Past

Graph products of groups were introduced by Green as a construction that encompasses both direct products and free products. Likewise, the notion of graph product of von Neumann algebras, introduced by Caspers and Fima, recovers both tensor products and free products. Camille Horbez will present rigidity theorems for graph products of tracial von Neumann algebras, and discuss the computation of their symmetries, drawing parallels with the case of groups. This is a joint work with Adrian Ioana. 

Random walks on graphs can mix slowly. To speed it up, imagine that at each step instead of choosing the neighbor at random, there is a small probability $\varepsilon > 0$ that we can choose it. We show that in this case, at least for graphs of bounded degree, there is a way to steer the walk so that we visit every vertex in $n^{1+o(1)}$ many steps. The key to this result is a way to decompose arbitrary graphs into small-diameter pieces.

I will discuss some recent progress on the Donaldson Segal programme, and in particular how calibrated cycles (coassociative submanifolds, special Lagrangians) arise from the large mass limit of $G_2$ and Calabi Yau monopoles.

Thompson’s groups, introduced by Thompson in 1965, have had a lot of attention in the last fifty years. Being finitely presented, a natural question is to compute their Dehn function. All three groups are conjectured to have quadratic Dehn function; this conjecture was confirmed for Thompson’s group 𝐹 by Guba in 2006. During Matteo Migliorini's talk, we show how to deduce from Guba’s result that Thompson’s group 𝑇 has a quadratic Dehn function as well.

Koszul algebras are positively graded algebras with very amenable homological properties. Typical examples include the polynomial ring over a field or the exterior and symmetric algebras of a vector space. A category is called Koszul if it has a grading with which it is equivalent to the category of graded modules over a Koszul algebra. A famous example (due to Soergel) is the principal block of category $\mathcal{O}$ for a semisimple Lie algebra. Koszulity is a very nice property, but often very difficult to check. In this talk, Thorsten Heidersdorf (Newcastle University) will give a criterion that allows to check Koszulity in case the category is a graded semi-infinite highest weight category (which is a structure that appears often in representation theory). This is joint work with Jonas Nehme and Catharina Stroppel.

Recent years have seen the expansion of the traditional notion of symmetry in quantum theory to so-called generalised or categorical symmetries, which may in particular be non-invertible. This seems to be at odds with Wigner's theorem, which asserts that quantum symmetries ought to be implemented by (anti)unitary -- and hence invertible -- operators on the Hilbert space. In this talk, we will try to resolve this puzzle for generalised symmetries that are described by (higher) fusion categories. After giving a gentle introduction to the latter, we will discuss how one can associate an inner-product-preserving operator to (possibly non-invertible) symmetry defects and illustrate our construction through concrete examples. Based on the recent work 2602.07110 with Gai and Schäfer-Nameki.

In this talk we describe several aspects related to the theory of some anisotropic parabolic equations. The anisotropy shown in such equations will appear in the form of porous medium, in the fast and porous medium diffusion regime. In particular, we show the existence of selfsimilar fundamental solutions, which is uniquely determined by its mass, and the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. 

The investigation of both models are objects of joint works with F. Feo and J. L. V´azquez.

Any two 1 by 1 real matrices commute.  This is in general not the case for 2 by 2 real matrices.  However, if A, B, C, and D are any 2 by 2 real matrices, then ABCD - ABDC - ACBD + ACDB + ADBC - ADCB - BACD + BADC + BCAD - BCDA - BDAC + BDCA + CABD - CADB - CBAD + CBDA + CDAB - CDBA - DABC + DACB + DBAC - DBCA - DCAB + DCBA = 0.  This identity is the first instance of a general result of Amitsur and Levitski; I will explain a simple graph-theoretic proof due to Swan.

