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.

 

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.

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.

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 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.

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.

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.

TBA

to follow

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. 

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.

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.

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.

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.

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.

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).

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).

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.

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!)

Associate 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?

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.

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).

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.

TBA

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 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.

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.

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.

to follow

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