Workshops

Workshops will be held in the following subject areas

WSAL1: 15:10-17:10, 23rd June 2025, Room FOR/SR1-3, presentation 15:10-15:50

Inna Capdeboscq (University of Warwick)

Title: Chevalley groups over non-archemadian local fields: some subgroups

Abstract: In this talk we discuss the structure and generation of some subgroups of Chevalley groups over non-archimedean local fields. This is a joint work with Bertrand Remy

WSAL1: 15:10-17:10, 23rd June 2025, Room FOR/SR1-3, presentation 15:50-16:30

Matthew Fayers (Queen Mary University London)

Title: Irreducible Specht modules

Abstract: Given a finite group G and a prime p, it is an interesting question to ask which ordinary irreducible representations of G remain irreducible in characteristic p. For the symmetric groups this question was answered a while ago, and in the meantime there has been considerable activity in extending this question to other groups and algebras. I will give a survey of these results.

WSAL1: 15:10-17:10, 23rd June 2025, Room FOR/SR1-3, TALK CANCELLED

~~Benjamin Briggs (Imperial College London)~~

~~Title: Moment angle manifolds and linear free resolutions~~

Abstract: This is a talk about some of the extremely close connections between the combinatorics of simplicial complexes, the homological algebra of monomial rings, and the topology of toric spaces. I’ll start by explaining how these three areas thread together by constructing the Stanley-Reisner ring and the moment angle space associated to a simplicial complex. Then I’ll survey some of the interactions between these objects, focusing on how linear resolutions affect the geometry of moment angle manifolds, as part of some joint work with Steve Amelotte.

WSAL2: 15:40-17:40, 24th June 2025, Room FOR/SR1-3, presentation 15:40-16:20

Lewis Topley (University of Bath)

Title: Quantizations of conjugacy classes in positive characteristic

Abstract: Poisson algebras provide a basic algebraic framework for Hamiltonian dynamics, and quantization is the process of finding a non-commutative algebra which approximates a Poisson algebra in a precise sense. In recent years it has become apparent that large families of rather complicated Poisson varieties in characteristic zero admit rigid quantization theories: their global quantizations can be classified. Over fields of positive characteristic the appearance of such phenomena has not even been enunciated as a conjecture. In this talk I will explain a work in progress with Filippo Ambrosio and Matt Westaway which classifies the quantizations of conjugacy classes of matrices, using methods from representation theory.

WSAL2: 15:40-17:40, 24th June 2025, Room FOR/SR1-3, presentation 16:20-17:00

Ehud Meir (University of Aberdeen)

Title: Geometric methods in braided vector spaces and their Nichols algebras

Abstract: The Nichols algebra of a braided vector space is a generalization of both the symmetric algebra and the exterior algebra.It can be realized as a Hopf algebra in some braided monoidal category. By the process of Bosonization, Nichols algebras provide an abundance of examples of finite dimensional non-semisimple Hopf algebras. This raises the question of the finite-dimensionality of Nichols algebras. In this talk I will explain how to prove that certain Nichols algebras are infinite dimensional, by using geometric methods and in particular reductive group actions. This is based on a joint work with Istvan Heceknberger and Leandro Vendramin. If time permits, I will also talk about more recent work with Ben Martin, about the geometry of the space of braiding in a given dimension.

WSAL2: 15:40-17:40, 24th June 2025, Room FOR/SR1-3, presentation 17:00-17:40

Ilaria Colazzo (University of Leeds)

Title: Matched Pairs of Groups and Combinatorial Solutions to the Pentagon Equation

Abstract: In this talk, I will present a complete classification of finite bijective set-theoretic solutions to the Pentagon Equation, uncovering a surprising connection with matched pairs of groups. We will introduce all necessary definitions. Next, we will focus on the fundamental components: the irretractable solutions, and we will examine how these solutions relate to matched pairs of groups. Finally, we will show how each irretractable solution can be lifted to classify all bijective solutions.

WSDS1: 15:10-17:10, 23rd June 2025, Room NEW/GRE, presentation 15:10-15:50

Cecilia Gonzalez Tokman (University of Queensland)

Title: Universal gap growth for Lyapunov exponents of perturbed matrix products

Abstract: In this talk, we address the question of quantitative simplicity of the Lyapunov spectrum of bounded matrix cocycles subjected to additive random perturbations. In dimensions 2 and 3, we show explicit lower bounds on the gaps between consecutive Lyapunov exponents of the perturbed cocycle, depending only on the scale of the perturbation. In arbitrary dimensions, we establish existence of a universal lower bound on these gaps. A novelty of this work is that the bounds provided are uniform over all choices of the original sequence of matrices, making no stationarity assumptions. Hence, our results apply to random and sequential dynamical systems alike. Joint work with Jason Atnip, Gary Froyland and Anthony Quas.