In the first part of the talk, I will present an overview of recent advances in the description of diffusion-reaction processes and their first-passage statistics, with the special emphasis on the role of the boundary local time and related spectral tools. The second part of the talk will illustrate the use of these tools for the analysis of boundary-catalytic branching processes. These processes describe a broad class of natural phenomena where the population of diffusing particles grows due to their spontaneous binary branching (e.g., division, fission, or splitting) on a catalytic boundary located in a complex environment. We investigate the possibility of the geometric control of the population growth by compensating the proliferation of particles due to catalytic branching events by their absorptions in the bulk or on boundary absorbing regions. We identify an appropriate Steklov spectral problem to obtain the phase diagram of this out-of-equilibrium stochastic process. The principal eigenvalue determines the critical line that separates an exponential growth of the population from its extinction. In other words, we establish a powerful tool for calculating the optimal absorption rate that equilibrates the opposite effects of branching and absorption events and thus results in steady-state behavior of this diffusion-reaction system. Moreover, we show the existence of a critical catalytic rate above which no compensation is possible, so that the population cannot be controlled and keeps growing exponentially. The proposed framework opens promising perspectives for better understanding, modeling, and control of various boundary-catalytic branching processes, with applications in physics, chemistry, and life sciences.

Vertex operator algebras provide a succinct mathematical description of the chiral sector of two-dimensional conformal field theories. Various extensions of the framework of vertex operator algebras have been proposed in the literature which are capable of describing full two-dimensional conformal field theories, including both chiral and anti-chiral sectors. I will explain how the notion of a full vertex operator algebra can be elegantly described using the modern language of factorisation algebras developed by Costello and Gwilliam. This talk will be mainly based on [arXiv:2501.08412].

In this week's Fridays@2, a panel of representatives from a range of companies who employ mathematics graduates will be here to answer your questions.A degree in mathematics opens doors far beyond academia, but what do those paths really look like? Join us for a panel event bringing together mathematicians working across Finance, Digital Services, Technology, Consulting, Data Analytics, and Teaching.

Our speakers will share their career journeys, how they moved from studying mathematics into industry roles, and what their day to day work involves. This is your opportunity to gain insight into the skills employers value, the challenges and opportunities in different sectors, and the many ways mathematical thinking shapes real world impact.

Whether you already have a clear goal or are still exploring your options, come along with your questions and curiosity and discover where maths could take you.

I will present joint work with M. Botnan, S. Oppermann, and J. Steen on multiparameter persistent homology in degree zero. It is known that arbitrary diagrams of vector spaces and linear maps can be realized as homology of diagrams of simplicial complexes in some positive degree. We study the more restrictive case of degree zero, which corresponds to diagrams freely generated from sets and set maps. Despite their seemingly simple combinatorial nature, a full understanding of the structure of these representations remains elusive. I will summarize our findings and discuss some conjectures.

I’ll discuss a top down example of holography at zero ’t Hooft coupling. The gauge side is self-dual N=4 SYM, whilst the gravitational side is the closed topological B-model on a certain Calabi-Yau 7-fold that fibres over twistor space. As an application, I’ll discuss an interpretation of correlation functions of certain determinant operators in sI’ll discuss a top down example of holography at zero ’t Hooft coupling. The gauge side is self-dual N=4 SYM, whilst the gravitational side is the closed topological B-model on a certain Calabi-Yau 7-fold that fibres over twistor space. As an application, I’ll discuss an interpretation of correlation functions of certain determinant operators in self-dual SYM in terms of giant graviton D-branes on this 7-fold.elf-dual SYM in terms of giant graviton D-branes on this 7-fold.

Joint with Atul Sharma arxiv:2512.04152.

We survey the life of a Turing pattern, from initial diffusive instability through the emergence of dominant spatial modes and to an eventual spatially heterogeneous pattern. While many mathematically ideal Turing patterns are regular, repeating in structure and remaining of a fixed length scale throughout space, in the real world there is often a degree of irregularity to patterns. Viewing the life of a Turing pattern through the lens of spatial modes generated by the geometry of the bounded space domain housing the Turing system, we discuss how irregularity in a Turing pattern may arise over time due to specific features of this space domain or specific spatial dependencies of the reaction-diffusion system generating the pattern.

Shimura varieties are highly symmetric algebraic varieties that play an important role in the Langlands program. In the first part of the talk, I will try to give you a sense of what they are like, with a focus on their different kinds of symmetries. In the second part of the talk, I will introduce Igusa stacks, a powerful new tool in the study of Shimura varieties. To illustrate their role, I will discuss how Igusa stacks can shed light on the many structures that exist on the intersection cohomology of Shimura varieties. This is joint work in progress with Linus Hamann and Mingjia Zhang.

