parameter bound of Leygonie and Tillmann as a special case.

Come along for free pizza and to hear about the Mathematrix events this term.

Junior Algebra & Representation Theory starts with a week 1 social. Join us in the Quillen Room!

In this talk, we introduce deep learning schemes for forward–backward systems whose backward component is a backward stochastic Volterra integral equation (BSVIE). BSVIEs generalize classical BSDEs by introducing a second time variable, and arise naturally in recursive utilities with memory, time-inconsistent stochastic control, and dynamic risk measures.

Building on deep solvers for BSDEs, we develop backward learning schemes adapted to the two-time structure of BSVIEs. When the forward process is Markovian, the solution is given by deterministic functions of the two time variables and the forward state, which we approximate with neural networks. We prove the algorithm's convergence by decomposing the error into a time-discretization error and a neural network approximation error, and validate it numerically.

When the forward process is itself of Volterra type, the Markovian structure is lost, and the solution depends on the entire past trajectory. Using a path-dependent PDE representation, we approximate the solution with neural networks taking path signatures as inputs, and test the method on examples with explicit solutions.

Based on joint works with Nacira Agram, and with Nacira Agram and Mounir Zebbar.

Ruth Misener will talk about: 'Optimizing over graphs: Challenges, Formulations, and Applications'

Applications involving optimization over graphs include molecular design, graph neural network verification, neural architecture search, etc. This talk discusses formulating graph spaces using mixed-integer optimization and incorporating application-specific constraints. We discuss computational challenges with these mixed-integer optimization formulations and zoom in on the practical implications for these applications. We mention what has been done (by both ourselves and others) and what other research still needs to be done.

Co-authors: Shiqiang Zhang, Yilin Xie, Christopher Hojny, Juan Campos, Jixiang Qing, Christian Feldmann, David Walz, Frederik Sandfort, Miriam Mathea, Calvin Tsay

This talk is hosted by Rutherford Appleton Laboratory, Harwell Campus

I will present my recent results on global regularity for a canonical shell model of the Navier-Stokes equations, covering the unforced equation and certain classes of forcing. I will discuss the distinction between the forced and unforced regularity problems in this model, with the corresponding Navier-Stokes questions as motivation.

Internal gravity waves (IGWs) are oscillatory modes of motion that exist in the interior of a stably stratified fluid with gravity as the restoring force. In Earth's oceans, these waves play a fundamental role in the transport and mixing of energy, momentum, heat, carbon and nutrients. Much like ocean surface waves, IGWs become unstable and break when their amplitude is large and density interfaces become steeply inclined. This talk will explore the evolution and fate of such waves in two different scenarios.

I will first focus on IGWs that propagate into horizontally sheared currents, explaining how the fate of these waves depends both on their geometry and the strength of the shear flow, and why the total energy dissipated by ensuing turbulence is unexpectedly large. Second, I will discuss the dynamics of internal bores – long, large-amplitude solitary waves that propagate along sharp density interfaces. I will outline the mathematical links between internal bores and gravity currents and, by studying observations off the coast of California, demonstrate their relevance to understanding transport and dissipation on the continental shelf.

David Niederkofler (TU Wien) is going to talk about: "Optimal Time-Adaptivity for Parabolic Problems"

Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems usually do not exhibit the necessary coercive structure that is used in optimality proofs to show a certain quasi-orthogonality, which is crucial for the theory. Recently, by using a new equivalence between quasi-orthogonality and inf-sup stability of the underlying problem, it was shown that an adaptive Crank-Nicolson scheme for the heat equation is optimal under a severe step size restriction. In this work, we use this new approach towards quasi-orthogonality together with Radau IIA methods of any order larger than one to obtain the first adaptive time stepping method for non-stationary PDEs that is provably rate optimal with respect to number of time steps vs. approximation error.

We show that if a group admits an action with no algebraicity, then all of its mixed identities are singular. This generalises a theorem of Abért showing such groups are lawless. Our result confirms a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups, and it also applies to several groups arising in topological contexts. I'll also discuss how ideas from independence relations in model theory arise in the proof and yield more general results, also applying to infinite-dimensional general (and projective) linear groups over arbitrary fields, recovering results of Bradford, Schneider, and Thom.

This is joint work with Michael Pinsker.

In this talk I will discuss homological and homotopical isoperimetric inequalities, including those of higher rank.  I will emphasise what is known in the non-positive curvature case.

The anyons of a (2+1)d topological order are described mathematically by a modular tensor category (MTC). Given an MTC, any n-component link becomes an n-tensor that stores some information about this MTC. For example, the 1-component unknot (twisted once) gives a vector "T" that stores the topological spins of all anyons; the 2-component Hopf link evaluates to matrix "S", from which the Verlinde formula recovers the fusion rules of the anyons. Do T and S store all the data? The answer is no: there are different MTCs with the same S and T. So the question becomes what other links to include in this list of tensors to tell MTCs apart? 
I will start by telling you how to turn any link you can drawn on a piece of paper (or a whiteboard) in to a tensor, then review the T and S story and then discuss some recent work where people considered adding the Whitehead link and the Borromean rings. There will be lots of drawings!

Kesten's theorem characterises amenability in terms of spectral radii: a symmetric random walk on a group G has spectral radius 1 if and only if it generates an amenable subgroup.

We prove a quantitative "1%" version of this result for G a free group, or a fully residually free group: for every ε > 0, there is δ > 0 such that for every probability measure μ on G with convolution operator norm at least ε, there is a co-set of an amenable subgroup of G of mass > δ under μ.

This answers a question of Emmanuel Breuillard. We believe the same result to hold in a broader setting of linear groups, with δ depending on the dimension. The proof uses uniform exactness and a generalisation of the Tits alternative.

Joint work with Emmanuel Breuillard.