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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

TBA

The dynamic nature of first order methods can be interpreted by means of continuous time models. In this survey talk, we explain how physical concepts like acceleration, inertia or momentum have been used to improve the performance of convex optimization algorithms. 

We give special attention to the historical evolution of complexity results, especially in the form of convergence rates, under the light of this connection. We also discuss different ways in which acceleration schemes can be applied when the smoothness or strong convexity parameters are unknown, and how these ideas extend to saddle point and constrained problems. 

The talk gives some survey about recent applications of finitely additive measures to Lebesgue integrable functions. After a short introduction to such measures and related integrals, purely finitely additive measures are of particular interest. Special examples are given and, as a first application, an integral representation for the precise representative of Lebesgue integrable functions is provided. Then, based on a general approach to traces, a new version of the Gauss-Green formula is introduced, where neither a pointwise trace nor a pointwise normal is needed on the boundary. This allows e.g. the treatment of inner boundaries and of concentrations on the boundary. A second boundary integral is used to handle singularities that hadnot been accessible before. Finally, weak versions of differentiability for Lebesgue integrable functions are discussed, a mean value formula for a class of Sobolev functions is given, and a new approach to the generalized derivatives in the sense of Clarke is provided.

TBA

In recent work with Jef Laga, we adapt a construction of Slodowy to build families of algebraic curves in graded Lie algebras (this generalizes earlier work of Thorne). This required an understanding of nilpotent orbits in Vinberg representations, and it raised some interesting questions about these orbits that we were able to answer. Our motivation comes from proofs in arithmetic statistics in which orbits in certain representations are used to parametrize rational points on curves. In this talk, Beth Romano gives an introduction to these ideas via examples.

to follow

The Poisson boundary of a probability measure on a countable group is a probability space endowed with a stationary group action that captures the asymptotic behavior of the associated random walk. Since its introduction by Furstenberg in the 1960s, the study of Poisson boundaries and stationary actions has become a powerful tool for understanding geometric and algebraic properties of groups.

In this talk, I will discuss connections between stabilizers of stationary actions, in particular, those arising from the Poisson boundary, and the C*-simplicity of the associated reduced group C*-algebra. I will also address the (seemingly unrelated) problem of realizing different Poisson boundaries on a common underlying topological model. The talk is based on joint work with Anna Cascioli and Martín Gilabert Vio, and with Josh Frisch.

Two hundred years ago, Robert Brown observed the statistics of the motion of grains of pollen in water. It took almost one hundred years for Einstein and others to develop an effective theory describing this motion as that of a random walker. In this talk, I will challenge a key implication of this well established theory. When studying systems with very large numbers of particles diffusing together, I will argue that the Einstein random walk theory breaks down when it comes to predicting the statistical behavior of extreme particles—those that move the fastest and furthest in the system. In its place, I will describe a new theory of extreme diffusion which captures the effect of the hidden environment in which particles diffuse together and allows us to interrogate that environment by studying extreme particles. I will highlight one piece of mathematics that led us to develop this theory—a non-commutative binomial theorem—and hint at other connections to integrable probability, quantum integrable systems and stochastic PDEs.