WSDS1: 15:10-17:10, 23rd June 2025, Room NEW/GRE, presentation 15:50-16:30

Kasun Fernando (Brunel University)

Title: A Dolgopyat-type estimate for a class of partially hyperbolic systems

Abstract: In this talk, I will discuss a Dolgopyat-type estimate for a class of partially hyperbolic systems on the 2-Torus, and its consequences: the CLT, its higher order corrections, and exponential decay of correlations for a class of three-dimensional skew products. This is a joint work with Roberto Castorrini.

WSDS1: 15:10-17:10, 23rd June 2025, Room NEW/GRE, presentation 16:30-17:10

Jens Rademacher (Universitat Hamburg)

Title: Interaction and intrinsic dynamics of localised patterns: kinks and fronts in excitable media

Abstract: The dynamics of spatially localised patterns is a characteristic phenomenon of nonlinear spatially extended systems. In the past decades many mathematical results have been obtained concerning the dynamics caused by the interaction of multiple localised patterns through spatial coupling. Much less attention has been paid to the possibility of intrinsic dynamics of a localised state. This talk discusses some aspects of both for excitable media, where the localised states are kinks or fronts. In the first part we consider a class of cellular automata, which do not feature long range spatial interaction and then turn to so-called weak interaction of related kinks in neural models. In the second part we discuss motion of single fronts by intrinsic dynamics, caused by scale-interaction between components, which admits provably chaotic dynamics. The first part is joint work with Marc Kesseböhmer (Bremen) and Dennis Ulbrich (Münster), the second with Martina Chirilus-Bruckner (Leiden) and Peter van Heijster (Wageningen)

WSDS2: 15:40-17:40, 24th June 2025, Room NEW/GRE, presentation 15:40-16:20

Jamie Walton (University of Nottingham)

Title: Hierarchy and cut and project sets

Abstract: I will introduce the two main methods of construction of aperiodically ordered patterns: by substitution (or ‘inflate, subdivide’) rules, creating patterns which may be thought of as ‘upside-down fractals’, and the cut and project method, wherein one takes an irrational slice of a periodic lattice. Many well-known aperiodic patterns, such as the Penrose tilings or recently discovered hat/spectre families of tilings (at least with particular parameters for edge-lengths), can be constructed through both procedures. I will introduce a generalised notion of a ‘pattern’ and its associated dynamical system, which gives a unified framework for studying tilings, point sets and more in the translationally FLC (finite local complexity) case. We also define when such a pattern is ‘substitutional’ in a general sense. I will explain recent results (in preparation) which establish an equivalence between unique composition of the hierarchy of such patterns and aperiodicity of their associated dynamical system, mildly generalising a well-known result of Solomyak (still only in the FLC case, but extending here to non-minimal patterns). I will then explain recent results, joint with Harriss and Koivusalo, on when a cut and project set with Euclidean total space is substitutional. We obtain a particularly simple and checkable criterion in the case that the window is assumed to be polytopal.

WSDS2: 15:40-17:40, 24th June 2025, Room NEW/GRE, presentation 16:20-17:00

Andrew Mitchell (Open University)

Title: Substitutions and their generalisations

Abstract: Dynamical systems arising from substitutions are the prototypical examples of aperiodic minimal systems and provide theoretical models for quasicrystals. In recent years, increased attention has been paid to generalisations of substitutions, including S-adic systems, staggered substitutions and random substitutions, for which the associated dynamical system can exhibit diverse properties not possible for classical substitution systems. In this talk, I will provide an overview of recent developments in the theory of substitutions and their generalisations, and discuss several pertinent open questions.

WSDS2: 15:40-17:40, 24th June 2025, Room NEW/GRE, presentation 17:00-17:40

Nataliya Balabanova (University of Birmingam)

Title: Adaptive dynamics for Prisoner’s Dilemma

Abstract: Adaptive dynamics is a method of determining the best way to alter the strategy of mutants in the population in order to maximise the profit. This mathematical apparatus can be applied in game theory to the repeated donation game, otherwise known as the prisoner’s dilemma. In this talk, we are going to cover the construction of the payoff function for the prisoner’s dilemma, its symmetries, the corresponding adaptive dynamics and some of their properties.