In the past years, deep learning algorithms have been applied to numerous classical problems from mathematical finance. In particular, deep learning has been employed to numerically solve high-dimensional derivatives pricing and hedging tasks. Theoretical foundations of deep learning for these tasks, however, are far less developed. In this talk, we start by revisiting deep hedging and introduce a recently developed adversarial training approach for making it more robust. We then present our recent results on theoretical foundations for approximating option prices, solutions to jump-diffusion PDEs and optimal stopping problems using (random) neural networks, allowing to obtain more explicit convergence guarantees. We address neural network expressivity, highlight challenges in analysing optimization errors and show the potential of random neural networks for mitigating these difficulties.

Dr Carolina Urzua Torres will talk about 'Paving the way to a T-coercive method for the wave equation'

Space-time Galerkin methods are gradually becoming popular, since they allow adaptivity and parallelization in space and time simultaneously. A lot of progress has been made for parabolic problems, and its success has motivated an increased interest in finding space-time formulations for the wave equation that lead to unconditionally stable discretizations. In this talk I will discuss some of the challenges that arise and some recent work in this direction.

In particular, I will present what we see as a first step toward introducing a space-time transformation operator $T$ that establishes $T$-coercivity for the weak variational formulation of the wave equation in space and time on bounded Lipschitz domains. As a model problem, we study the ordinary differential equation (ODE) $u'' + \mu u = f$ for $\mu>0$, which is linked to the wave equation via a Fourier expansion in space. For its weak formulation, we introduce a transformation operator $T_\mu$ that establishes $T_\mu$-coercivity of the bilinear form yielding an unconditionally stable Galerkin-Bubnov formulation with error estimates independent of $\mu$. The novelty of the current approach is the explicit dependence of the transformation on $\mu$ which, when extended to the framework of partial differential equations, yields an operator acting in both time and space. We pay particular attention to keeping the trial space as a standard Sobolev space, simplifying the error analysis, while only the test space is modified.
The theoretical results are complemented by numerical examples.  

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or RandomizedSVD. Although these SVD approaches---especially RandomizedSVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work: (1) It is adaptive in the sense that it takes as an input parameter the desired tolerance, rather than an a priori guess of the numerical rank. (2) It typically runs significantly faster than both existing CUR algorithms and techniques such as RandomizedSVD, in particular when these methods are run in an adaptive rank mode. Its asymptotic complexity is  $\mathcal{O}(mn + (m+n)r^2 + r^3)$ for an $m\times n$ matrix of numerical rank $r$. (3) It relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds.

You will likely be familiar with the notion of a hydrogen atom, having seen something about its discrete energy levels and orbitals at some point or another. This is an example of a quantum system. In this talk, we explore what transpires when taking a quantum system and placing it into a three-dimensional container having some prescribed geometry. In the limit where the container is large (relative to the natural lengthscale of the quantum system), its influence over the quantum system is negligible; yet, as the container is made small (comparable to the aforementioned lengthscale), geometric information intrinsic to the container plays an important role in determining the energy and orbital structure of the system. We describe how to do (numerically-assisted) perturbation theory in this small-container limit and then match it to the large-box regime, using a combination of these asymptotics and direct simulations to tell the story of geometrically confined quantum systems. Much of our focus will be on linear Schrödinger equations governing single-particle quantum systems; however, time permitting, we will briefly discuss how to do similar things to study geometrically confined nonlinear Schrödinger equations, with geometric confinement of Bose-Einstein condensates being a primary motivation. Geometric confinement of an attractive Bose-Einstein condensate can, for instance, modify the collapse threshold and enhance stability, with the particular choice of confining geometry shifting the boundary of instability, staving off the collapse which is prevalent in three-dimensional attractive condensates.

A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; a core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, if they exist, they are unique. Moreover, $\omega$-categorical theories always have a core companion, which is also $\omega$-categorical.

In the first part of this talk, we show that many model-theoretic properties, such as stability, NIP, simplicity, and NSOP, are preserved when moving to the core companion of a complete theory.

