Past

In the definition of the skein lasagna module of a $4$-manifold $X$, it is essential that the input TQFT be fully functorial for link cobordisms in $S^3 \times [0, 1]$. I will describe an approach to resolve existing sign ambiguities in Kronheimer and Mrowka's spectral sequence from Khovanov homology to singular instanton link homology. The goal is to obtain a theory that is fully functorial for link cobordisms in $S^3 \times [0,1]$, and where the $E_2$ page carries a canonical isomorphism to Khovanov-Rozansky $\mathfrak{gl}_2$ link homology. Possible applications include non-vanishing theorems for $4$-manifold Khovanov skein lasagna modules à la Ren-Willis.

We begin with motivation on how the study of SPDEs are relevant when interested in fluctuations of particle systems. 

We then present a law of large numbers, central limit theorem and large deviations principle for the generalised Dean--Kawasaki SPDE with truncated noise. 

Our main contribution is the ability to consider the equation on a general $C^2$-regular, bounded domain with Dirichlet boundary conditions. On the particle level the boundary condition corresponds to absorption and injection of particles at the boundary.

The work is based on discussions with Benjamin Fehrman and can be found at https://arxiv.org/pdf/2504.17094 

A G-kernel is a group homomorphism from a (discrete) group G to Out(A), the outer automorphism group of a C*-algebra A. There are cohomological obstructions to lifting such a G-kernel to a group action. In the setting of von Neumann algebras, G-kernels on the hyperfinite II_1-factor have been completely understood via deep results of Connes, Jones and Ocneanu. 

In the talk I will explain how G-kernels on C*-algebras and the lifting obstructions can be interpreted in terms cohomology with coefficients in crossed modules. G-kernels, group actions and cocycle actions then give rise to induced maps on classifying spaces. For strongly self-absorbing C*-algebras these classifying spaces turn out to be infinite loop spaces creating a bridge to stable homotopy theory.

The talk is based on joint work with S. Giron Pacheco and M. Izumi, and with my PhD student V. Bianchi.
 

Perturbative expansions in quantum theory, particularly in quantum field theory and string theory, are typically factorially divergent due to underlying non-perturbative sectors. Resurgence provides a universal toolbox to access the non-perturbative effects hidden within the perturbative series, producing a collection of exponentially small corrections. Under special assumptions, the non-perturbative data extracted via resurgent methods exhibit intrinsic number-theoretic structures that are deeply rooted in the symmetries of the theory. The framework of modular resurgence aims to formalise this observation. In this talk, I will first introduce the systematic, algebraic approach of resurgence to the problem of divergences and describe the emerging bridge between the resurgence of q-series and the analytic and number-theoretic properties of L-functions and quantum modular forms. I will then apply it to the spectral theory of quantum operators associated with toric Calabi-Yau threefolds. Here, a complete realisation of the modular resurgence paradigm is found in the study of the spectral trace of local P^2, where the asymptotics at weak and strong coupling are captured by certain q-series, and is generalised to all local weighted projective planes. This talk is based on arXiv:2212.10606, 2404.10695, 2404.11550, and work to appear soon.

I will describe some recent efforts to recreate the miraculous properties of perverse sheaves on complex analytic spaces in the setting of real stratified spaces.

Virtually planar groups (that is, those groups with a finite-index subgroup admitting a planar Cayley graph) exhibit many fairly unique coarse geometric properties. Often, we find that any one of these properties completely characterises this class of groups. 

(EFC-DGAs) lead to an algebraic approach to derived analytic geometry, pioneered for more general Fermat theories by Carchedi and Roytenberg.
 
They are well-suited to modelling finite-dimensional analytic spaces, and classical theorems in analysis ensure they give a largely equivalent theory to Lurie's more involved approach via pregeometries. DG dagger affinoid spaces provide a well-behaved class of geometric building blocks whose homotopy theory is governed by the underlying EFC-DGAs. 

Time permitting, I might also say a little about non-commutative generalisations.
 

Dirac’s quantum theory of magnetic monopoles requires a Dirac string attached to each monopole, and it is important that the field equations do not depend on the positions of the Dirac strings, provided that they comply with the Dirac veto: they must not intersect the worldliness of electrically charged particles. This theory is revisited, and it is shown that it has generalised symmetries related to the freedom of moving the Dirac strings. The Dirac veto is interpreted as an anomaly and the possibility of cancelling the anomaly by embedding in a higher-dimensional theory will be discussed. 

