Past

I will report on a joint work in progress with Egor Morozov proving that for generic elements in several families of Laplace-type operators invariant under a finite group action, all eigenspaces are irreducible representations. In particular, for the case of Laplace-Beltrami operators, this provides a natural generalization of Uhlenbeck's result on the generic simplicity of the spectrum to the equivariant setting. Moreover, this extends previous work of Zelditch and solves the finite group case of a well-known question raised by Guillemin and Yau. For Schrödinger operators, our results rigorously underpin the notion of accidental degeneracy for certain quantum-mechanical systems with finite symmetry. Our approach involves modern methods of equivariant transversality which we extend to higher dimensions.

For any manifold, we can assign its mapping class group, that is, the group of its diffeomorphisms modulo isotopies. Although such a group can be studied for manifolds of any dimension, the mapping class groups of surfaces draw special attention. They are isomorphic to the outer automorphism groups of $\pi_1(S)$ and have many properties similar to lattices in semisimple Lie groups, as well as connections with the theory of moduli of curves.

One of the most important parts of the research on mapping class groups is the study of their representation. In particular, in the general situation, we still don't know if they have a faithful representation into $\operatorname{GL}_n(\mathbb{C})$.

In my talk, I will show basic facts about mapping class groups and briefly describe a few known methods for constructing their representations and discuss their properties. In particular, I will present recent results classifying low-dimensional representations of the mapping class group.

Many cells exhibit exponential growth not only at the population level but also at the single-cell level. However, single-cell growth rates fluctuate over time. We distinguish between two conceptually distinct sources of growth rate fluctuations: intrinsic continuous fluctuations resulting from intracellular processes, and fluctuations that originate at division events, which we refer to as kicks. We use a simple model to describe single-cell growth and identify the signatures of continuous noise and division kicks. To infer the true biological behavior reliably from experiments, it is crucial to account for measurement noise. We derive analytical expressions for the statistics of meaningful observables, accounting for continuous fluctuations, division kicks, and measurement noise. Importantly, we find that ignoring measurement noise can lead to incorrect biological conclusions. Our results provide insights into how different sources of growth rate variability and measurement errors influence observed cell size dynamics, offering an interpretable framework for analyzing experimental data in cellular biology. 

Professor Luis Nunes Vicente will talk about 'Reducing Sample Complexity in Stochastic Derivative-Free Optimization via Tail Bounds and Hypothesis Testing';

We introduce and analyze new probabilistic strategies for enforcing sufficient decrease conditions in stochastic derivative-free optimization, with the goal of reducing sample complexity and simplifying convergence analysis. First, we develop a new tail bound condition imposed on the estimated reduction in function value, which permits flexible selection of the power used in the sufficient decrease test, q in (1,2]. This approach allows us to reduce the number of samples per iteration from the standard O(delta^{−4}) to O(delta^{-2q}), assuming that the noise moment of order q/(q-1) is bounded. Second, we formulate the sufficient decrease condition as a sequential hypothesis testing problem, in which the algorithm adaptively collects samples until the evidence suffices to accept or reject a candidate step. This test provides statistical guarantees on decision errors and can further reduce the required sample size, particularly in the Gaussian noise setting, where it can approach O(delta^{−2-r}) when the decrease is of the order of delta^r. We incorporate both techniques into stochastic direct-search and trust-region methods for potentially non-smooth, noisy objective functions, and establish their global convergence rates and properties. 

This is joint work with Anjie Ding, Francesco Rinaldi, and Damiano Zeffiro.

Quantum magic quantifies the computational resources needed for quantum operations that cannot be easily performed classically. This requires unitaries, known as "Non-Clifford gates", that map Pauli operators to outside the Pauli group. I will first provide a pedagogical introduction to these concepts following [arXiv:quant-ph/9807006] and then summarise the recent results of [arXiv:2604.14271] constructing non-Clifford gates from path integrals in Chern-Simons theories, whose magic-generating properties are determined by the algebraic data of the topological field theory.

