We all sleep. But what determines when and for how long? In this talk I’ll describe some of the fundamental mechanisms that regulate sleep. I’ll introduce the nonsmooth coupled oscillator systems that form the basis of current mathematical models of sleep-wake regulation and discuss their dynamical behaviour. I will describe how we are using models to unravel environmental, societal and physiological factors that determine sleep timing and outline how constructing digital-twins could enable us to create personalised light interventions for sleep timing disorders.

Professor Kaibo Hu will be talking about: 'Finite element form-valued forms'

Some of the most successful vector-valued finite elements in computational electromagnetics and fluid mechanics, such as the Nédélec and Raviart-Thomas elements, are recognized as special cases of Whitney’s discrete differential forms. Recent efforts aim to go beyond differential forms and establish canonical discretizations for more general tensors. An important class is that of form-valued forms, or double forms, which includes the metric tensor (symmetric (1,1)-forms) and the curvature tensor (symmetric (2,2)-forms). Like the differential structure of forms is encoded in the de Rham complex, that of double forms is encoded in the Bernstein–Gelfand–Gelfand (BGG) sequences and their cohomologies. Important examples include the Calabi complex in geometry and the Kröner complex in continuum mechanics.
These constructions aim to address the problem of discretizing tensor fields with general symmetries on a triangulation, with a particular focus on establishing discrete differential-geometric structures and compatible tensor decompositions in 2D, 3D, and higher dimensions.
 

Approximate subrings are subsets $A$ of a ring $R$ satisfying \[ A + A + AA \subset F + A \] for some finite $F \subset R$. They encode the failure of sum-product phenomena, much like approximate subgroups encode failure of growth in groups.

I will discuss how approximate subrings mirror approximate subgroups and how model-theoretic tools, such as a stabilizer lemma for approximate subrings due to Krupiński, lead to structural results implying a general, non-effective sum-product phenomenon in arbitrary rings: either sets grow rapidly under sum and product, or nilpotent ideals govern their structure. I will also outline related results for infinite approximate subrings and conjectures unifying known (effective) sum-product phenomena.

Based on joint work with Krzysztof Krupiński.

Based on mathematical ecological models, this report reviews the impact of spatially heterogeneous environments on the persistence and distribution of biological populations. The report aims to elucidate the interplay between population density and key factors, including diffusion coefficients, resource availability, and habitat structure. The study first investigates the ecological consequences of dispersal strategies within environments characterized by uneven resource distribution, demonstrating the monotonic dependence of peak population densities on diffusion rates. Furthermore, analysis of a consumer-resource system indicates that under resource-limited conditions, the ecosystem converges to a globally asymptotically stable state of coexistence. Building on these findings, the report additionally explores the constraints imposed by domain geometry on the spatial patterning of populations.

• as holomorphic functions on certain symplectic spaces
• as matrix coefficients of the Heisenberg (and metaplectic) representation,
• as sections of line bundles on abelian varieties.

Details to be finalised

Speaker Aaron Mishkin will talk about; 'Convex Analysis of Non-Convex Neural Networks' 

One of the key themes in modern optimization is the boundary between convex and non-convex problems. While convex problems can often be solved efficiently, many non-convex programs are NP-Hard and formally difficult. 

In this talk, we show how to break the barrier between convex and non-convex optimization by reformulating, or "lifting", neural networks into high-dimensional spaces where they become convex. These convex reformulations serve two purposes: as algorithmic tools to enable fast, global optimization for two-layer ReLU networks; and as a convex proxy to study variational properties of the original non-convex problem. In particular, we show that shallow ReLU networks are equivalent to models with simple "gated ReLU" activations, derive the set of all critical points for two-layer ReLU networks, and give the first polynomial-time algorithm for optimal neuron pruning. We conclude with extensions to ReLU networks of arbitrary depth using a novel layer-elimination argument.

Partition diversity refers to the concept that for some networks there may be multiple, similarly plausible ways to group the nodes, rather than one single best partition. In this talk, I will present two projects that address this idea from different but complementary angles. The first introduces the benchmark stochastic cross-block model (SCBM), a generative model designed to create synthetic networks with two distinct 'ground-truth' partitions. This allows us to study the extent to which existing methods for partition detection are able to reveal the coexistence of multiple underlying structures. The second project builds on this benchmark and paves the way for a Bayesian inference framework to directly detect coexisting partitions in empirical networks. By formulating this model as a microcanonical variant of the SCBM, we can evaluate how well it fits a given network compared to existing models. We find that our method more reliably detects partition diversity in synthetic networks with planted coexisting partitions, compared to methods designed to detect a single optimal partition. Together, the two projects contribute to a broader understanding of partition diversity by offering tools to explore the ambiguity of network structure.

