For every positive integer N and every α ∈ [0,1), let B(N, α) denote the probabilistic model in which a random set A of (1,...,N) is constructed by choosing independently every element of (1,...,N) with probability α. We prove that, as N → +∞, for every A in B(N, α) we have |AA| ~ |A|^2/2 with probability 1-o(1), if and only if (log(α^2(log N)^{log 4-1}))(√loglog N) → ∞. This improves on a theorem of Cilleruelo, Ramana and Ramar\'e, who proved the above asymptotic between |AA| and |A|^2/2 when α =o(1/√log N), and supplies a complete characterization of maximal product sets of random sets.

Line patterns in free groups are collections of lines in the Cayley graph of a non-abelian free group F, which correspond to finite sets of words in F. Following Cashen and Macura, we will discuss line patterns by looking at the topology of Decomposition Spaces, which are quotients of the boundary of F that correspond to the different line patterns. Given a line pattern, we will also construct a cube complex whose isometry group is isomorphic to the group of quasi isometries of F which (coarsely) preserve the line pattern. This is a useful tool for studying the quasi isometric rigidity of related groups.

For a group G and a finite dimensional linear representation σ : G → GLn(D) over a skew field (division ring) D, the agrarian invariants with respect to σ are the homological invariants of G with coefficient module Dn. In this talk I will discuss the relationship between agrarian invariants, L 2 -invariants, Thurston norm and twisted Alexander polynomials. I will also discuss an ongoing work with Dawid Kielak.

The conformal dimension of a hyperbolic group is a powerful numeric quasi-isometry invariant associated to its boundary.

As an invariant it is finer than the topological dimension and allows us to distinguish between groups with homeomorphic boundaries.

I will start by talking about what conformal geometry even is, before discussing how this connects to studying the boundaries of hyperbolic groups.

I will probably end by saying how jolly hard it is to compute.

In this talk I will introduce the study of lattices in locally compact groups through their actions CAT(0) spaces. This is an extremely rich class of groups including S-arithmetic groups acting on products of symmetric spaces and buildings, right angled Artin and Coxeter groups acting on polyhedral complexes, Burger-Mozes simple groups acting on products of trees, and the recent CAT(0) but non biautomatic groups of Leary and Minasyan. If time permits I will discuss some of my recent work related to the Leary-Minasyan groups.

The graph complex is a remarkable object with very rich structure and many, sometimes mysterious, connections to topology. To illustrate one such connection, I will attempt to construct a “self-linking” invariant of knots and expand on the ideas behind it.

Riche—Williamson recently proved that the characters of tilting modules for GL_h are given by non-singular p-Kazhdan—Lusztig polynomials providing p>h.  This is equivalent to calculating the decomposition numbers for symmetric groups labelled by partitions with at most h columns.  We discuss how this result can be generalised to all cyclotomic quiver Hecke algebras via a new and explicit isomorphism between (truncations of) quiver Hecke algebras and Elias–Williamson’s diagrammatic endomorphism algebras of Bott–Samelson bimodules. 

This allows us to give an elementary and explicit proof of the main theorem of Riche–Williamson’s recent monograph and extend their categorical equivalence to all cyclotomic quiver Hecke algebras, thus solving Libedinsky–Plaza’s categorical blob conjecture.  Furthermore, it allows us to classify and construct the homogeneous simple modules of quiver Hecke algebras via BGG resolutions.   
 
This is joint work with A. Cox, A. Hazi, D.Michailidis, E. Norton, and J. Simental.  
 

I will describe joint work with Abouzaid which constructs a stable homotopy theory refinement of Floer homology that has coefficients in the Morava K-theory spectra. The classifying spaces of finite groups satisfy Poincare duality for the Morava K-theories, which allows us to use this version of Floer homology to produce virtual fundamental chains for moduli spaces of Floer trajectories. As an application, we prove the Arnold conjecture for ordinary cohomology with coefficients in finite fields.

A celebrated result in geometry is the Kobayashi-Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite-Einstein metric if and only if the bundle is slope polystable. Recently, Dervan and Sektnan have conjectured an analogue of this correspondence for fibrations whose fibres are compact Kähler manifolds admitting Kähler metrics of constant scalar curvature. Their conjecture is that such a fibration is polystable in a suitable sense, if and only if it admits an optimal symplectic connection. In this talk, I will provide an introduction to this theory, and describe my recent work on the conjecture. Namely, I show that existence of an optimal symplectic connection implies polystability with respect to a large class of fibration degenerations. The techniques used involve analysing geodesics in the space of relatively Kähler metrics of fibrewise constant scalar curvature, and convexity of the log-norm functional in this setting. This is work for my PhD thesis, supervised by Frances Kirwan and Ruadhaí Dervan.