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 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.

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 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.

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.

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.

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.

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.

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.

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. 

(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). 

A central goal in the study of quantum chaos is being able to make universal statements about the dynamics of generic Hamiltonian systems. Under time evolution, an initially local operator progressively explores the Hilbert space of a system becoming increasingly non-local in the process. We will see that this idea lends itself to a natural notion of operator complexity measured (in the Hilbert space of operators) by the overlap of a time-evolving operator with a basis naturally adapted to time evolution and stratified by the growth in the operator's support. The information contained in this so-called Krylov basis is encoded in a sequence called the Lanczos coefficients which quantify the rate at which an operator is "pushed" along the Krylov basis to successively more complex elements. The universal operator growth hypothesis is then the conjecture that the Lanczos coefficients grow asymptotically linearly in any quantum chaotic system. In this talk, I will present an overview of these ideas and see how they manifest in the example of the well-studied SYK model. This talk is primarily based on 1812.08657.

I will introduce the concept of Booleanization of a theory and state some examples, including ring of adeles of number fields and sheaves of structures, and discuss some model theoretic properties.

This is joint work with Ehud Hrushovski from
Jamshid Derakhshan and Ehud Hrushovski, Imaginaries, Products, and the Adele Ring, https://arxiv.org/abs/2309.11678v3

In this talk I will present a few topics of recent interest that centre around the theme of “driven interfacial hydrodynamics”: fluid mechanical systems in which droplets and particles are self-propelled through interaction with the environment. I will also present some very recent work on using differentiable physics (a branch of physics-informed machine learning) to determine constitutive relations for highly plasticised metals.

This talk will contain elements of fluid dynamics, experimental mechanics, dynamical systems, statistical physics, and machine learning.

Neurons interact via spikes, which is a pulse-like, discontinuous mechanism. Their mean-field PDE description gives Fokker-Planck equations with novel nonlinearities. From a probability point of view, these give rise to Mckean-Vlasov equations involving hitting times. Similar mechanisms also arise in models for systemic risk in mathematical finance, and the supercooled Stefan problem. In this talk, we will first present models for spiking neurons: both microscopic particle models and macroscopic PDE models, with an emphasis on the general mathematical structure. A central question for these equations is the finite-time blow-up of the firing rate, which scientifically corresponds to the synchronization of a neuronal network. We will discuss how to continue the solution physically after the blow-up, by introducing a new timescale. The new timescale also helps us to understand the long term behavior of the equation, as it reveals a hidden contraction structure in the hyperbolic case. Finally, we will present a recently developed numerical solver based on this framework. Numerical tests show that during the synchronization the standard microscopic solver suffers from a rather demanding time step requirement, while our macro-mesoscopic solver does not.

Timely optimization problems are high-dimensional, calling for dimensionality reduction techniques to solve them efficiently. The random embedding strategy, which optimizes the objective along a low-dimensional subspace of the search space, is arguably the simplest possible dimensionality reduction method. Recent works quantify the probability of success of this strategy to solve the original problem by lower bounding the probability of a random subspace to intersect the set of approximate global minimizers. These works showed that, when the objective has low effective dimension (i.e., is only varying along a low-dimensional subspace of the search space), random embeddings of sufficiently large dimension solve the original high-dimensional problem with probability one. In this work, we relax the low effective dimension assumption by considering objectives with anisotropic variability, namely, Lipschitz continuous functions whose Lipschitz constant is small (though nonzero) when the function is restricted to a high-dimensional subspace. Exploiting tools from stochastic geometry, we lower bound the probability for a random subspace to intersect the set of approximate global minimizers of these objectives, hence, the probability of random embeddings to succeed in solving (approximately) the original global optimization problem. Our findings offer deeper insights into the role of the dimension of the optimization problem in this probability of success.

TBA

Over 30 years has passed since the original proof of Fermat's Last Theorem by Wiles and Taylor—Wiles. There are now several proofs known to humanity, and I'm currently teaching one of them to a computer. This made me try to find out what the most ergonomic route was nowadays, and I found it by asking Richard Taylor what it was. In the talk I will summarise how to prove Fermat's Last Theorem in 2026, highlighting the differences between the modern method and the original route discovered by Wiles (we do use p=3, but in a different way). I won't talk much at all about Lean and essentially none of the work I will present is my own; this will just be a standard number theory seminar, and probably everything in it will already be known to the experts, but hopefully younger people will learn something.

