Past

In this talk, we consider the coefficients of the Fourier series obtained by exponentiating a logarithmically correlated holomorphic function on the open unit disc, whose Taylor coefficients are independent complex Gaussian variables, the variance of the coefficient of degree k being theta/k where theta > 0 is an inverse temperature parameter. In joint articles with Paquette, Simm and Vu, we show a randomized version of the central limit theorem in the subcritical phase theta < 1, the random variance being related to the Gaussian multiplicative chaos on the unit circle. We also deduce, from results on the holomorphic multiplicative chaos, other results on the coefficients of the characteristic polynomial of the Circular Beta Ensemble, where the parameter beta is equal to 2/theta. In particular, we show that the central coefficient of the characteristic polynomial of the Circular Unitary Ensembles tends to zero in probability, answering a question asked in an article by Diaconis and Gamburd.

A landmark result in the study of locally compact, abelian groups is the Pontryagin duality. In simple terms, it says that for a given locally compact, abelian group G, one can uniquely associate another locally compact, abelian group called the Pontryagin dual of G. In the realm of C*-algebras, whenever such an abelian group G acts on a C*-algebra A, there is a canonical action of the dual group of G on the crossed product of A by G. In particular, it is natural to ask to what extent one can relate properties of the given G-action to those of the dual action. 

In this talk, I will first introduce a property for actions of locally compact abelian groups called the abelian Rokhlin property and then state a duality type result for this property. While the abelian Rokhlin property is in general weaker than the known Rokhlin property, these two properties coincide in the case of the acting group being the real numbers. Using the duality result mentioned above, I will give new examples of continuous actions of the real numbers which satisfy the Rokhlin property. Part of this talk is based on joint work with Johannes Christensen and Gábor Szabó.

In this talk I will describe a possible strategy to obtain new solution to LMCF in the Kummer K3 surface by a fixed point argument. The key idea is that the regions where curvature concentrates in the Kummer K3 surface are modeled on the Eguchi-Hanson space. 

Property (T) (respectively aTmenability) is equivalent to admitting only a trivial action (respectively, a proper action) on a median space, and is also equivalent to admitting only a trivial action (respectively, a proper action) on a Hilbert space (so some L2). For p>2 I will investigate an analogous equivalent characterisation.

In this talk I will discuss a new lower bound for the off-diagonal Ramsey numbers $R(3,k)$. For this, we develop a version of the triangle-free process that is significantly easier to analyse than the original process. We then 'seed' this process with a carefully chosen graph and show that it results in a denser graph that is still sufficiently pseudo-random to have small independence number.

This is joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.

Nichols algebras, also known as small shuffle algebras, are a family of graded bialgebras including the symmetric algebras, the exterior algebras, the positive parts of quantized enveloping algebras, and, conjecturally, Fomin-Kirillov algebras. As the case of Fomin-Kirillov algebra shows, it can be very
difficult to determine the maximum degree of a minimal generating set of relations of a Nichols algebra. 

Building upon Kapranov and Schechtman’s equivalence between the category of perverse sheaves on Sym(C) and the category of graded connected bialgebras,  we describe the geometric counterpart of the maximum degree of a generating set of relations of a graded connected bialgebra, and we show how this specialises to the case o Nichols algebras.

The talk is based on joint work with Francesco Esposito and Lleonard Rubio y Degrassi.
 

We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature--it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic point-like excitations. The fermionic toric code, importantly, possesses an anomalous 2-form symmetry, associated with the space-like Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to an equilibrium phase of matter that only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous 2-form symmetries. Our work therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.

Agents in social networks with threshold-based dynamics change opinions when influenced by sufficiently many peers. Existing literature typically assumes that the network structure and dynamics are fully known, which is often unrealistic. In this work, we ask how to learn a network structure from samples of the agents' synchronous opinion updates. Firstly, if the opinion dynamics follow a threshold rule where a fixed number of influencers prevent opinion change (e.g., unanimity and quasi-unanimity), we give an efficient PAC learning algorithm provided that the number of influencers per agent is bounded. Secondly, under standard computational complexity assumptions, we prove that if the opinion of agents follows the majority of their influencers, then there is no efficient PAC learning algorithm. We propose a polynomial-time heuristic that successfully learns consistent networks in over 97% of our simulations on random graphs, with no failures for some specified conditions on the numbers of agents and opinion diffusion examples.

