Past

The fundamental group of a hyperbolic surface has an infinite number of rank k subgroups. What does it mean, therefore, to pick a 'random' subgroup of this type? In this talk, I will introduce a method for counting subgroups and discuss how counting allows us to study the properties of a random subgroup and its associated cover.

We will present some of the remarkable properties of eigenvalue trajectories for rank-one perturbations of random matrices, with an emphasis on two models of particular interest, namely weakly non-Hermitian and weakly non-unitary matrices. In both cases, precise estimates can be obtained for the critical timescale at which an outlier can be observed with high probability. We will outline the proofs of these results and highlight their significance in connection with quantum chaotic scattering. (Based on joint works with L. Erdös and J. Reker)

The notion of semi-uniform stability of a strongly continuous semi-group refers to the stability of classical solutions of a linear evolution equation, and this has analogues with the classical Katznelson-Tzafriri theorem. The co-generator of a strongly continuous semigroup is a bounded linear operator that comes from a particular discrete approximation to the semigroup. After reviewing some background on (quantified) stability theory for semigroups and the Katznelson-Tzafriri theorem, I will present some results relating the stability of a strongly continuous semigroup with that of its cogenerator. This talk is based on joint work with David Seifert.

The conjugacy growth function of a finitely generated group is a variation of the standard growth function, counting the number of conjugacy classes intersecting the n-ball in the Cayley graph. The asymptotic behaviour is not a commensurability invariant in general, but the conjugacy growth of finite extensions can be understood via the twisted conjugacy growth function, counting automorphism-twisted conjugacy classes. I will discuss what is known about the asymptotic and formal power series behaviour of (twisted) conjugacy growth, in particular some relatively recent results for certain groups of polynomial growth (i.e. virtually nilpotent groups).

Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as the maximal minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees form a regular matroid. In this talk, I will give a short historical overview of the tree-counting problem and a related quantity from electrical circuit theory: the effective resistance. I will describe a characterization of effective resistances in terms of a certain polytope and discuss some recent applications to discrete notions of curvature on graphs. More details can be found in the recent preprint: https://arxiv.org/abs/2410.07756

The classical lower-tail problem for triangles in random graphs asks the following: given $\eta\in[0,1)$, what is the probability that $G(n,p)$ contains at most $\eta$ times the expected number of triangles?  When $p=o(n^{-1/2})$ or $p = \omega(n^{-1/2})$ the asymptotics of the logarithm of this probability are known via Janson's inequality in the former case and regularity or container methods in the latter case.

We prove for the first time asymptotic formulas for the logarithm of the lower tail probability when $p=c n^{-1/2}$ for $c$ constant.  Our results apply for all $c$ when $\eta \ge 1/2$ and for $c$  small enough when $\eta < 1/2$.  For the special case $\eta=0$ of triangle-freeness, our results prove that a phase transition occurs as $c$ varies (in the sense of a non-analyticity of the rate function), while for $\eta \ge 1/2$ we prove that no phase transition occurs.

Our method involves ingredients from algorithms and statistical physics including rapid mixing of Markov chains and the cluster expansion.  We complement our asymptotic formulas with efficient algorithms to approximately sample from $G(n,p)$ conditioned on the lower tail event.

Joint work with Will Perkins, Aditya Potukuchi and Michael Simkin.

Let G be a reductive group over a finite field F of characteristic p. I will present work with Tzu-Jan Li in which we determine the endomorphism algebra of the Gelfand-Graev representation of the finite group G(F) where the coefficients are taken to be l-adic integers, for l a good prime of G distinct from p. Our result can be viewed as a finite-field analogue of the local Langlands correspondence in families. 

We will be joined by Dr Brigitte Stenhouse from the Open University to discuss harassment, particularly in academic settings. 

We prove the local Lipschitz continuity of energy minimizing harmonic maps between singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.  Joint work with Yaoting Gui and Jürgen Jost.