India Marsden will talk about: 'Expanding the definition of a finite element: groups, complexes and software'

The finite element method is a flexible framework to discretise and solve partial differential equations which has been applied to many problems across science and engineering, for example weather modelling and battery design. A core feature of the success of the finite element method, the Ciarlet definition of the components of a finite element has been used for many years. The experience of these decades (and the subsequent implementations) has exposed several key deficiencies. In particular, Ciarlet’s definition is missing information about the global continuity of the mesh and how the degrees of freedom map to each other under the relative orientation of the mesh entities. This information is necessary to implement the finite element method, leaving scope for a new definition.

We propose a new definition to handle these issues and incorporate the constantly growing landscape of new elements. This new definition also aims to encapsulate more information about the elements, such as the symmetries, incorporating ideas from Group Theory. Through this work, we hope to produce a robust, thorough definition that allows processes such as implementation-independent serialisation of finite element data.

Alongside this new definition, we will discuss the new software FUSE, which provides a domain specific language for the definition and enables elements defined in this way to be used in high performance simulation using the finite element package Firedrake. 

Core-annular flows are often proposed to reduce frictional losses in industrial pipeline transport processes. Traditionally, a low-viscosity lubricating film is placed around a more viscous core to reduce the drag on the core. However, maintaining stable pipelining, where the core and the lubricant remain separated has proved challenging.
In this talk we present an alternative approach using three-layer, horizontal core-annular pipe flow, in which two fluids are separated by a deformable elastic solid. In the experiments, an elastic solid created by an in-situ chemical reaction maintains the separation of the core and annular fluids. Corrugations of the elastic interface are observed and stable pipelining, where the elastic shell created separating the two fluids remains intact, is successfully demonstrated even when the core fluid is buoyant. We also develop a theoretical model combining lubrication theory for the fluids with standard shell theory for the elastic solid, to predict the buckling states resulting from radial compression of the shell.
The self-sculpting of the shell by buckling cannot by itself generate hydrodynamic lift owing to symmetry in the direction of flow. Instead, we demonstrate that hydrodynamic lift can be achieved by other elastohydrodynamic effects, when that symmetry becomes broken during the bending of the shell.

Abstract elementary classes (AECs) provide an extension of first order model theory in which we can still attempt a classification theory. The question of when limit models (a kind of surrogate for saturated models for AECs) are isomorphic has connections to important open problems in AECs, such as Shelah's categoricity conjecture. Most work in this area is towards 'positive' results - that is, showing limit models are isomorphic. The question of when limit models are not isomorphic is less explored.

In this talk we give a full characterisation of the spectrum of limit models under reasonable assumptions in a stable AEC - that is, describe completely which limit models are isomorphic and which are not. In particular this applies to the first order stable setting. Given time we will discuss applications, a more general result in the 'positive' direction, and touch on a recent result which says that all high cofinality limit models are disjoint amalgamation bases. Based largely on joint work with Marcos Mazari-Armida.

I will give a short introduction to knotted surfaces in 4-space and discuss some recent developments. First, I will give some motivation, briefly discuss methods for distinguishing knotted surfaces (such as the Khovanov TQFT), and talk about connections with 4-manifolds. Then, I will introduce Artin’s spinning construction, variants of which were defined by Zeeman, Fox, Litherland, and Price-Roseman. Finally, I will specialize to knotted RP^2’s in S^4 and construct a knotted RP^2 in S^4 that cannot be decomposed as the connected sum of an unknotted RP^2 and a knotted S^2. This last result on RP^2’s is joint with Hughes, Kim, and Miller.

Two hundred years ago, Robert Brown observed the statistics of the motion of grains of pollen in water. It took almost one hundred years for Einstein and others to develop an effective theory describing this motion as that of a random walker. In this talk, I will challenge a key implication of this well established theory. When studying systems with very large numbers of particles diffusing together, I will argue that the Einstein random walk theory breaks down when it comes to predicting the statistical behavior of extreme particles—those that move the fastest and furthest in the system. In its place, I will describe a new theory of extreme diffusion which captures the effect of the hidden environment in which particles diffuse together and allows us to interrogate that environment by studying extreme particles. I will highlight one piece of mathematics that led us to develop this theory—a non-commutative binomial theorem—and hint at other connections to integrable probability, quantum integrable systems and stochastic PDEs.