In the second part of this talk, we study the notion of core interpretability, which arises by taking the core companions of structures interpretable in a given structure. We show that there are structures which are core interpretable but not interpretable in $(\mathbb{N};=)$ or $(\mathbb{Q};<)$. We conjecture that the class of structures which are core interpretable in $(\mathbb{N};=)$ equals the class of $\omega$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We present some partial results in this direction, including the answer a question of Walsberg.

This is joint work with Manuel Bodirsky and Bertalan Bodor.

 Given a group acting on a metric space X, one is often interested in the kernel of the action, consisting of those elements that fix every point of X. From a coarse geometric perspective, however, this notion is unsatisfactory, as the kernel is generally not invariant under G-equivariant quasi-isometries. To address this, one can instead consider the coarse kernel, defined as the collection of group elements that move every point of X by a uniformly bounded amount. In this talk, we study this coarse kernel under various assumptions on the action. 

When the action is geometric, we give a purely algebraic characterisation of the coarse kernel as the FC-centre of the group. We then specialise to actions on CAT(0) spaces, where we investigate the coarse kernel via the curtain model, a hyperbolic space associated to a CAT(0) space introduced by Petyt, Spriano, and Zalloum. Along the way, we will meet centralisers, boundaries, and actions on hyperbolic spaces! This is based on my summer project supervised by Davide Spriano and Harry Petyt.

Given a Galois representation attached to a regular algebraic cuspidal automorphic representation, the Hodge--Tate weight of the Galois representation is matched with the weight of the automorphic representation. Serre weight conjectures are mod p analogue of such a correspondence, relating ramification at p of a mod p Galois representation and Serre weights of mod p algebraic automorphic forms. In this talk, I will discuss how to understand Serre weight conjectures and modularity lifting as a relationship between representation theory of finite groups of Lie type (e.g. GSp4(Fp)) and the geometry of p-adic local Galois representations. Then I will explain the proof idea in the case of GSp4. This is based on a joint work with Daniel Le and Bao V. Le Hung.

AF-embeddability, i.e., the question whether a given C*-algebra can be realised as a subalgebra of an AF-algebra, has been studied for a long time with prominent early results by Pimsner and Voicuescu who constructed such embeddings for irrational rotation algebras in 1980. Since then, many AF-embeddings have been constructed for concrete examples but also many non-constructive AF-embeddability results have been obtained for classes of algebras typically assuming the UCT. 

In this talk by Joachim Zacharias, we will consider a separable unital C*-algebra A of decomposition rank at most 1 and construct from a suitable system of 1-decomposable cpc-approximations an AF-algebra E together with an embedding of A into E and a conditional expectation of E onto A without assuming the UCT. We also consider some extensions of this inclusion and indicate some applications.

The asymptotic large N limit of random matrices often transforms classical concepts (independence, cumulants, partitions of sets) into their free counter-parts (free independence, free cumulants, non-crossing partitions) and the limit of random matrices gives rise to interesting operator algebras. I will explain these relations, with a particular emphasis on the effect of non-linear functions on the entries of random matrices

How much does the derived ($\infty$-)category of a scheme remember? In this talk, I will consider this question in the context of deformation theory and make precise the close relationship between the deformation theory of a scheme and its derived category. Along the way, I will also introduce some basics of derived deformation theory and pay special attention to mixed and positive characteristic phenomena. This talk is based on my recent work https://arxiv.org/abs/2512.24347.

I'll discuss how to show 3D Poincaré duality and residual finiteness are together incompatible with property (T).

An edge-colouring of a graph $G$ can fail to be rainbow for two reasons: either it contains a monochromatic cherry (a pair of incident edges), or a monochromatic matching of size two. A colouring is a proper colouring if it forbids the first structure, and a star-colouring if it forbids the second structure. I will talk about the problem of determining the maximum number of colours in a star-colouring of a large complete graph which does not contain a rainbow copy of a given graph $H$. This problem is a special case of one studied by Axenovich and Iverson on generalised Ramsey numbers.

Joint work with Allan Lo, Klas Markström, Dhruv Mubayi, Maya Stein and Lea Weber.