The notion of Lip(gamma) Functions, for a parameter gamma > 0, introduced by Stein in the 1970s (building on earlier work of Whitney) is a notion of smoothness that is well-defined on arbitrary closed subsets (including, in particular, finite subsets) that is instrumental in the area of Rough Path Theory initiated by Lyons and central in recent works of Fefferman. Lip(gamma) functions provide a higher order notion of Lipschitz regularity that is well-defined on arbitrary closed subsets, and interacts well with the more classical notion of smoothness on open subsets. In this talk we will survey the historical development of Lip(gamma) functions and illustrate some fundamental properties that make them an attractive class of function to work with from a machine learning perspective. In particular, models learnt within the class of Lip(gamma) functions are well-suited for both inference on new unseen input data, and for allowing cost-effective inference via the use of sparse approximations found via interpolation-based reduction techniques. Parts of this talk will be based upon the works https://arxiv.org/abs/2404.06849 and https://arxiv.org/abs/2406.03232.

I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to an oriented fibration with Poincaré duality fiber. We construct a generalized fiber integration map associated to each Lie graph homology class and the main result is that this gives a faithful representation of graph homology. I will discuss how this leads to new possible interpretations of Lie graph homology classes as obstructions to, on one hand, smoothness of Poincaré duality fibrations, and, on the other hand, the existence of Poincaré duality algebra resolutions of the cochains of the total space as a dg module over the cochains of the base space.

I will discuss how some ideas from Geometric Langlands can be used to obtain new results in birational geometry and on the topology of algebraic varieties.

I will review recent progess on a topic of common interest to many
physicists https://www.arxiv.org/abs/2410.12043. Time permitting, I will
also comment about new sum rules for protected operators along RG flows:
https://arxiv.org/pdf/2409.09006. 

In this interactive workshop, we'll discuss what mathematicians are looking for in written solutions. How can you set out your ideas clearly, and what are the standard mathematical conventions? Please bring a pen or pencil!

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

I will describe a holomorphic-topological field theory in eleven-dimensions which captures a 1/16-BPS subsector of eleven-dimensional supergravity. Remarkably, asymptotic symmetries of the theory on flat space and on twisted versions of the AdS_4 x S^7 and AdS_7 x S^4 backgrounds recover three of the five infinite dimensional exceptional simple super-Lie algebras. I will discuss some applications of this fact, including character formulae for indices counting multigravitons and the contours of a program to holographically describe 1/16-BPS local operators in the 6d (2,0) SCFTs of type A_{N-1}. This talk is based on joint work, much in progress, with Fabian Hahner, Ingmar Saberi, and Brian Williams.

The Junior Algebra and Representation Theory Seminar will kick-off the start of the academic year with a social event in the common room. Come to catch up with your fellow students and maybe play a board game or two. Afterwards we'll have lunch together.

Turing patterns have long been proposed as a mechanism for spatial organization in biology, but their relevance remains controversial due to the stringent fine-tuning often required. In this talk, I will present recent efforts to engineer synthetic Turing systems in bacterial colonies, highlighting both successes and limitations. While our three-node gene circuit generates patterns, challenges remain in extending these results to broader contexts. Additionally, I will discuss our exploration of machine learning methods to address the inverse problem of pattern formation, helping the design process down the road. This work addresses the ongoing task in translating theory into robust biological applications, offering insights into both current capabilities and future directions.

The important problem of backtesting financial models over long horizons inevitably leads to overlapping returns, giving rise to correlated samples. We propose a new method of dealing with this problem by decorrelation and show how this increases the discriminatory power of the resulting tests.


About the speaker
Nikolai Nowaczyk is a Risk Management & AI consultant who has advised multiple institutional clients in  projects around counterparty credit risk and xVA as well as data science and machine learning. 
Nikolai holds a PhD in mathematics from the University of Regensburg and has been an Academic Visitor at Imperial College London.
 

Registration for in-person attendance is required in advance.

Register here.

Von Neumann algebras which are not matrix algebras, yet still possess a unique trace, form a basic class called II$_1$ factors. The set of asymptotically commuting elements (or, the relative commutant of the algebra within its own ultrapower), dubbed the central sequence algebra, can take many different forms. In this talk, we discuss an elementary class of II$_1$ factors whose central sequence algebra is again a II$_1$ factor. We show that the class of infinitely generic II$_1$ factors possess this property, and ask some related questions about properties of other existentially closed II$_1$ factors. This is based on joint work with Isaac Goldbring, David Jekel, and Srivatsav Kunnawalkam Elayavalli.