By using the ratios conjecture, we study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for \(\bigl(\zeta'/\zeta\bigr)^{k}\) near the critical line. We explain the connection between this problem and the variance of the convoluted von Mangoldt function in short intervals. We obtain an explicit leading piecewise polynomial in the length parameter which is consistent with the microscopic-shift results of Fan Ge. We also discuss other RMT results for moments of the logarithmic derivative of characteristic polynomials and their relation to trace-average problems over U(N). 

The Poisson boundary of a probability measure on a countable group is a probability space endowed with a stationary group action that captures the asymptotic behaviour of the associated random walk. Since its introduction by Furstenberg in the 1960s, the study of Poisson boundaries and stationary actions has become a powerful tool for understanding geometric and algebraic properties of groups.

In this talk, I will discuss connections between stabilizers of stationary actions, in particular, those arising from the Poisson boundary, and the C*-simplicity of the associated reduced group C*-algebra. I will also address the (seemingly unrelated) problem of realizing different Poisson boundaries on a common underlying topological model. The talk is based on joint work with Anna Cascioli and Martín Gilabert Vio, and with Josh Frisch.

I will discuss some recent and ongoing works on DT invariants of quivers associated to local Calabi-Yau 3-folds, and on conjectural DT4 invariants of local Calabi-Yau 4-folds, in the spirit of "physical mathematics" --- physics computations leading to potentially interesting mathematics. In the CY3 case, I will explain a recently proposed covering formula for quiver DT invariants [arXiv:2603.15334], wherein the DT invariants of some quiver Q are expressed as a sum of DT invariants of a "larger" Galois-covering quiver. I will aim to explain our partial, physics-based derivation of the covering formula. In the CY4 case, I will look at graded quivers associated to exceptional collections of coherent sheaves on local CY 4-folds and discuss what their "DT4 invariants" should look like according to our current physics intuition. These DT4 invariants are generally rational functions of various equivariant parameters of the local geometry.

A Groethendieck pair consists of a finitely generated residually finite group G, with a finitely generated subgroup N such that the inclusion N -> G induces an isomorphism of profinite completions. I will present a new method to produce Groethendieck pairs with peculiar properties, using iterated group theoretic Dehn filling on hyperbolic virtually special groups. Such pairs witness the profinite non-invariance of quasimorphisms, stable commutator length, and actions on hyperbolic spaces and finite-dimensional CAT(0) cube complexes.

Researchers across disciplines rely on clustering to uncover meaningful patterns in noisy similarity data. Standard two-step pipelines reduce noise before clustering, introducing arbitrary parameters that often produce misleading structure. We unite noise reduction and clustering through Bayesian community detection, using information theory to balance model complexity and fit. This one-step approach automatically determines the number of clusters, avoids detecting patterns in random data, and makes full use of limited samples. Testing on synthetic benchmarks and gene expression data shows the approach yields more reliable and interpretable results than widely used alternatives, improving data-driven discovery across scientific disciplines where samples are limited or expensive.

In recent work with Jef Laga, we adapt a construction of Slodowy to build families of algebraic curves in graded Lie algebras (this generalizes earlier work of Thorne). This required an understanding of nilpotent orbits in Vinberg representations, and it raised some interesting questions about these orbits that we were able to answer. Our motivation comes from proofs in arithmetic statistics in which orbits in certain representations are used to parametrize rational points on curves. In this talk, Beth Romano gives an introduction to these ideas via examples.

An anomaly for a global symmetry G says “no”. It stops us from driving the theory to a trivially gapped phase while preserving G. Relatedly, it also prevents us from constructing boundary conditions that preserve G, without adding additional boundary degrees of freedom.

Does a vanishing anomaly say “yes”? It has been proposed that both of these statements can be upgraded to “if and only if” statements. We probe both of these proposals in the simplest theory in which they are non-trivial: the theory of two Dirac fermions in two dimensions, with G chiral. 

Along the way, we will construct all self-duality defects of two free Weyl fermions that arise from gauging an invertible symmetry. These play a central role then in the construction of symmetric boundaries for two Dirac fermions.