In 1962, Horn raised the following problem: Let A and B be n-by-n Hermitian matrices with respective eigenvalues a_1,...,a_n and b_1,...,b_n. What can we say about the possible eigenvalues c_1,...,c_n of A + B?

The deterministic perspective is that the set of possible values for c_1,...,c_n are described by a collection of inequalities known as the Horn inequalities.

Free probability offers the following alternative perspective on the problem: if (A_n) and (B_n) are independent sequences of n-by-n random matrices with empirical spectra converging to probability measures mu and nu respectively, then the random empirical spectrum of A_n + B_n converges to the free convolution of mu and nu.

But how are these two perspectives related?

In this talk Samuel Johnston will discuss approaches to free probability that bridge between the two perspectives. More broadly, Samuel will discuss how the fundamental operations of free probability (such as free convolution and free compression) arise out of statistical physics mechanics of corresponding finite representation theory objects (hives, Gelfand-Tsetlin patterns, characteristic polynomials, Horn inequalities, permutations etc).

This talk is based on joint work with Octavio Arizmendi (CIMAT, Mexico), Colin McSWiggen (Academia Sinica, Taiwan) and Joscha Prochno (Passau, Germany).

Fracture is usually treated as an outcome to be avoided; here we see it as something we may write into a lattice's microstructure. Maxwell lattices sit at the edge of mechanical stability, where robust topological properties provide a way on how stress localises and delocalises across the structure with directional preference. Building on this, we propose a direct relationship between lattice topology and damage propagation. We identify a set of topology- and geometry-dependent parameters that gives a simple, predictive framework for nonideal Maxwell lattices and their damage processes. We will discuss how topological polarisation and domain walls steer and arrest damage in a repeatable way. Experiments confirm the theoretical predicted localisation and the resulting tuneable progression of damage and show how this control mechanism can be used to enhance dissipation and raise the apparent fracture energy.

I will present a very short introduction to the mathematics behind the scientific imaging technique known as ptychography, starting with a brief overview of the physics model and the various simplifications required, before moving on to the main ptychography inverse problem and the three principal classes of optimization algorithms currently being used in practice. 

Speaker Estefania Loayza Romero will talk about:  A Riemannian Approach for PDE-Constrained Shape Optimization Using Outer Metrics

In PDE-constrained shape optimisation, shapes are traditionally viewed as elements of a Riemannian manifold, specifically as embeddings of the unit circle into the plane, modulo reparameterizations. The standard approach employs the Steklov-Poincaré metric to compute gradients for Riemannian optimisation methods. A significant limitation of current methods is the absence of explicit expressions for the geodesic equations associated with this metric. Consequently, algorithms have relied on retractions (often equivalent to the perturbation of identity method in shape optimisation) rather than true geodesic paths. Previous research suggests that incorporating geodesic equations, or better approximations thereof, can substantially enhance algorithmic performance. This talk presents numerical evidence demonstrating that using outer metrics, defined on the space of diffeomorphisms with known geodesic expressions, improves Riemannian gradient-based optimisation by significantly reducing the number of required iterations and preserving mesh quality throughout the optimisation process.

This talk is hosted at RAL. 

I will present new results, joint with Krzysztof Klosin (CUNY), on the modularity of residually reducible Galois representations with 3 residual pieces. This will be applied to prove the p-adic modularity of Picard curves.

Certain models of collective dynamics exhibit deceptively simple patterns that are surprisingly difficult to explain. These patterns often arise from phase transitions within the underlying dynamics. However, these phase transitions can be explained only when one derives continuum equations from the corresponding individual-based models. In this talk, I will explore this subtle yet rich phenomenon and discuss advances and open problems.

Bring interesting problems (relating to your research or otherwise) for a unique brainstorming session

Dr Jenny Power (Heriot-Watt University) will share some of the personal and unexpected lessons she learnt while completing her PhD. We’ll then discuss how a “perfect PhD student” doesn’t exist!

Session to be run by Five Rings

Speaker Lucas Theis will talk about: 'What makes an image realistic ?'