WSDS3: 15:40-17:40, 25th June 2025, Room NEW/GRE, presentation 15:40-16:00

Demetris Hadjiloucas (European University – Cyprus)

Title: False bubbling in Invariant Graphs of Quasiperiodically Forced Maps

Abstract: The invariant subspace of a class of quasiperiodically forced maps is studied as a parameter is varied. It is shown that the invariant subspace is a unique non-trivial smooth curve which is attracting and is the graph of a smooth function. However, as the parameter is varied, a phenomenon we call false bubbling is observed. While simple iteration of the dynamics as well as Newton’s method fail to converge to the true invariant curve, an algorithm based on Singular Value Decomposition is devised and shown to converge to the true invariant curve. This is then used to make a conjecture on the existence and regularity of this invariant subspace as it loses stability beyond some critical value of the parameter. This is joint work with Paul Glendinning.

WSDS3: 15:40-17:40, 25th June 2025, Room NEW/GRE, presentation 16:00-16:20

Alejandro Rodriguez Sponheimer (Lund University)

Title: Strong Borel–Cantelli Lemmas for Recurrence

Abstract: The study of recurrence in dynamical systems can be dated back to the famous Poincar\’e Recurrence Theorem. One formulation of it is that, under some mild assumptions on the measure-preserving dynamical system $(X,T,\mu,d)$, almost every point is recurrent. In other words, \[ \liminf_{n \to \infty}d(T^{n}x,x) = 0 \] for $\mu$-a.e.\ $x$. A shortcoming of this result is that we do not obtain any quantitative information about the recurrence. For example, what can be said about the rate of convergence? What about the rate of recurrence? In 1993, Boshernitzan proved that, under some additional mild assumptions on $(X,T, mu,d)$, one can obtain a rate of convergence for the limit $\liminf_{n \to \infty}d(T^{n}x,x) = 0$. Since then, there have been various improvements to Boshernitzan’s result. In this talk, I will present some recent improvements in the form of dynamical strong Borel–Cantelli lemmas, giving conditions for which the limit \[ \lim_{n \to \infty} \frac{\sum_{k=1}^{n}\charfun_{B(x,r_k(x))}(T^{k}x)}{\sum_{k=1}^{n} \mu(B(x,r_k(x))} = 1 \] holds. Such statements can be viewed as recurrence versions of dynamical strong Borel–Cantelli lemmas for the shrinking targets problem. This talk is based on joint work with Tomas Persson.

WSDS3: 15:40-17:40, 25th June 2025, Room NEW/GRE, presentation 16:20-16:40

Jiayao Shao (University of Warwick)

Title: Stochastic Dynamical System Methods Applied to Ship Capsize Problem

Abstract: Large Deviation Theory is widely applied in the field of rare events. For example, on top of some deterministic dynamical systems, we add small noise to disturb the system to allow transition from a stable sink to a saddle. In the limitation of small enough noise, this transition would be a rare stochastic event. Under this setting, we proposed a method to investigate the probability of such an event. If it happens, the most likely trajectory of transition could be deduced. Moreover, we isolate the two fixed points by ‘Method of Division’ which largely saves computation cost and allows the case of infinite time scale. The above method will be illustrated by an engineering problem-the ship capsize problem.

WSDS3: 15:40-17:40, 25th June 2025, Room NEW/GRE, presentation 16:40-17:00

Andrew Burbanks (University of Portsmouth)

Title: Existence of periodic orbits of renormalisation operators for coupled systems

Abstract: Universal scaling behaviour is observed in the bifurcation structure of families of maps. The best known example concerns period-doubling cascades in which asymptotic scaling relationships occur in parameter space, state space, and time, as the system approaches the accumulation of period-doublings. This can be explained by considering renormalisation operators that encode the scaling. The existence of periodic points of these operators, and the properties of the associated tangent maps, help to explain quantitative universality of the dynamics. We present work on the existence of renormalisation fixed points and cycles for period doubling in coupled pairs of maps. Our work involves obtaining rigorous computer-assisted bounds on operations in Banach spaces of pairs of functions. Example applications concern the FQ-type fixed point and the C-type period-two cycle, that help to explain universality of scaling in qualitatively different classes of coupled dynamical systems.