We review recent developments in the theory of weak convergence of pde-constrained sequences. We consider the weak lower semicontinuity problem along weakly convergent A-free sequences, where A is a linear pde system of constant rank, and provide improvements to the A-quasiconvexity theory of Fonseca--Müller and the compensated compactness theory of Murat--Tartar. Special emphasis will be placed on concentration effects of weak convergence, in particular by presenting the resolution of a question due to Coifman--PL Lions--Meyer--Semmes and a recent connection between quasiconcavity and higher integrability, generalizing an old result of Müller. Time permitting, we will present the characterization of Young measures generated by A-free sequences by duality with A-quasiconvex functions and recent advances in the regularity theory for A-quasiconvex variational problems. 

Joint work with Christopher Irving, André Guerra, Jan Kristensen, Zhuolin Li, and Matthew Schrecker.

The Lindelöf hypothesis is known to be weaker than the Riemann hypothesis and one way to assess the difference in their strength is to consider what can be said about the zeroes of the zeta function under the assumption of the Lindelöf hypothesis. Viewing this question in the context of zero density estimates, we prove that $N(\sigma,T) \leq T^{\frac{4(5-6\sigma)}{3(3-2\sigma)} + o(1)}$. This improves the currently known estimate conditional on the Lindelöf hypothesis, $N(\sigma,T) \leq T^{2(1-\sigma)+o(1)}$ based on the mean value theorem, for $\sigma$ near $3/4$.

The talk gives some survey about recent applications of finitely additive measures to Lebesgue integrable functions. After a short introduction to such measures and related integrals, purely finitely additive measures are of particular interest. Special examples are given and, as a first application, an integral representation for the precise representative of Lebesgue integrable functions is provided. Then, based on a general approach to traces, a new version of the Gauss-Green formula is introduced, where neither a pointwise trace nor a pointwise normal is needed on the boundary. This allows e.g. the treatment of inner boundaries and of concentrations on the boundary. A second boundary integral is used to handle singularities that hadnot been accessible before. Finally, weak versions of differentiability for Lebesgue integrable functions are discussed, a mean value formula for a class of Sobolev functions is given, and a new approach to the generalized derivatives in the sense of Clarke is provided.

Weighted flag varieties are generalizations of flag varieties and weighted projective spaces.  Although they are not usually homogeneous varieties, they are orbifolds and admit a torus action with isolated fixed points, and like ordinary flag varieties, their equivariant cohomology admits a Schubert basis.  This talk will be an introduction to weighted flag varieties, and will also discuss positivity.  Abe and Matsumura proved that the equivariant cohomology of weighted Grassmannians has a positivity property analogous to that for ordinary (non-weighted) flag varieties.  We prove a strengthened version of this result for arbitrary weighted flag varieties, along the way providing a geometric interpretation of the weighted roots of Abe and Matsumura.  This is joint work with Scott Larson.

The dynamic nature of first order methods can be interpreted by means of continuous time models. In this survey talk, we explain how physical concepts like acceleration, inertia or momentum have been used to improve the performance of convex optimization algorithms. 

We give special attention to the historical evolution of complexity results, especially in the form of convergence rates, under the light of this connection. We also discuss different ways in which acceleration schemes can be applied when the smoothness or strong convexity parameters are unknown, and how these ideas extend to saddle point and constrained problems. 

Given a cyclic subgroup G of GL(2,C) acting on C^2, it was first noticed by Wunram in the 80s that there is a correspondence between certain special representations of G and the exceptional curves appearing in the minimal resolution Y of the surface singularity C^2/G. In modern terms, this was reformulated by Ishii and Ueda as the existence of a fully faithful functor from the derived category of sheaves of Y to the G-equivariant derived category of C^2. In this talk, I will describe a mirror symmetric interpretation of this which exhibits the fully faithful inclusion in algebraic geometry as a sequence of positive Lefschetz stabilizations in symplectic geometry.

Fersztand, Jacquard, Nanda, and Tilmann ('24) introduced the Skyscraper Invariant, a filtration of the classical rank-invariant, for multiparameter persistence modules. It is defined by considering the Harder-Narasimhan (HN) filtration of the module along a special set of stability conditions.