Given a singularity with a crepant resolution, a symmetry of the derived 
category of coherent sheaves on the resolution may often be constructed 
using the formalism of spherical functors. I will introduce this, and 
new work (arXiv:2409.19555) on general constructions of such symmetries 
for hypersurface singularities. This builds on previous results with 
Segal, and is inspired by work of Bodzenta-Bondal.

Let $K$ be an unramified extension of $\mathbb{Q}_p$ for a prime $p > 3$. The reduced part of the Emerton-Gee stack for $\mathrm{GL}_{2}$ can be viewed as parameterizing two-dimensional mod $p$ Galois representations of the absolute Galois group of $K$. In this talk, we will consider the extremely non-generic irreducible components of this reduced part and see precisely which ones are smooth or normal, and which have Gorenstein normalizations. We will see that the normalizations of the irreducible components admit smooth-local covers by resolution-rational schemes. We will also determine the singular loci on the components, and use these results to update expectations about the conjectural categorical $p$-adic Langlands correspondence. This is based on recent joint work with Ben Savoie.

Roe type algebras are operator algebras designed to catch the large-scale behaviour of metric spaces. This talk focuses on the following question: if two Roe type algebras associated to spaces (X,d_X) and (Y,d_Y) are isomorphic, how similar are X and Y? We provide positive results proved in the last 5 years, and, if time allows it, we show that sometimes answers to this question are subject to set theoretic considerations

Joint work with Paul Hacking (U Mass Amherst). We first explain how to 
prove homological mirror symmetry for a maximal normal crossing 
Calabi-Yau surface Y with split mixed Hodge structure. This includes the 
case when Y is a type III K3 surface, in which case this is used to 
prove a conjecture of Lekili-Ueda. We then explain how to build on this 
to prove an HMS statement for K3 surfaces. On the symplectic side, we 
have any K3 surface (X, ω) with ω integral Kaehler; on the algebraic 
side, we get a K3 surface Y with Picard rank 19. The talk will aim to be 
accessible to audience members with a wide range of mirror symmetric 
backgrounds.

Following the 2012 breakthrough in deep learning for classification and visions problems, the last decade has seen tremendous raise of interest in machine learning in a wider mathematical research community from foundational research through field specific analysis to applications. 

As data is at the core of any inverse problem, it was a natural direction for the field to investigate how machine learning could aid various aspects of inversion yielding numerous approaches from somewhat ad-hoc but very effective like learned unrolled methods to provably convergent learned regularisers with everything in between. In this talk I will review some on these developments through a lens of the research of our group.   

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

A number of moduli problems are, via Hodge theory, closely related to 
ball quotients. In this situation there is often a choice of possible 
compactifications such as the GIT compactification´and its Kirwan 
blow-up or the Baily-Borel compactification and the toroidal 
compactificatikon. The relationship between these compactifications is 
subtle and often geometrically interesting. In this talk I will discuss 
several cases, including cubic surfaces and threefolds and 
Deligne-Mostow varieties. This discussion links several areas such as 
birational geometry, moduli spaces of pointed curves, modular forms and 
derived geometry. This talk is based on joint work with S. 
Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.

The celebrated Splitting Theorem by Cheeger-Gromoll states that a manifold with non-negative Ricci curvature which contains a line is isometric to a product, where one of the factors is the real line. A related result was later proved by Kasue. He showed that a manifold with non-negative Ricci curvature and two mean convex boundary components, one of which is compact, is also isometric to a product. In this talk, I will present a variant of Kasue’s result based on joint work with Andrea Mondino. We consider manifolds with non-negative Ricci curvature and disconnected mean convex boundary. We show that if one boundary component is parabolic and convex, then the manifold is a product, where one of the factors is an interval of the real line. The result is an application of recently developed tools in synthetic geometry and exploits the interplay between Ricci curvature and optimal transport.