A subset $A$ of $A_n$ is $k$-product-free if for all $a_1,a_2,\dots,a_k\in A$, $a_1a_2\dots a_k$ $\notin A$.
We determine the largest $3$-product-free and $4$-product-free subsets of $A_n$ for sufficiently large $n$. We also obtain strong stability results and results on multiple sets with forbidden cross products. The principal technical ingredient in our approach is the theory of hypercontractivity in $S_n$. Joint work with Peter Keevash.

Given a space with some kind of geometry, one can ask how the geometry of the space relates to its homology. This talk will survey some comparisons of geometric notions of complexity with homological notions of complexity. We will then focus on hyperbolic 3-manifolds and the main result will replace a spectral gap problem related to torsion in homology with a geometric version involving geodesic length and stable commutator length. As an application, we provide "bad" examples of hyperbolic 3-manifolds with bounded geometry but extremely small (1-form) spectral gaps.

Scaling arguments give a natural guess at the regularity condition on the noise in a stochastic PDE for a local solution theory to be possible, using the machinery of regularity structures or paracontrolled distributions. This guess of ``subcriticality'' is often, but not always, correct. In cases when it is not, a the blowup of the variance of certain nonlinear functionals of the noise necessitates a different, multiplicative renormalisation. This led to a general prediction and the first results in the case of the KPZ equation in [Hairer '24]. We discuss recent developments towards confirming this prediction. Based on joint works with Fabio Toninelli and Yueh-Sheng Hsu.

I will present some ongoing work on solving parametric linear systems arising from the application of the finite elements method on elliptic partial differential trial equations. The focus of the talk will be on leveraging randomised numerical linear algebra to solve these equations in high-dimensional parameter spaces with special emphasis on the multi-query context where optimal sampling is not practical. In this context I will discuss some ideas on choosing a suitable low-dimensional approximation of the solution, as well as reducing the variance of the sketched systems. This research aims at exploring the potential of randomisation as a probabilistic framework for model order reduction, with potential applications to online simulations, uncertainty quantification and inverse problems, via the research grant EPSRC EP/V028618/1

Bio: Nick Polydorides is a professor in computational engineering at the University of Edinburgh and has interests in randomised numerical linear algebra, inverse problems and edge computing. Previously, he was a faculty at the Cyprus Institute, and a postdoctoral fellow at MIT’s lab for Information and Decision Systems. He has a PhD in Electrical Engineering from the University of Manchester.  

The rise to fame of supersymmetry since the 1970s shook the world. It held much promise—from explaining naturalness, unifying fundamental forces, to being the ideal candidate for dark matter. But since the LHC (arguably even a bit before that), many of these dreams have been shattered by experiments. Today, the pursuit of supersymmetric theories by the physics community is a mere shadow of its former self.

This symposium is not to discuss whether supersymmetry is useful in the fields of physics and mathematics—it clearly is. Rather, this is a debate about whether its death is natural. We’ve had a crack at it for half a century. Is this the limit of what we can do? Are we any closer to achieving the original goals we set out? Is the death premature, accelerated by a negative campaign from SUSY critics? Or is it the other way around—has it been at death’s door for decades, kept alive only because authoritative figures cannot let go?

Twenty years ago, this wouldn’t even be a debate. Twenty years from now, there may not be any young people working on SUSY at all. This seems like the right time to talk.

Explicit examples of the AdS/CFT correspondence where both bulk and boundary theories are tractable are hard to come by, but the minimal tension string on AdS_3 x S^3 x T^4  is one notable example. In this paper, we discuss how one can construct sigma models on twistor space, with a particular focus on applying these techniques to the aforementioned string theory. We derive novel incidence relations, which allow us to understand to what extent the minimal tension string encodes information about the bulk. We identify vertex operators in terms of bulk twistor variables and a map from twistor space to spacetime is presented. We also demonstrate the presence of a partially broken global supersymmetry algebra in the minimal tension string and we argue that this implies that there exists an N=2 formulation of the theory. The implications of this are studied and we demonstrate the presence of an additional constraint on physical states. This is based on work with Ron Reid-edwards https://arxiv.org/abs/2411.08836.