This talk will begin with a post-hoc motivation for considering stability conditions on persistence modules. To compute an approximation of the Skyscraper Invariant we present a technique which, exploiting the geometry of low-dimensional bifiltrations, lets us perform a brute-force computation. We compare it against Cheng's algorithm [Cheng24] which can compute HN filtrations of arbitrary acyclic quiver representations in polynomial time in the total dimension.

To avoid unnecessary recomputation in our algorithm, we ask for which stability conditions the HN filtrations are equivalent. This partition of the space of stabililty conditions is called the wall-and-chamber structure. We show that for a finitely presented d-parameter module it is given by the lower envelopes of a set of multilinear polynomials of degree d-1. For d=2 it is then easy to compute this, enabling a faster algorithm to compute the Skyscraper Invariant up to arbitrary accuracy. As a proof of concept for data analysis, we use it to compute a filtered version of the Multiparameter Landscape for large modules from real world data.

The Springer correspondence parameterises the irreducible representations of the Weyl group of a complex semisimple Lie algebra by nilpotent orbits. A key ingredient in the construction is the convolution operation, which appears in various forms throughout geometric representation theory. In this talk, we'll introduce the geometry of the Springer resolution, describe the convolution operation, and illustrate how it gives rise to a geometric construction of Weyl group representations.

The characterisation of the physics of the M5-brane remains an important open problem in string theory. While the superconformal field theory that resides on a planar M5-brane in flat space is poorly understood, other configurations involving M5-branes wrapped on certain manifolds have well-known superconformal field theory descriptions, including N=2 class S field theories. In this talk, I will use new methods based on exceptional generalised geometry to describe the gravity duals of class S field theories, compute a universal sector of their light-operator spectrum, and provide, for the first time, a holographic match of their superconformal index.

Heterogeneity is a fundamental feature of biological systems. Oncology is one of the fields in which this feature is most evident, as its key players are characterised by mutability, plasticity, and often “uncontrolled” dynamics. Whether heterogeneity arises from spatial structure, environmental variability, or cellular traits, effective therapeutic strategies must explicitly account for it in order to eradicate or control tumours.

From a modern perspective, this requires balancing the hit-hard / keep-it-sensitive trade-off, while also considering not only medical but also broader patient-related side effects of treatments. Contemporary medicine is increasingly exploring ways to exploit the very characteristics that have historically made cancer so dangerous, turning them into potential advantages for therapy.

The multiscale nature of tumour systems, together with the need to predict the combined effects of multiple, non-parallelisable processes, makes the development of optimised mathematical tools particularly compelling. Such tools can address questions that are both scientifically challenging and highly relevant from a clinical and humanitarian perspective.

In this seminar, we will analyse tumour masses from a structured population perspective, focusing on the role of heterogeneity in shaping therapeutic strategies. We will first discuss how heterogeneity in phenotypic composition and nutrient distribution influences the eco-evolutionary dynamics of tumour growth. We will then consider more specifically its impact on radiotherapy.

In particular, we will highlight the advantages of mathematically rigorous modelling in bridging theory and biology. We will also adopt a more exploratory perspective, using these models to illustrate how mathematics can serve as a potential decision-support tool for the selection and optimisation of treatment protocols, within an image- and model-driven framework.

The final part of the seminar will focus on potential future developments, with the aim of fostering an open and collaborative discussion on novel perspectives to improve understanding, prediction, and therapeutic optimisation.

The Zilber—Pink conjecture describes the points on an algebraic variety which have 'special' properties. In this talk, I will discuss some new results which can be proved about this, focusing on the examples of subvarieties of a torus, an abelian variety, or a product of modular curves. The method of proof is a generalisation of the Buium—Coleman proof of the Manin—Mumford conjecture. Parts of this are joint work with Sudip Pandit (KCL) and with Arnab Saha (IIT Gandhinagar).

The path from classic Black& Scholes quant finance to AI-driven trading and hedging. We review a number of recent results and put them in context of a wider strategy.