WSDS3: 15:40-17:40, 25th June 2025, Room NEW/GRE, presentation 17:00-17:20

Saeed Shaabanian (University of St Andrews)

Title: A natural connection between recurrence Laws in $\phi$-mixing dynamical systems

Abstract: Recurrence in dynamical systems can be analyzed via two key laws: hitting time statistics, which study the return times to shrinking neighborhoods of a point, and escape rates, which measure how quickly orbits leave through a fixed subset of the space, interpreted as a hole. A foundational result by Bruin, Demers, and Todd (2018) established a general formula connecting these two perspectives, showing that results for one law can be transferred to the other by tuning certain parameters. This formula was originally derived in the context of interval maps. In this talk, we discuss how to extend this framework to symbolic dynamical systems with a $\phi$-mixing measure, with one of the key applications being the generalization of the results to other dynamical systems, such as hitting of Anosov diffeomorphisms to certain subspaces, or interval maps to Cantor sets.

WSDS3: 15:40-17:40, 25th June 2025, Room NEW/GRE, presentation 17:20-17:40

Yiwei Zhang (Anhui Univeristy of Science and Technology)

Title: Recent progresses on typically periodic optimization problems in ergodic optimization

Abstract: I will report some recent progresses on typically periodic optimization problems in ergodic optimziation theory. This is a joint work with Wen Huang, Oliver Jenkinson, Leiye Xu.

WSFM1: 15:10-17:10, 23rd June 2025, Room NEW/COL, presentation 15:10-16:10

Bhavik Mehta (Imperial College London)

Title: Title TBC

Abstract: Abstract TBC

WSFM1: 15:10-17:10, 23rd June 2025, Room NEW/COL, presentation 16:10-17:10

Athina Thoma (University of Southampton)

Title: Using Lean in university mathematics teaching: Insights and challenges

Abstract: Interactive Theorem Provers (ITPs) like Lean are being integrated into university mathematics teaching, either as compulsory or optional addition. This talk presents educational research on how first-year undergraduates engage with Lean, drawing on analysis from their pen-and-paper proofs, activity with Lean, and their reflections. I will discuss the challenges students encounter, the learning opportunities Lean affords, and insights from lecturers who integrated Lean into an introductory proof course.

WSFM2: 15:40-17:40, 24th June 2025, Room NEW/COL, presentation 15:40-16:40

Christopher Birkbeck (University of East Anglia)

Title: Formalising modular forms and sphere packings in Lean

Abstract: I will discuss some work on formalising modular forms in Lean and how it is used in our (ongoing) formalization of Maryna Viazovska’s Fields Medal-winning paper proving that no packing of unit balls in Euclidean space R^8 has density greater than that of the E8-lattice packing. Part of this is joint work with Sidharth Hariharan, Gareth Ma, Bhavik Mehta, Seewoo Lee and Maryna Viazovska.

WSFM2: 15:40-17:40, 24th June 2025, Room NEW/COL, presentation 16:40-17:40

Yaël Dillies (Stockholm University)

Title: Title TBC

Abstract: Abstract TBC

WSGT1: 15:10-17:10, 23rd June 2025, Room NEW/PUR, presentation 15:10-15:50

Chunyi Li (University of Warwick)

Title: Bridgeland Stability Conditions and Their Applications in Algebraic Geometry

Abstract: Inspired by work in mathematical physics, Bridgeland introduced the concept of stability conditions on triangulated categories. Over the past two decades, Bridgeland stability conditions have developed into a powerful tool for addressing problems across multiple mathematical fields. The existence of such stability conditions forms the foundation for several important theories in this area. In this talk, I will discuss recent progress in constructing stability conditions and their applications in algebraic geometry.

WSGT1: 15:10-17:10, 23rd June 2025, Room NEW/PUR, presentation 15:50-16:30

Cristina Manolache (Sheffield University)

Title: Reduced Gromov–Witten invariants in any genus

Abstract: Gromov-Witten (GW) invariants ideally give counts of curves of genus g in a given variety. However, GW invariants with g greater than one, have a more subtle enumerative meaning: curves of lower genus also contribute to GW invariants. In genus one this problem was corrected by Vakil and Zinger, who defined more enumerative numbers called “reduced GW invariants”. More recently Hu, Li and Niu gave a construction of reduced GW invariants in genus two. I will define reduced Gromov–Witten invariants in all genera. This is work with A. Cobos-Rabano, E. Mann and R. Picciotto.

