Normal heart function relies of the fine-tuned synchronization of cellular components. In healthy hearts, calcium oscillations and physical contractions are coupled across a synchronised network of 3 billion heart cells. When the process of functional isolation of rogue cells isn’t successful, the network becomes maladapted, resulting in cardiovascular diseases, including heart failure and arrythmia. To advance knowledge on this normal-to-disease transition we must first address the lack of a mechanistic understanding of the plastic readaptation of these networks. In this talk I will explore coupling and loss of synchronisation using a mathematical model of calcium oscillations informed by experimental data. I will show some preliminary results pointing at the heterogeneity hidden behind seemingly uniform cell populations, as a causative mechanism behind disrupted dynamics in maladapted networks.

In this talk, I'm going to give an introduction to my area of research, which concerns automorphic L-functions. We're going to start by introducing the ring of adeles and how it leads us to an integral representation of the Riemann zeta function. We'll see how this can be generalised for an arbitrary automorphic representation and pose general conjectures which resemble the Riemann Hypothesis. I'll finish by presenting the statement and an idea behind my recent result related to those conjectures.

theories on the lattice based on symmetry disentanglers: constant-depth
circuits of local unitaries that transform not-on-site symmetries into on-
site ones. When chiral symmetry can be realized not-on-site and such a
disentangler exists, the symmetry can be implemented in a strictly local
Hamiltonian and gauged by standard lattice methods. Using lattice ro-
tor models, we realize this idea in 1+1 and 3+1 spacetime dimensions
for U (1) symmetries with mixed ’t Hooft anomalies, and show that sym-
metry disentanglers can be constructed when anomalies cancel. As an
example, we present an exactly solvable Hamiltonian lattice model of the
(1+1)-dimensional “3450” chiral gauge theory, and we argue that a related
construction applies to the U (1) hypercharge symmetry of the Standard
Model fermions in 3+1 dimensions. Our results open a new route toward
fully local, nonperturbative formulations of chiral gauge theories.

Zigzag persistence enables tracking topological changes in time-dependent data such as video streams. Nevertheless, traditional methods face severe computational and memory bottlenecks. In this talk, I show how the zigzag persistence of image sequences can be reduced to a graph problem, making it possible to leverage the near-linear time algorithm of Dey and Hou. By invoking Alexander duality, we obtain both H₀ and H₁ at the same computational cost, enabling fast computation of homological features. This speed-up brings us close to real-time analysis of dynamical systems, and, if time permits, I will outline how it opens the door to new applications such as the study of PDE dynamics using zigzag persistence, with the Gray-Scott diffusion equation as a motivating example.

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

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.

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.

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 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.

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.

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.

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.

to follow

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

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.

TBA

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.

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.

TBA

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.  

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.

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.

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.

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.

TBA

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.

to follow

(Joint seminar with Random Matrix Theory)

I will discuss a 2-dimensional model of random walk in random environment known as line model. The environment is described by two independent families of i.i.d. random variables dictating rates of jumps in vertical, respectively horizontal directions, and whose values are constant along vertical, respect. horizontal lines. When jump rates are heavy-tailed in one of the directions, the random walk becomes superdiffusive in that direction, with an explicit scaling limit written as a two-dimensional Brownian motion time-changed (in one of the components) by a process introduced by Kesten and Spitzer in 1979. I will present ideas of the proof of this result, which relies on appropriate time-change arguments.  In the case of a fully degenerate environment, I will present a sufficient condition for non-explosion of the process (which is also believed to be sharp), as well as conjectures on the associated scaling limit.

This is based on joint work with J.-D. Deuschel (TU Berlin).