Professor Colbrook is going to talk about: 'A Computational Framework for Infinite-Dimensional Nonlinear Spectral Problems' 

Nonlinear spectral problems -- where the spectral parameter enters operator families nonlinearly -- arise in many areas of analysis and applications, yet a systematic computational theory in infinite dimensions remains incomplete. In this talk, I present a unified framework based on a solve-then-discretise philosophy (familiar, for example, from Chebfun!), ensuring that truncation preserves convergence. The setting accommodates unbounded operators, including differential operators with spectral-parameter-dependent boundary conditions. 
In the first part, I introduce a provably convergent method for computing spectra and pseudospectra under the minimal assumption of gap-metric continuity of operator graphs -- the weakest natural setting in which the resolvent norm remains continuous. 
In the second part, I develop a contour-based framework for discrete spectra of holomorphic operator families, with a complete analysis of stability, convergence, and randomised sketching based on Gaussian probes. This perspective unifies and extends many existing contour integral methods. Examples throughout highlight practical effectiveness and subtle phenomena unique to infinite dimensions, including the perhaps unexpected sensitivity to probe selection when seeking to avoid spectral pollution.

It is in general nontrivial to construct a 2d worldsheet model whose correlators evaluate to the amplitudes of a target theory. In this talk I will go through a neat, self contained (and to my knowledge, isolated) example in which the matter content and vertex operators of the dual 2d theory can be straightforwardly read off from the action of a 4d theory. Specifically, we will see that a genus 0 worldsheet model whose correlators compute all the tree amplitudes for pure 4d N=4 super Yang-Mills can be essentially derived from the twistor action in elementary steps. We will then discuss the limitations of this approach. There are no twistorial prerequisites assumed.

Jing-Yuan Wang is going to talk about: 'A Runtime-Data-Driven Enhancement Preconditioner for PCG for a Sequence of SPD Linear Systems'
 

In this work, we propose a runtime-data-driven enhancement preconditioner for improving the convergence of a preconditioned conjugate gradient method for solving a sequence of symmetric positive definite linear systems of equations. The methodology is designed for the situation where a subset of the systems has been solved and the convergence is considered too slow. In such a situation, data generated from the solved problems (residual vectors, intermediate solution vectors, approximate error vectors) are first analyzed by an unsupervised learning algorithm as a 3-step process: (1) dimension reduction; (2) classification of the slow features; (3) construction of projections to each of the feature subspaces. Based on the results of the analysis, one or more enhancement preconditioners are constructed using projection matrices corresponding to the features extracted from the slow convergence subspaces. The enhancement preconditioners are additively incorporated into the existing preconditioners and are employed to solve other systems in the sequence. The enhancement preconditioner can be further enhanced when necessary by repeating this process. Numerical experiments for time-dependent problems, including parabolic and hyperbolic equations, and stochastic elliptic equations demonstrate that the proposed approach improves the convergence considerably for other systems in the sequence when classical preconditioners are insufficient.

For more than a hundred years, scientists have carefully analysed the apparently random fluctuations in Brownian trajectories to learn about soft systems. In a more general sense, however, the information hidden within experimental fluctuations is typically underexploited, due to challenges in unambiguously linking fluctuation signatures to underlying physical mechanisms. In this talk, I will discuss our recent work developing new approaches to interpreting fluctuations in experimental data from a variety of soft systems, and thereby turn ‘noise’ into signal. In particular, I will share some recent results taking a fresh look at fluctuations in equilibrium colloidal monolayers. Here, we have combined experiment, simulation and theory to explore how simply counting colloids can reveal details of self and collective dynamics in interacting systems [1,2,3]. I will then discuss ongoing work to extend this understanding to confined driven systems [4], with the long-term goal of elucidating characteristic fluctuations in our synthetic nanopore experiments [5].

[1] E. K. R. Mackay, B. Sprinkle, S. Marbach, A. L. Thorneywork, Phys. Rev X. (2024)

[2] A. Carter, ALT et al., Soft Matter, 21, 3991, (2025)

[3] E. K. R. Mackay, ALT et al., arXiv:2512.17476, (2025)

[4] S. F. Knowles, E. K. R. Mackay, A. L. Thorneywork, J. Chem. Phys., (2024)

[5] S. F. Knowles, A. L. Thorneywork et al., Phys. Rev. Lett, 127, 137801, (2021)

We reformulate the statement that the theory of the free group is stable in terms of simplicial diagram chasing and profinite sets, without any terminology from logic. This includes three characterisations of stability (via indiscernible sequences, counting types, and definable types), and the notions of a first order theory and a model.