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on measures. Therefore, operations on generalized valuations can be used to define integral transforms that unify both classical Radon transforms and their topological analogues based on the Euler characteristic, which have been successfully used in shape analysis. However, this unification is proven under rather restrictive assumptions in Alesker's original paper, leaving key aspects conjectural. In this talk, I will present a recent result obtained with A. Bernig that significantly closes this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof relies on a comparison between these operations and operations on characteristic cycles.

I will present a novel correspondence between the gravitational phase space at null infinity and the subleading phase space for finite-distance null hypersurfaces, such as black hole horizons. Utilizing the Newman-Penrose formalism and an off-shell Weyl transformation, this construction transfers key structures from asymptotic boundaries to null surfaces in the bulk—for instance, a notion of radiation. Imposing self-duality conditions, I will identify the celestial symmetries and construct their canonical generators for finite-distance null hypersurfaces. This framework provides new observables for black hole physics.

In the 1970s, Serre conjectured that any continuous, irreducible and odd mod p representation of the absolute Galois group G_Q is modular. Serre furthermore conjectured that there should be an explicit minimal weight "k" such that the Galois representation is modular of this weight, and that this weight only depends on the restriction of the Galois representation to the inertial subgroup I_p. This is often called the weight part of Serre's conjecture. Both the weight part, and the modularity part, of the Serre's conjecture are nowadays known to be true. In this talk, I want to explain how to rephrase the conjecture in representation theoretic terms (for k >= 2), so that the weight k is replaced by a certain (mod p) irreducible representation of GL_2(F_p), and how upon rephrasing the conjecture one can realize it as a statement about local-global compatibility with the mod p local Langlands correspondence.

The rheological (deformation and flow) properties of biological tissues  are important in processes such as embryo development, wound healing and 
tumour invasion. Indeed, processes such as these spontaneously generate  stresses within living tissue via active process at the single cell level. 
Tissues are also continually subject to external stresses and deformations  from surrounding tissues and organs. The success of numerous physiological 
functions relies on the ability of cells to withstand stress under some conditions, yet to flow collectively under others. Biological tissue is 
furthermore inherently viscoelastic, with a slow time-dependent mechanics.  Despite this rich phenomenology, the mechanisms that govern the 
transmission of stress within biological tissue, and its response to bulk deformation, remain poorly understood to date.

This talk will describe three recent research projects in modelling the rheology of biological tissue. The first predicts a strain-induced 
stiffening transition in a sheared tissue [1]. The second elucidates the interplay of external deformations applied to a tissue as a whole with 
internal active stresses that arise locally at the cellular level, and shows how this interplay leads to a host of fascinating rheological 
phenomena such as yielding, shear thinning, and continuous or discontinuous shear thickening [2]. The third concerns the formulation of 
a continuum constitutive model that captures several of these linear and nonlinear rheological phenomena [3].

[1] J. Huang, J. O. Cochran, S. M. Fielding, M. C. Marchetti and D. Bi, 
Physical Review Letters 128 (2022) 178001

[2] M. J. Hertaeg, S. M. Fielding and D. Bi, Physical Review X 14 (2024) 
011017.

[3] S. M. Fielding, J. O. Cochran, J. Huang, D. Bi, M. C. Marchetti, 
Physical Review E (Letter) 108 (2023) L042602.

Solving large-scale continuous-time algebraic Riccati equations is a significant challenge in various control theory applications. 

This work demonstrates that when the matrix coefficients of the equation are quasiseparable, the solution also exhibits numerical quasiseparability. This property enables us to develop two efficient Riccati solvers. The first solver is applicable to the general quasiseparable case, while the second is tailored to the particular case of banded coefficients. Numerical experiments confirm the effectiveness of the proposed algorithms on both synthetic examples and case studies from the control of partial differential equations and agent-based models. 

Low-rank approximation of parameter-dependent matrices A(t) is an important task in the computational sciences, with applications in areas such as dynamical systems and the compression of series of images. In this talk, we introduce AdaCUR, an efficient randomised algorithm for computing low-rank approximations of parameter-dependent matrices using the CUR decomposition. The key idea of our approach is the ability to reuse column and row indices for nearby parameter values, improving efficiency. The resulting algorithm is rank-adaptive, provides error control, and has complexity that compares favourably with existing methods. This is joint work with Yuji Nakatsukasa.

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.