This talk is based on arXiv:2411.18741.

For hyperbolic systems of conservation laws in 1-D, fundamental questions about uniqueness and blow up of weak solutions still remain even for the apparently “simple” systems of two conserved quantities such as isentropic Euler and the p-system. Similarly, in the multi-dimensional case, a longstanding open question has been the uniqueness of weak solutions with initial data corresponding to the compressible vortex sheet. We address all of these questions by using the lens of convex integration, a general method of constructing highly irregular and non-unique solutions to PDEs. Our proofs involve computer-assistance. This talk is based on joint work with László Székelyhidi, Jr.
 

Dirichlet polynomials are useful in the study of the Riemann zeta function & Dirichlet L functions, serving as approximations to them via the approximate functional equation. Understanding how often they can be large gives bounds on the number of zeroes of these functions in vertical strips - known as zero density estimates - which are relevant to the distribution of primes in short intervals. Based on Guth-Maynard, we study large values of Dirichlet polynomials with characters, relevant to Dirichlet L functions. Joint work with Yung Chi Li. 

I will discuss an enriched version of Shimizu's characterizations of non-degeneracy for finite braided tensor categories. Using these characterizations, it follows that an enriched finite braided tensor category is invertible as an object of the Morita 4-category of enriched braided tensor categories if and only if it is non-degenerate. As an application, I will explain how to extend the full dualizability result of Brochier-Jordan-Synder by showing that a finite braided tensor category is fully dualizable in the Morita 4-category of braided tensor categories if its symmetric center is separable.
 

In this talk, we consider the Cauchy problem for a number of semilinear PDEs, subject to initial data distributed according to a family of Gaussian measures.  

We first discuss how the flow of Hamiltonian equations transports these Gaussian measures. When the transported measure is absolutely continuous with respect to the initial measure, we say that the initial measure is quasi-invariant. 

In the high-dispersion regime, we exploit quasi-invariance to build a (unique) global flow for initial data with negative regularity, in a regime that cannot be replicated by the deterministic (pathwise) theory.  

In the 0-dispersion regime, we discuss the limits of this approach, and exhibit a sharp transition from quasi-invariance to singularity, depending on the regularity of the initial measure. 

We will also discuss how the same techniques can be used in the context of stochastic PDEs, and how they provide information on the invariant measures for their flow. 

This is based on joint works with  J. Coe (University of Edinburgh), J. Forlano (Monash University), and M. Hairer (EPFL).

In the last three decades, a fruitful way to approximate the area functional in low codimension is to interpret submanifolds as the nodal sets of maps (or sections of vector bundles), critical for suitable physical energies or well-known lagrangians from gauge theory. Inspired by the situation in codimension two, where the abelian Higgs model has provided a successful framework, we look at the non-abelian SU(2) model as a natural candidate in codimension three. In this talk we will survey the new key difficulties and some recent partial results, including a joint work with D. Parise and D. Stern and another result by Y. Li.

String-inspired methods have revealed deep connections between seemingly unrelated field theories. A striking example is the double copy structure, rooted in the string theory Kawai–Lewellen–Tye (KLT) relations. In this talk, we will explore how a variety of theories—including colored scalars, pions, and gluons—emerge from a single, unifying object: the KLT kernel. We will argue that this kernel is not only a powerful computational tool, but also a conceptually rich structure worthy of independent study.

Based mainly on https://arxiv.org/abs/1610.04230 and the recent work https://arxiv.org/abs/2505.01501.

It’s a busy and stressful term for a lot of us so come and take a break and do some colouring and origami with us. Venting is very much encouraged.

We will review physics-informed neural networks (NNs) and summarize available extensions for applications in computational science and engineering. We will also introduce new NNs that learn functionals and nonlinear operators from functions and corresponding responses for system identification. 

These two key developments have formed the backbone of scientific machine learning that has disrupted the path of computational science and engineering and has created new opportunities for all scientific domains. We will discuss some of these opportunities in digital twins, autonomy, materials discovery, etc.

Moreover, we will discuss bio-inspired solutions, e.g., spiking neural networks and neuromorphic computing.