Graph causal optimal transport is a recent generalisation of causal optimal transport in which the allowed couplings satisfy causal restrictions given by a directed graph. Inspired by applications to structural causal models, it was originally introduced in Eckstein and Cheridito (2023). We study fundamental properties of graph causal optimal transport, with a particular focus on its induced Wasserstein distance. Our main result is a full characterisation of the directed graphs for which this associated Wasserstein distance is indeed a metric, an open problem in the original paper. We fully characterise the gluing properties of graph causal couplings, prove denseness of Monge maps, and provide a dynamic programming principle. Finally, we present an application to continuity of stochastic team problems. Based on joint work with Jan Obloj.

Alday & Maldacena conjectured an equivalence between string amplitudes in AdS5 ×S5 and null polygonal Wilson loops together with a duality with amplitudes for planar N = 4 super-Yang-Mills (SYM).  At strong coupling this identifies SYM amplitudes with (regularized) areas of minimal surfaces in AdS.  They reformulated the minimal surface problem as a Hitchin system and in collaboration with Gaiotto, Sever & Vieira they introduced a Y-system and a thermodynamic Bethe ansatze (TBA) expressing the complete integrability that could in principle be used to solve for the amplitude at strong coupling. This lecture will review the parts of this material that we need and use them to identify new geometric structures on the spaces of kinematics for super Yang-Mills amplitudes/null polygonal Wilson loops.   In AdS3, the kinematic space is the cluster variety  M_{0.n} X M_{0,n}, where M_{0,n} is the moduli space of n points on the Riemann sphere moduli Mobius transformations.   The nontrivial part of these amplitudes at strong coupling, the remainder function,  turns out to be the (pseudo-)K ̈ahler scalar for a (pseudo-)hyper-Kaher geometry. It satisfies an integrable system and we give its its Lax form. The result follows from a new perspective on Y-systems more generally as defining the natural twistor space associated to the hyperkahler geometry of the Hitchin moduli space for these minimal surfaces. These connections in particular allows us to prove that  the amplitude at strong coupling satisfies the Plebanski equations for a hyperKahler scalar for  these pseudo-hyperk ̈ahler and related geometries. These hyperkahler geometries are nontrivial, (not semiflat) with a nontrivial TBA that encodes the mutations of the cluster structure.  These new structures underpinning the N=4 SYM amplitudes  will be important beyond strong coupling.  This is based on joint work with Hadleight Frost and Omer Gurdogan, https://arxiv.org/abs/2306.17044.

Gaussian fields arise in a variety of contexts in both pure and applied mathematics. While their geometric properties are well understood, their topological features pose deeper mathematical challenges. In this talk, I will begin by highlighting some motivating examples from different domains. I will then outline the classical theory that describes the geometric behaviour of Gaussian fields, before turning to more recent developments aimed at understanding their topology using the Wiener chaos expansion.

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The Bogomolov-Miyaoka-Yau inequality for minimal compact complex surfaces of general type was proved in 1977 independently by Miyaoka, using methods of algebraic geometry, and by Yau, as an outgrowth of his proof of the Calabi conjectures. In this talk, we outline our program to prove the conjecture that symplectic 4-manifolds with $b^+>1$ obey the Bogomolov-Miyaoka-Yau inequality. Our method uses Morse theory on the gauge theoretic moduli space of non-Abelian monopoles, where the Morse function is a Hamiltonian for a natural circle action and natural two-form.  We shall describe generalizations of Donaldson’s symplectic subspace criterion (1996) from finite to infinite dimensions. These generalized symplectic subspace criteria can be used to show that the natural two-form is non-degenerate and thus an almost symplectic form on the moduli space of non-Abelian monopoles. This talk is based on joint work with Tom Leness and the monographs https://arxiv.org/abs/2010.15789  (to appear in AMS Mathematical Surveys and Monographs), https://arxiv.org/abs/2206.14710 and https://arxiv.org/abs/2410.13809. 

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Arham Farid (MI EDI Officer) will join us to chat about current EDI initiatives and to hear our thoughts about ways EDI can improve in the Maths Institute.

Rare and extreme events are notoriously hard to handle in any complex stochastic system: They are simultaneously too rare to be reliably observable in experiments or numerics, but at the same time often too impactful to be ignored. Large deviation theory provides a classical way of dealing with events of extremely small probability, but generally only yields the exponential tail scaling of rare event probabilities. In this talk, I will discuss theory, and algorithms based upon it, that improve on this limitation, yielding sharp quantitative estimates of rare event probabilities from a single computation and without fitting parameters. Notably, these estimates require the computation of determinants of differential operators, which in relevant cases are not traceclass and require appropriate renormalization. We demonstrate that the Carleman--Fredholm operator determinant is the correct choice. Throughout, I will demonstrate the applicability of these methods to high-dimensional real-world systems, for example coming from atmosphere and ocean dynamics.