We introduce a new methodology based on the multirevolution idea for constructing integrators for stochastic differential equations in the situation where the fast oscillations themselves are driven by a Stratonovich noise. Applications include in particular highly-oscillatory Kubo oscillators and spatial discretizations of the nonlinear Schrödinger equation with fast white noise dispersion. We construct a method of weak order two with computational cost and accuracy both independent of the stiffness of the oscillations. A geometric modification that conserves exactly quadratic invariants is also presented. If time allows, we will discuss ongoing work on uniformly accurate methods for such systems. This is a joint work with Gilles Vilmart.

Slender elastic filaments with intrinsic helical geometry are encountered in a wide range of physical and biological settings, ranging from coil springs in engineering to bacteria flagellar filaments. The equilibrium configurations of helical filaments under a variety of loading types have been well studied in the framework of the Kirchhoff rod equations. These equations are geometrically nonlinear and so can account for large, global displacements of the rod. This geometric nonlinearity also makes a mathematical analysis of the rod equations extremely difficult, so that much is still unknown about the dynamic behaviour of helical rods under external loading.

An important class of simplified models consists of 'equivalent-column' theories. These model the helical filament as a naturally-straight beam (aligned with the helix axis) for which the extensional and torsional deformations are coupled. Such theories have long been used in engineering to describe the free vibrations of helical coil springs, though their validity remains unclear, particularly when distributed forces and moments are present. In this talk, we show how such an effective theory can be derived systematically from the Kirchhoff rod equations using the method of multiple scales. Importantly, our analysis is asymptotically exact in the small-wavelength limit and can account for large, unsteady displacements. We then illustrate our theory with two loading scenarios: (i) a heavy helical rod deforming under its own weight; and (ii) axial rotation (twirling) in viscous fluid, which may be considered as a simple model for a bacteria flagellar filament. More broadly, our analysis provides a framework to develop reduced models of helical rods in a wide variety of physical and biological settings, as well as yielding analytical insight into their tensile instabilities.

A group is coherent if all its finitely generated subgroups are finitely presented. Aside from some easy cases, it appears that coherence is a phenomenon that occurs only among groups of cohomological dimension 2. In this talk, we will give many examples of coherent and incoherent groups, discuss techniques to prove a group is coherent, and mention some open problems in the area.

This is a joint event by the Mathematrix and Mirzakhani Societies for all women and non-binary people in the Maths department. Join us in the South Mezzanine for some hot drinks and sweet treats. 

We explore the weak coupling limit for stochastic Burgers type equation in critical dimension, and show that it is given by a Gaussian stochastic heat equation, with renormalised coefficient depending only on the second order Hermite polynomial of the nonlinearity. We use the approach of Cannizzaro, Gubinelli and Toninelli (2024), who treat the case of quadratic nonlinearities, and we extend it to polynomial nonlinearities. In that sense, we extend the weak universality of the KPZ equation shown by Hairer and Quastel (2018) to the two dimensional generalized stochastic Burgers equation. A key new ingredient is the graph notation for the generator. This enables us to obtain uniform estimates for the generator. This is joint work with Nicolas Perkowski.

The classification program of C*-algebras aims to classify simple, separable, nuclear C*-algebras by their K-theory and traces, inspired by analogous results obtained for von Neumann algebras. A landmark result in this project was obtained in 2015, building upon the work of numerous researchers over the past 20 years. More recently, Carrión, Gabe, Schafhauser, Tikuisis, and White developed a new, more abstract approach to classification, which connects more explicitly to the von Neumann algebraic classification results. In their paper, they carry out this approach in the stably finite setting, while for the purely infinite case, they refer to the original result obtained by Kirchberg and Phillips. In this talk, I provide an overview of how the new approach can be adapted to classify purely infinite C*-algebras, recovering the Kirchberg-Phillips classification by K-theory and obtaining Kirchberg's absorption theorems as corollaries of classification rather than (pivotal) ingredients. This is joint work with Jamie Gabe.