Geometry and topology call tell us about the shape of data. In this talk, I will give an introduction to my work on learning stratified spaces from samples, look at the use of persistent homology in cell biophysics, and apply persistence in understanding material structures.

The self-energies in Quantum Electrodynamics (QED) are not only fundamental physical quantities but also well-suited for investigating the mathematical structure of perturbative Quantum Field Theory. In this talk, I will discuss the QED self-energies up to the fourth order in the loop expansion. Going beyond one loop, where the integrals can be expressed in terms of multiple polylogarithms, we encounter functions associated with an elliptic curve, a K3 surface and a Calabi-Yau threefold. I will review the method of differential equations and apply it to the scalar Feynman integrals appearing in the self-energies. Special emphasis will be placed on the concept of canonical bases and on how to generalize them beyond the polylogarithmic case, where they are well understood. Furthermore, I will demonstrate how canonical integrals may be identified through a suitable integrand analysis.

This talk will be a case study on the recently discovered boundary Carrollian conformal algebra (BCCA) in theoretical physics. It is an infinite-dimensional subalgebra of an abelian extension of the Witt algebra. A striking feature of this is that it is not integer graded; this already puts us in a rather novel setting, since infinite-dimensional Lie algebras almost exclusively appear with integer grading in physics. But this means that there is new ground to be broken in this direction of research. In this talk, I will present some very early results from our attempt at studying the representations of the BCCA. Any thoughts and comments are very welcome as they could be immensely helpful for us to navigate these unfamiliar waters!

During the talk I will describe my research on host-pathogen interactions during lung infections. Various modelling approaches have been used, including a hybrid multiscale individual-based model that we have developed, which simulates pulmonary infection spread, immune response and treatment within in a section of human lung. The model contains discrete agents which model the spatio-temporal interactions (migration, binding, killing etc.) of the pathogen and immune cells. Cytokine and oxygen dynamics are also included, as well as Pharmacokinetic/Pharmacodynamic models, which are incorporated via PDEs. I will also describe ongoing work to develop a continuum model, comparing the spatial dynamics resulting from these different modelling approaches.  I will focus in the most part on two infectious diseases: Tuberculosis and COVID-19.

I am going to start by reviewing axioms of quantum mechanics, which in fact give a description of a Hilbert space. I will argue that the language that Dirac and his followers developed is that of continuous logic and the form of axiomatisation is that of "algebraic logic" in the sense of A. Tarski's cylindric algebras. In fact, Hilbert spaces can be seen as a continuous model theory version of cylindric algebras.

We study a multi-agent setting in which brokers transact with an informed trader. Through a sequential Stackelberg-type game, brokers manage trading costs and adverse selection with an informed trader. In particular, supplying liquidity to the informed traders allows the brokers to speculate based on the flow information. They simultaneously attempt to minimize inventory risk and trading costs with the lit market based on the informed order flow, also known as the internalization-externalization strategy. We solve in closed form for the trading strategy that the informed trader uses with each broker and propose a system of equations which classify the equilibrium strategies of the brokers. By solving these equations numerically we may study the resulting strategies in equilibrium. Finally, we formulate a competitive game between brokers in order to determine the liquidity prices subject to precommitment supplied to the informed trader and provide a numerical example in which the resulting equilibrium is not Pareto efficient.

Let G be an infinite discrete group; e.g. free group, surface groups, or hyperbolic 3-manifold group.

Finite dimensional unitary representations of G of fixed dimension are usually very hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps. 

The talk is a broadly accessible discussion on these themes, based on joint works with W. Hide, L. Louder, D. Puder, J. Thomas, R. van Handel.

If $E/\mathbb{Q}$ is an elliptic curve, and $F/\mathbb{Q}$ is a finite Galois extension, then $E(F)$ is not merely a finitely generated abelian group, but also a Galois module. If we fix a finite group $G$, and let $F$ vary over all $G$-extensions, then what can we say about the statistical behaviour of $E(F)$ as a $\mathbb{Z}[G]$-module? In this talk I will report on joint work with Adam Morgan, in which we investigate the simplest non-trivial special case of this very general question. Our work has surprising connections to questions about frequency of failure of the Hasse principle for genus 1 hyperelliptic curves, and to work of Heath-Brown on 2-Selmer group distributions in quadratic twist families.