WSGT1: 15:10-17:10, 23rd June 2025, Room NEW/PUR, presentation 16:30-17:10

Anne-Sophie Kaloghiros (Brunel University)

Title: Explicit K-stability and moduli construction of Fano 3-folds

Abstract: The Calabi problem asks which compact complex manifolds are Kähler-Einstein (KE)- i.e. can be endowed with a canonical metric that satisfies both an algebraic condition (being Kähler) and the Einstein (partial differential) equation. Such manifolds always have a canonical class of definite sign. The existence of such a metric on manifolds with positive and trivial canonical class (general type and Calabi Yau) was proved in the 70s. In the case of Fano manifolds the situation is more subtle – Fano manifolds may or may not have a KE metric. We now know that a Fano manifold admits a KE metric precisely when it satisfies a sophisticated algebro-geometric condition called K-polystability. Surprisingly, K-polystability also sheds some light on another important problem: while Fano varieties do not behave well in families, K-polystable Fano varieties do, and form K-moduli spaces. Our explicit understanding of K-polystability is still partial, and few examples of K-moduli spaces are known. In dimension 3, Fano manifolds were classified into 105 deformation families by Mori-Mukai and Iskovskikh. In this talk, I will present an overview of the Calabi problem in dimension 3. Knowing – as we do – which families in the classification of Fano 3-folds have K-polystable members is a starting point to investigate the corresponding K-moduli spaces. I will describe explicitly some K-moduli spaces of Fano 3-folds of small dimension.

WSGT2: 15:40-17:40, 24th June 2025, Room NEW/PUR, presentation 15:40-16:20

Emanuele Dotto (University of Warwick)

Title: Characteristic polynomials in homotopy theory

Abstract: The properties of the characteristic polynomial of matrices can be encoded into a ring homomorphism from the cyclic K-group to the ring of Witt vectors. I will give an overview of this characterisation, due to Almkvist and Grayson, and explain how to obtain this homomorphism formally by extending the Witt vectors construction to bimodules. In particular, this construction makes sense also when R is not commutative, and can be refined for matrices self-adjoint with respect to a symmetric form. Most of this work is joint with Achim Krause, Thomas Nikolaus and Irakli Patchkoria.

WSGT2: 15:40-17:40, 24th June 2025, Room NEW/PUR, presentation 16:20-17:00

Joao Faria Martins (University of Leeds)

Title: A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from discrete higher gauge theory

Abstract: Quinn’s Finite Total Homotopy TQFT is a topological quantum field theory defined for any dimension n of space, depending on the choice of a homotopy finite space B. For instance, B can be the classifying space of a finite group or a finite 2-group. In this talk, I will report on recent joint work with Tim Porter on once-extended versions of Quinn’s Finite Total Homotopy TQFT, taking values in the symmetric monoidal bicategory of groupoids, linear profunctors, and natural transformations between linear profunctors. The construction works in all dimensions, yielding (0,1,2)-, (1,2,3)-, and (2,3,4)-extended TQFTs, given a homotopy finite space B. I will shown how to compute these once-extended TQFTs when B is the classifying space of a homotopy 2-type, represented by a crossed module of groups. Reference: Faria Martins J, Porter T: “A categorification of Quinn’s finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids.” arXiv:2301.02491 [math.CT]

WSGT2: 15:40-17:40, 24th June 2025, Room NEW/PUR, presentation 17:00-17:40

Ka Man Yim (Cardiff University)

Title: An algebraic derivation of Morse complexes for poset-graded chain complexes

Abstract: The Morse-Conley complex is a central object in the application of homological algebra to analysing dynamical systems, as well as information compression in topological data analysis. Given a poset-graded chain complex, its Morse-Conley complex is the optimal chain-homotopic reduction of the initial complex that respects the poset grading. In this work, we give a purely algebraic derivation of the Conley complex using homological perturbation theory. Our algebraic formulation presents an alternative to combinatorial approaches in Forman’s discrete Morse theory. We show how this algebraic perspective also yields efficient algorithms for computing the Conley complex. This talk features joint work with Álvaro Torras Casas and Ulrich Pennig in “Computing Connection Matrices of Conley Complexes via Algebraic Morse Theory” (arXiv:2503.09301).

