Cambridge University

I will report on work joint with Florian Naef in which we produce, for a map f of spaces over a space B such that f has compact fibers, a rational model for the fiberwise transfer of fiberwise topological Hochschild homology, considered as a map of parametrized spectra over B. This is motivated by applications to moduli spaces of manifolds: in particular we can detect the vanishing of certain cohomology classes originating from a graph complex via the classifying space of homotopy automorphisms.
 

In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between a smoothing, which is a type of surgery on a curve that resolves a self-intersection, and k-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay. There will be lots of pictures. Based on joint work with Max Neumann-Coto.

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.

I will review how the equations of general relativity near a spacetime singularity map onto an arithmetic hyperbolic billiard dynamics. The semiclassical quantum states for this dynamics are Maaβ cusp forms on fundamental domains of modular groups. For example, gravity in four spacetime dimensions leads to PSL(2,Z) while five dimensional gravity leads to PSL(2,Z[w]), with Z[w] the Eisenstein integers. The automorphic forms can be expressed, in a dilatation (Mellin transformed) basis as L-functions. The Euler product representation of these L-functions indicates that these quantum states admit a dual interpretation as a "primon gas" partition function. I will describe some physically motivated mathematical questions that arise from these observations.

I will relate two notorious open questions in low-dimensional topology.  The first asks whether every hyperbolic group is residually finite. The second, the congruence subgroup property, relates the finite-index subgroups of mapping class groups to the topology of the underlying surface. I will explain why, if every hyperbolic group is residually finite, then mapping class groups enjoy the congruence subgroup property. Time permitting, I may give some further applications to the question of whether hyperbolic 3-manifolds are determined by the finite quotients of their fundamental groups.

Schwarzian Theory is a quantum field theory which has attracted a lot of attention in the physics literature in the context of two-dimensional quantum gravity, black holes and AdS/CFT correspondence. It is predicted to be universal and arise in many systems with emerging conformal symmetry, most notably in Sachdev--Ye--Kitaev random matrix model and Jackie--Teitelboim gravity.

In this talk we will discuss our recent progress on developing rigorous mathematical foundations of the Schwarzian Field Theory, including rigorous construction of the corresponding measure, calculation of both the partition function and a natural class of correlation functions, and a large deviation principle.

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd (integral) distance from each other. We will also discuss some further results with Rose McCarty and Michal Pilipczuk concerning prime and polynomial distances.

I will discuss joint work with S. Galatius and A. Kupers in which we investigate the homology of general linear groups over a ring $A$ by considering the collection of all their classifying spaces as a graded $E_\infty$-algebra. I will first explain diverse results that we obtained in this investigation, which can be understood without reference to $E_\infty$-algebras but which seem unrelated to each other: I will then explain how the point of view of cellular $E_\infty$-algebras unites them.

One of the main characterisations of word-hyperbolic groups is that they are the groups with a linear isoperimetric function. That is, for a compact 2-complex X, the hyperbolicity of its fundamental group is equivalent to the existence of a linear isoperimetric function for disc diagrams D -->X.
It is likewise known that hyperbolic groups have a linear annular isoperimetric function and a linear homological isoperimetric function. I will talk about these isoperimetric functions, and about a (previously unexplored)  generalisation to all homotopy types of surface diagrams. This is joint work with Dani Wise.