We introduce a graph renormalization procedure based on the coarse-grained Laplacian, which generates reduced-complexity representations across scales. This method retains both dynamics and large-scale topological structures, while reducing redundant information, facilitating the analysis of large graphs by decreasing the number of vertices. Applied to graphs derived from electroencephalogram recordings of human brain activity, our approach reveals collective behavior emerging from neuronal interactions, such as coordinated neuronal activity. Additionally, it shows dynamic reorganization of brain activity across scales, with more generalized patterns during rest and more specialized and scale-invariant activity in the occipital lobe during attention.

Yotam Hendel will discuss how polynomial maps can be studied by examining the analytic behavior of pushforwards of regular measures under them over finite and local fields. 

The guiding principle is that bad singularities of a map are reflected in poor analytic behavior of its pushforward measures. Yotam will present several results in this direction, as well as applications to areas such as counting points over finite rings and representation growth. 

Based on joint work with I. Glazer, R. Cluckers, J. Gordon, and S. Sodin.

The flow of viscous fluid through a soft porous medium exerts drag on the matrix and induces non-uniform deformation. This behaviour can become increasingly complicated when the medium has a complex rheology, such that deformations exhibit elastic (reversible) and plastic (irreversible) behaviour, or when the rheology has a viscous component, making the response of the medium rate dependent. This is perhaps particularly the case when compaction is repeated over many cycles, or when additional forces (e.g. gravity or an external load) act simultaneously with flow to compact the medium, as in many industrial and geophysical applications. Here, we explore the interaction of viscous effects with elastic and plastic media from a theoretical standpoint, focussing on unidirectional compaction. We initially consider how the medium responds to the reversal of flow forcing when some of its initial deformation is non-recoverable. More generally, we explore how spatial variations in stress arising from fluid flow interact with the stress history of the sample when some element of its rheology is plastic and rate-dependent, and characterise the response of the medium depending on the nature of its constitutive laws for effective stress and permeability.

Consider a scalar conservation law with a spatially discontinuous flux at a single point x = 0, and assume that the flux is uniformly convex when x ̸= 0. I will discuss controllability problems for AB-entropy solutions associated to the so-called (A, B)-interface connection. I will first present a characterization of the set of profiles of AB-entropy solutions at a time horizon T > 0, as fixed points of a backward-forward solution operator. Next, I will address the problem of identifying the set of initial data driven by the corresponding AB-entropy solution to a given target profile ω T, at a time horizon T > 0. These results rely on the introduction of proper concepts of AB-backward solution operator, and AB-genuine/interface characteristics associated to an (A, B)-interface connection, and exploit duality properties of backward/forward shocks for AB-entropy solutions.
 

Based on joint works with Luca Talamini (SISSA-ISAS, Trieste)

The classical Leopoldt conjecture predicts that the global units of a number field (tensored with Qp) inject into the local units at p. In this talk, I'll discuss some non-abelian generalisations of this in the setting of Galois representations.

The talk will be about a natural percolation model built from the so-called Brownian loop soup. We will give sense to studying its phase transition in dimension d = 2 + epsilon, with epsilon varying in [0,1], and discuss how to perform a rigorous „epsilon-expansion“ in this context. Our methods give access to a whole family of universality classes, and elucidate the behaviour of critical exponents etc. near the (lower-)critical dimension, which for this model is d=2. 

Based on joint work with Wen Zhang.

By work of Goodwillie-Weiss, given any manifold $M$ with boundary, there is a cosimplicial space whose totalization is a close approximation to the space of embedding of $[0,1]$ in $M$ with fixed behaviour at the boundary. The resulting homology spectral sequence is known to collapse rationally for $M=\mathbb{R}^n$ by work of Lambrechts-Turchin and Volic. I will explain a new proof of this result which can be generalized to a manifold of the form $M=X\times[0,1]$ with $X$ a smooth and proper complex algebraic variety. This involves constructing an action of some Galois group on the completion of the cosimplicial space. This is joint work with Pedro Boavida de Brito and Danica Kosanovic.

We consider a class of Lagrangian sections L contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: L is isomorphic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. On
surfaces and threefolds, under more restrictive assumptions, this result can be used to show a precise relation with Bridgeland stability, as predicted by Joyce. Based on arXiv:2505.07228 and arXiv:2508.17709.

Dominik Lukeš from the AI Competency Centre will give an introductory survey of AI in relation to programming.