WSGT3: 15:40-17:40, 25th June 2025, Room NEW/PUR, presentation 15:40-16:10

Oli Gregory (Imperial College London)

Title: Griffiths groups of supersingular varieties

Abstract: The 2nd Griffiths group Griff^2(X) of a variety X is an invariant which measures the difference between homological equivalence and algebraic equivalence for codimension 2 cycles on X. It plays an important role in the theory of motives, but is still mysterious. Over the complex numbers Griff^2(X) interacts with Hodge theory, and over global fields it is related to special values of L-functions. I will investigate the situation in characteristic p>0. Here, a consequence of the Beilinson-Bloch conjecture says that Griff^2(X) should be ‘large’ if X is ordinary, and ‘small’ if X is supersingular. I will show that Griff^2(X) actually vanishes for supersingular varieties over the algebraic closure of a finite field.

WSGT3: 15:40-17:40, 25th June 2025, Room NEW/PUR, presentation 16:10-16:40

Katerina Santicola (University of Warwick)

Title: Sharp interpolation of rational objects

Abstract: Interpolation is the art of building an object that passes through a given set of points. For instance, if we are given two points, we can always pass a line through them, and if we are given five points, we can always pass a conic through them. A recent breakthrough in algebraic geometry relates the minimal degree of the curve to the number of points we want to interpolate. What if we wanted to interpolate a set of points belonging in a set, and ensure the object passes through no other points in the set? The aim of this talk is to give an overview of how this question is answered when we ask for sharp interpolation instead.

WSGT3: 15:40-17:40, 25th June 2025, Room NEW/PUR, presentation 16:40-17:10

Dmytro Karvatskyi (Institute of Mathematics of the NAS of Ukraine (visiting the University of St. Andrews))

Title: Topological and fractal properties of homogeneous self-similar sets

Abstract: We study the topological and fractal properties of attractors of homogeneous IFSs on the real line with a rational contraction factor and translations. In simple cases, the generated self-similar set is either a closed interval or homeomorphic to the classical Cantor set. The most intriguing scenario arises when the attractor has a non-empty interior and a fractal boundary, which will be the main focus of the talk. We further compute the Lebesgue measure of the interior and the Hausdorff dimension of the boundary for these complex self-similar sets within a specific family.

WSGT3: 15:40-17:40, 25th June 2025, Room NEW/PUR, presentation 17:10-17:40

Marija Jelic Milutinovic (University of Belgrade, Faculty of Mathematics (Serbia))

Title: Matching Complexes

Abstract: Simplicial complexes constructed from graphs form a rich and diverse research area, establishing numerous interesting connections to a variety of mathematical fields. These complexes provide a natural combinatorial framework, bridging graph-theoretic properties with concepts from algebraic topology. Among them, matching complexes represent a particularly significant class. For a simple graph G, the matching complex M(G) is the simplicial complex on the set of edges, whose faces correspond to matchings in the graph, where a matching is a set of edges no two of which share a vertex. The topology of matching complexes has been extensively studied over the years, leading to many applications in topological combinatorics. However, for most classes of graphs, much about this topology remains mysterious. This talk will offer a brief overview and then present our recent results on matching complexes of outerplanar and related graphs. We prove that the matching complex of an outerplanar graph is either contractible or homotopy equivalent to a wedge of spheres. Furthermore, we introduce a subcomplex of the matching complex, called the perfect matching complex, whose facets correspond to perfect matchings in a graph. We then discuss several results concerning the homotopy type of perfect matching complexes. The talk is based on papers co-authored with Margaret Bayer (University of Kansas, USA) and Julianne Vega (Maret School, Washington, USA), as part of a project supported by the NSF Grant No. DMS-1440140 (MSRI, Berkeley).

WSNT1: 15:10-17:10, 23rd June 2025, Room NEW/RED, presentation 15:10-15:50

Andrew Pearce-Crump (University of Bristol)

Title: Number Theory versus Random Matrix Theory: the joint moments story

Abstract: It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height T, is $\frac{1}{2} (e^2-5) \log T$ as $T \rightarrow \infty$. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer.