We do so by generalising slightly and allowing the universe of a first order structure/model to be an arbitrary (symmetric) simplicial set: formulas and basic predicates now may denote sets of simplices of an arbitrary (symmetric) simplicial set rather than sets of tuples of elements of a set. In this generalised sense the type space functor of a theory is its universal model classifying its usual models: taking the type of a tuple gives a map from a usual model of a theory to its type space functor. We define a property of simplicial maps weaker then being a fibration, and find it appears in the conditions characterising which maps correspond to models, when the generalised semantics is well-behaved, and which symmetric simplicial profinite sets correspond to first order theories.

In 1982, Conway introduced the angel-devil game, which is played on an infinite chess board.  For fixed k, the angel moves at most distance k from its current position on its turn.  The devil then blocks a square permanently.  The devil wins if the angel eventually has no legal moves left.  Berlekamp showed the devil wins against the 1-angel.  Conway asked whether there exists k such that the k-angel has a winning strategy against the devil.  This was resolved independently by Kloster, Máthé, and Bowditch in 2006.  Bowditch proposed playing the game on Cayley graphs of finitely generated groups.  A group for which the devil beats the k-angel for every k is called diabolical.  We will explore the ends of these diabolical groups.

 I will talk about recent progress on (a) a conjecture of Fyodorov and Keating on supercritical asymptotics of moments of moments of characteristic polynomials of the circular beta ensemble and (b) on the correlation functions of the sine beta point process. This is joint work with Joseph Najnudel.

Let G be a finite Chevalley group, i.e., the group of F_q points of a reductive group over F_q. Virtual representations of G in defining characteristic can be constructed in two ways, either by Brauer-Nesbitt reduction of complex representations, or by restricting an algebraic representation. G. Lusztig conjectured the shape of formulas connecting the two procedures; I will discuss a realization of his proposal related to decomposition of the class of diagonal for G/B coming from summands in the push-forward of the structure sheaf under Frobenius. 

Time permitting I will discuss a different, unrelated at present, way to describe such virtual representations linking it to homology of an affine Springer fiber. This found application in the work of Tony Feng and Viet Bao Le Hung on Breuil-Mezard conjectures. 

Based on joint works with Finkelberg, Kazhdan and Morton-Ferguson and with Boixeda Alvarez, McBreen and Yun respectively.

Let G be a finite Chevalley group, i.e., the group of F_q points of a reductive group over F_q. Virtual representations of G in defining characteristic can be constructed in two ways, either by Brauer-Nesbitt reduction of complex representations, or by restricting an algebraic representation. G. Lusztig conjectured the shape of formulas connecting the two procedures; I will discuss a realization of his proposal related to decomposition of the class of diagonal for G/B coming from summands in the push-forward of the structure sheaf under Frobenius. 

Time permitting I will discuss a different, unrelated at present, way to describe such virtual representations linking it to homology of an affine Springer fiber. This found application in the work of Tony Feng and Viet Bao Le Hung on Breuil-Mezard conjectures. 

Based on joint works with Finkelberg, Kazhdan and Morton-Ferguson and with Boixeda Alvarez, McBreen and Yun respectively.

It is known for a long time, due to a celebrated theorem of Ornstein and Weiss, that (classical/plain) orbit equivalence offers no information about ergodic probability measure preserving actions of amenable groups. On the other hand, conjugacy is too intractable, and effectively hopeless to study in full generality. Quantitative orbit equivalence aims to bridge this gap by adding intermediate layers of rigidity— a strategy that has borne fruit already in the late 1960s but was used as a general framework only semi-recently. In this talk, Spyridon Petrakos will introduce aspects of quantitative orbit equivalence and present a complete picture of it for integer odometers. This is joint work with Petr Naryshkin.