The study of the rationality of the $L^2$-Betti numbers of a countable group has led to the development of a rich theory in $L^2$-homology with deep implications in structural properties of the groups. For decades almost nothing has been known about the general question of whether the strong Atiyah conjecture passes to free products of groups or not. In this talk, we will confirm that the strong and algebraic Atiyah conjectures are stable under the graph of groups construction provided that the edge groups are finite. Moreover, we shall see that in this case the $\ast$-regular closure of the group algebra is precisely a universal localization of the associated graph of rings

Creating networks of statistical dependencies between brain regions is a powerful tool in neuroscience that has resulted in many new insights and clinical applications. However, recent interest in higher-order interactions has highlighted the need to address beyond-pairwise dependencies in brain activity. Multivariate information theory is one tool for identifying these interactions and is unique in its ability to distinguish between two qualitatively different modes of higher-order interactions: synergy and redundancy. I will present results from applying the O-information, the partial entropy decomposition, and the local O-information to resting state fMRI data. Each of these metrics indicate that higher-order interactions are widespread in the cortex, and further that they reveal different patterns of statistical dependencies than those accessible through pairwise methods alone. We find that highly synergistic subsystems typically sit between canonical functional networks and incorporate brain regions from several of these systems. Additionally, canonical networks as well as the interactions captured by pairwise functional connectivity analyses, are strongly redundancy-dominated. Finally, redundancy/synergy dominance varies in both space and time throughout an fMRI scan with notable recurrence of sets of brain regions engaging synergistically. As a whole, I will argue that higher-order interactions in the brain are an under-explored space that, made accessible with the tools of multivariate information theory, may offer novel insights.

We give an exponential improvement on the upper bound for the $r$-colour diagonal Ramsey number for all $r$. The proof relies on geometric insights and offers a simplified proof in the case of $r=2$.

Joint Work with: Paul Ballister, Béla Bollobás, Marcelo Campos, Simon Griffiths, Rob Morris, Julian Sahasrabudhe and Marius Tiba.

Plethysms lie at the intersection of representation theory and algebraic combinatorics. We give a recursive formula for a family of plethysm coefficients encompassing those involved in Foulkes' Conjecture. We also describe some applications, such as to the stability of plethysm coefficients and Sylow branching coefficients for symmetric groups. This is joint work with Y. Okitani.

In this talk I will discuss asymmetric orbifolds and will focus on their application to toroidal compactifications of heterotic string theory. I will consider theories in 6 and 4 dimensions with 16 supercharges and reduced rank. I will present a novel formalism, based on the Leech lattice, to construct ‘islands’ without vector multiplets.

We propose a density functional theory of Thomas-Fermi-(von Weizsacker) type to describe the response of a single layer of graphene to a charge some distance away from the layer. We formulate a variational setting in which the proposed energy functional admits minimizers. We further provide conditions under which those minimizers are unique. The associated Euler-Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. For a class of special potentials, we also establish a precise universal asymptotic decay rate, as well as an exact charge cancellation by the graphene sheet. In addition, we discuss the existence of nodal minimizers which leads to multiple local minimizers in the TFW model. This is a joint work with Cyrill Muratov (University of Pisa).

We show that any finite coloring of an amenable group contains 'many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables.  This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem.  Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.

In this talk, I will describe recent work in the application of machine learning to explore questions in algebraic geometry, specifically in the context of the study of Q-Fano varieties. These are Q-factorial terminal Fano varieties, and they are the key players in the Minimal Model Program. In this work, we ask and answer if machine learning can determine if a toric Fano variety has terminal singularities. We build a high-accuracy neural network that detects this, which has two consequences. Firstly, it inspires the formulation and proof of a new global, combinatorial criterion to determine if a toric variety of Picard rank two has terminal singularities. Secondly, the machine learning model is used directly to give the first sketch of the landscape of Q-Fano varieties in dimension eight. This is joint work with Tom Coates and Al Kasprzyk.

Multiparameter persistence is an area of topological data analysis that synthesises the geometric information of a topological space via filtered homology. Given a topological space and a function on it, one can consider a filtration given by the sublevel sets of the space induced by the function and then take the homology of such filtration. In the case when the filtering function assumes values in the real plane, the homological features of the filtered object can be recovered through a "curved" grid on the plane called the extended Pareto grid of the function. In this talk, we explore how the computation of the biparameter matching distance between regular filtering functions on a regular manifold depends on the extended Pareto grid of these functions.