WSNT1: 15:10-17:10, 23rd June 2025, Room NEW/RED, presentation 15:50-16:30

Jesse Pajwani (University of Bath)

Title: Symmetric powers of trace forms

Abstract: Fix k to be a field of characteristic not 2 and let L/k be a finite separable field extension. The Galois trace map Tr: L -> k gives rise to a quadratic form Tr_L on L, viewed as a k vector space, given by sending x to Tr(x^2). This construction extends to finite étale algebras over k, so it is a natural question to ask whether this is compatible with operations on finite étale algebras over k. In particular, if A is a finite étale algebra, we can form the symmetric product A^(n), which is also a finite étale algebra. In this talk, we investigate the notion of a “symmetric power of quadratic forms” and show that Tr_{A^(n)} is given by a strange symmetric power of Tr_A, using results by Garibaldi-Merkurjev-Serre. If there is time at the end, we may discuss a higher dimensional analogue of this question, which concerns an arithmetic enhancement of the Euler characteristic of a variety. This is joint work with Ambrus Pál.

WSNT1: 15:10-17:10, 23rd June 2025, Room NEW/RED, presentation 16:30-17:10

Chris Williams (University of Nottingham)

Title: The exceptional zero conjecture for GL(3)

Abstract: The BSD conjecture predicts that a rational elliptic curve E has infinitely many points if and only if its L-function vanishes at s=1. Iwasawa theory is concerned with p-adic analogues of this phenomena. If E is p-ordinary, there is, for example, a p-adic analytic analogue L_p(E,s) of the L-function; and if E has good reduction at p, then L_p(E,1) = 0 if and only if L(E,1) = 0. However if E has split multiplicative reduction at p, then L_p(E,1) = 0 for trivial reasons, regardless of L(E,1); it has an ‘exceptional zero’. Mazur–Tate–Teitelbaum ‘s ‘exceptional zero conjecture’ (proved by Greenberg–Stevens in ’93) states that in this case the first derivative L_p'(E,1) = 0 if and only L(E,1) = 0. They used this to formulate a precise p-adic BSD conjecture. In this talk I will discuss exceptional zero phenomena, and describe joint work in progress with Daniel Barrera and Andy Graham proving the exceptional zero conjecture for GL(3).

WSNT2: 15:40-17:40, 24th June 2025, Room NEW/RED, presentation 15:40-16:20

Celine Maistret (University of Bristol)

Title: Galois module structure of Tate-Shafarevich groups of elliptic curves

Abstract: For an elliptic curve $E/ \mathbb{Q}$, computing its Tate-Shafarevich group is a fundamental but hard problem. Assuming its finiteness, the Birch and Swinnerton-Dyer conjecture (BSD) provides a formula for its size in terms of the special value of the L-function of $E/\mathbb{Q}$ and some arithmetic invariants of the curve. If $K/\mathbb{Q}$ is a Galois extension, one can ask if a BSD-like formula can be used to link the Galois module structure of the Tate-Shafarevich group of $E/K$ to the special value of the twisted L-function of $E/\mathh{Q}$ by an Artin representation associated to $K/\mathbb{Q}$. In this talk, I will present a joint work with Himanshu Shukla, where we give an explicit conjecture for such a link in the case of primitive Dirichlet characters and provide numerical verifications obtained via 11-descent procedures over $C_5$-extensions.

WSNT2: 15:40-17:40, 24th June 2025, Room NEW/RED, presentation 16:20-17:00

Natalie Evans (King’s College London)

Title: Correlations of almost primes

Abstract: The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime, where $h$ is any non-zero even integer. While this conjecture remains wide open, Matom\”{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

WSNT2: 15:40-17:40, 24th June 2025, Room NEW/RED, presentation 17:00-17:40

Robert Kurinczuk (University of Sheffield)

Title: Finiteness for Hecke algebras of p-adic groups

Abstract: Let G be a p-adic group and K a compact open subgroup of G. Over forty years ago, Bernstein proved that the Hecke algebra $\mathbb{C}[K\backslash G/K]$ is a finitely generated module over its centre and that its centre is a finitely generated $\mathbb{C}$-algebra. It has long been expected that analogues of Bernstein’s result should hold over other coefficient rings. In joint work with Dat, Helm, and Moss, we established such analogues of Bernstein’s finiteness results after inverting $p$. The key ingredient which unlocked the problem was Fargues and Scholze’s geometrization of the local Langlands correspondence. We will explain some of the main ideas from this approach, and time permitting discuss some current work in progress on Local Langlands in Families.