The last decade has seen tremendous progress in our ability to generate realistic-looking data, be it images, text, audio, or video. 
In this presentation, we will look at the closely related problem of quantifying realism, that is, designing functions that can reliably tell realistic data from unrealistic data. This problem turns out to be significantly harder to solve and remains poorly understood, despite its prevalence in machine learning and recent breakthroughs in generative AI. Drawing on insights from algorithmic information theory, we discuss why this problem is challenging, why a good generative model alone is insufficient to solve it, and what a good solution would look like. In particular, we introduce the notion of a universal critic, which unlike adversarial critics does not require adversarial training. While universal critics are not immediately practical, they can serve both as a North Star for guiding practical implementations and as a tool for analyzing existing attempts to capture realism.

In this talk, I will present a result proved in my recent paper arXiv:2502.13580. I will discuss biharmonic maps between (and submanifolds of) conformally compact manifolds, a large class of complete manifolds generalizing hyperbolic space. After an introduction to conformally compact geometry, I will discuss one of the main results of the paper: if S is a properly embedded sub-manifold of a conformally compact manifold (N,h), and moreover S is transverse to the boundary and (N,h) has non-positive curvature, then S must be minimal. This result confirms a conjecture known as the Generalized Chen’s Conjecture, in the conformally compact context.

TBA

We study a very general class of first-order linear hyperbolic
systems that both become weakly hyperbolic and contain singular
lower-order coefficients at a single time t = 0. In "critical" weakly
hyperbolic settings, it is well-known that solutions lose a finite
amount of regularity at t = 0. Here, we both improve upon the analysis
in the weakly hyperbolic setting, and we extend this analysis to systems
containing critically singular coefficients, which may also exhibit
modified asymptotics and regularity loss at t = 0.

In particular, we give precise quantifications for (1) the asymptotics
of solutions as t approaches 0, (2) the scattering problem of solving
the system with asymptotic data at t = 0, and (3) the loss of regularity
due to the degeneracies at t = 0. Finally, we discuss a wide range of
applications for these results, including weakly hyperbolic wave
equations (and equations of higher order), as well as equations arising
from relativity and cosmology (e.g. at big bang singularities).

This is joint work with Bolys Sabitbek (Ghent).

In this talk, I will examine the dynamics of the fermion–rotor system, originally introduced by Polchinski as a toy model for monopole–fermion scattering. Despite its simplicity, the system is surprisingly subtle, with ingoing and outgoing fermion fields carrying different quantum numbers. I will show that the rotor acts as a twist operator in the low-energy theory, changing the quantum numbers of excitations that have previously passed through the origin to ensure scattering consistent with all symmetries, thereby resolving the long-standing Unitarity puzzle. I will then discuss generalizations of this setup with multiple rotors and unequal charges, and demonstrate how the system can be viewed as a UV-completion of boundary states for chiral theories, establishing a connection to the proposed resolution of the puzzle using boundary conformal field theory.

Persistent homology (PH) is an operation which, loosely speaking, describes the different holes in a point cloud via a collection of intervals called a barcode. The two most frequently used variants of persistent homology for point clouds are called Čech PH and Vietoris-Rips PH. How much information is lost when we apply these kinds of PH to a point cloud? We investigate this question by studying the subspace of point clouds with the same barcodes under these operations. We establish upper and lower bounds on the dimension of this space, and find that the question of when the persistence map is identifiable has close ties to rigidity theory. For example, we show that a generic point cloud being locally identifiable under Vietoris-Rips persistence is equivalent to a certain graph being rigid on the same point cloud.

to follow

In this talk, I will introduce the nilpotent cohomological Hall algebra COHA(S, Z) of coherent sheaves on a smooth quasi-projective complex surface S that are set-theoretically supported on a closed subscheme Z. This algebra can be viewed as the "largest" algebra of cohomological Hecke operators associated with modifications along a subscheme Z of S. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S, Z) in terms of the Yangian of the corresponding affine ADE quiver Q (based on joint work with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot, arXiv:2502.19445). More generally, I will discuss nilpotent COHAs arising from Bridgeland stability conditions on the bounded derived category of nilpotent representations of the preprojective algebra of Q, following joint work with Olivier Schiffmann and Parth Shimpi (arXiv:2511.08576).

to follow

Join us to discuss x+y: A Mathematician’s Manifesto for Rethinking Gender by Eugenia Cheng.