WSNT3: 15:40-17:40, 25th June 2025, Room NEW/RED, presentation 15:40-16:00

Thomas Karam (University of Oxford)

Title: The amount and type of structure of a function in terms of its small images

Abstract: Abstract: It is well known that if G is an abelian group and A is a finite subset of G which minimises the size of A+A for a fixed size of A, then A must essentially be as close as possible to being a coset of some subgroup of G: for instance, if G is finite then the set A must be a coset in G (provided that G has subgroups with size |A|), and if G=Z then the set A must be an arithmetic progression. One may wish to describe an extension of this phenomenon to more arbitrary functions than addition on abelian groups. If X is a set and F: X x X -> X is a function, then can the structure of F (to the extent that there is some) be read from the structure of the subsets A of X minimising the size of the image F(A,A) for a fixed size of A, and can the “amount of stucture” of F be read from how quickly that minimal size grows with the size of A ? We will discuss the progress that has been made in this direction, starting with the results of Elekes-Ronyai, and what remains to be understood.

WSNT3: 15:40-17:40, 25th June 2025, Room NEW/RED, presentation 16:05-16:25

Madhuparna Das (University of Exeter)

Title: Exponential Sums with Additive Coefficients and its Consequences to Weighted Partitions

Abstract: In this article, we consider the weighted partition function $p_f(n)$ given by the generating series $\sum_{n=1}^{\infty} p_f(n)z^n = \prod_{n\in\mathbb{N}^{*}}(1-z^n)^{-f(n)}$, where we restrict the class of weight functions to strongly additive functions. Originally proposed in a paper by Yang, this problem was further examined by Debruyne and Tenenbaum for weight functions taking positive integer values, which has recently been generalized by Bridges et al. We establish an asymptotic formula for this generating series in a broader context, which notably can be used for the class of multiplicative functions. Moreover, we employ a classical result by Montgomery-Vaughan to estimate exponential sums with additive coefficients, supported on minor arcs.

WSNT3: 15:40-17:40, 25th June 2025, Room NEW/RED, presentation 16:30-16:50

Zain Kapadia (Queen Mary University London)

Title: Some Uniserial Specht Modules

Abstract: The Representation Theory of the Symmetric Groups is a classical and rich area of combinatorial representation theory. Key objects of study include Specht modules, the irreducible ordinary representations, which can be reduced modulo p (for p prime). In general, these are no longer irreducible and finding their decomposition numbers and submodule structures are key questions in the area. We give sufficient and necessary conditions for a Specht module in characteristic 2, labelled by a hook partition to be a direct sum of uniserial summands.

WSNT3: 15:40-17:40, 25th June 2025, Room NEW/RED, presentation 16:55-17:15

David Kurniadi Angdinata (London School of Geometry and Number Theory)

Title: Elliptic divisibility sequences in Lean

Abstract: Elliptic divisibility sequences are integer sequences satisfying a certain cyclic recurrence relation, which are crucial in the algebraic theory of division polynomials used in elliptic curve cryptography algorithms. In this talk, I will explain their construction and our subsequent formalisation in the Lean theorem prover, including how we inadvertently discovered a massive gap in the literature. This is joint work with Junyan Xu.

WSNT3: 15:40-17:40, 25th June 2025, Room NEW/RED, presentation 17:20-17:40

Harvey Gray (University of Exeter)

Title: Employment of AI on pure mathematics education

Abstract: The artificial intelligence sector has been proven on numerous occasions to be a viable way of aiding students’ educational experience by serving as a digital personal tutor. Our presentation investigates to what effect ChatGPT can have on pure mathematics education. This presentation investigates ChatGPT’s fluency with LEAN. LEAN is an interactive theorem prover that enables mathematical proof to be written and interpreted on a computer. LEAN has been credited to be a breakthrough in the future of pure mathematics education and is gradually being enrolled into the curriculum of mathematics programs at universities. In the initial stages of primary research, we challenge ChatGPT with different pure mathematics problems in the realm of propositional logic, a topic that beginning mathematics undergraduates are introduced to. We then investigate the LEAN solutions it provides and verify their validity. This leads to a firm understanding of ChatGPT’s fluency with LEAN, allowing us to understand the challenges that are associated with developing a CustomGPT that will help students write mathematical proof in LEAN. Rigorous development and testing is performed on the CustomGPT in the form of flowcharts and written instructions in order to assess its capability and its limitations. From this process, we conclude to what extent the CustomGPT can act as a digital personal tutor for the students’ education in pure mathematics.