Past Meetings: 2018–19

Alex Evetts — Growth in the Heisenberg group

The combinatorial and geometric notion of growth in a finitely generated group has been studied intensively since the 1980s, both asymptotically and in terms of formal power series. I will give a brief overview of growth and the related idea of conjugacy growth, focussing on the question of rationality of the formal power series. Next, I will attempt to describe the beautiful geometry behind the (integer) Heisenberg group, which gives us insight into its growth series, and explain how this relates to the work I am doing on conjugacy growth. If time allows I will talk about a generalisation of the direct product of groups that allows us to transfer information about growth from the Heisenberg group to other nilpotent groups.

Niall Hird — The Characters of GL_2(F_q)

This talk will explain how to calculate all of the irreducible complex characters for the group GL2(Fq). We begin by proving some necessary facts concerning GL2(Fq) such as its order and the conjugacy classes. Then we will introduce each irreducible character and provide a proof of irreducibility. 

Juliet Cooke — The Relative Tensor Product of Skein Categories

In this talk I will generalise the definition of a relative tensor product of modules to define a relative tensor product of k-linear categories. I will apply this definition to skein categories, categories of tangles surfaces up to 'skein relations', and sketch how the relative tensor product of skein categories is simply the skein category of the surfaces glued together topologically.

Kellan Steele — Determining the Divisor Class group: type A singularities

The ADE Dynkin diagrams appear in many different areas of mathematics and have been studied for quite some time. In algebraic geometry, the ADE Dynkin diagrams are related to Kleinian surface singularities. To start, we will recall the definition of divisor class group and an important theorem about their behaviour. Through an example, we will show how one may determine the divisor class group of the affine coordinate ring of a surface with singularities of type A. We will then explain how to generalise this method and, if there is time, discuss how one might hope to apply it to type D singularities. 

Alex Levine — The Normal Form of Thompson's Group F

Given a tree pair, it is possible to find an expression in terms of the generators of F, for the element represented by that tree pair, using the exponents of the tree pair, which are the lengths of specific paths in each tree. We give a brief attempt to use this to classify certain sets of elements of F using finite sequences of integers induced by the lists of the exponents. We also attempt to count elements of F by counting the pairs of sequences.

Ross Paterson — Elliptic Curves and the Congruent Number Problem

This talk will discuss elliptic curves defined over number fields, and their surprising link to triangles. This should be a largely self contained talk, and in particular no background in algebraic number theory will be assumed. We will survey this background, before going on to discuss the group structure on elliptic curves, first over the complex numbers and then over number fields. We will use this to show the equivalence between finding triangles with integer area and bounding the `rank' of certain elliptic curves over the rational numbers. Time permitting we may introduce the cohomological machinery used to study these curves, and the arithmetic statistics which arise from this. 

Emily Roff — The Magnitude of an Enriched Category

How should we measure the size of a category? If we regard categories as categorified sets, to measure them requires a categorical analogue of cardinality. It can be argued that cardinality is really a specialised Euler characteristic - so what is the Euler characteristic of a category? In this talk I'll outline an answer to that question. I'll tell the story of magnitude: a numerical invariant of enriched categories which generalises topological Euler characteristic, and whose specialisation to metric spaces turns out to encode a rich variety of classical geometric information. If time permits, I'll also discuss connections to graph theory and applications in mathematical ecology.

Okke van Garderen — Numerical Donaldson-Thomas Invariants in Representation Theory

At a young age, we have all been introduced to mathematics by learning how to count. In this talk I will show how to apply this valuable technique to algebra and (virtually) count representations. It turns out that these counts can be reduced to a comprehensive list of numbers, which are called Donaldson-Thomas invariants. Properly defined, these invariants seem to govern how the algebra can be "deformed" into simpler pieces, which we will see in a number of examples. 

Recently the definition of Geigle-Lenzing Weighted Projective Lines (WPL) has been generalised to higher dimensions. One of the interesting things about WPLs is that we can produce the ADE singularities, and in fact, these are the only instances where we also have an NCCR. We will look at the higher-dimensional analogue, Weighted Projective Planes and whether similar results exist. Under mild conditions we are able to construct a partial resolution of the Veronese subrings. We will look at some examples and obstructions to results which no longer hold. 

Peter Haine — The branched topology

Melvin Hochster's thesis shows that the category of spectral topological spaces (i.e., spaces underlying qcqs schemes) and quasicompact continuous maps is equivalent to the category of pro-objects in the category of finite posets. The latter category has a nontrivial involution induced by sending a poset to the same set with the opposite order relation. Hochster's thesis then provides a nontrivial involution on the category of spectral topological spaces. In particular, there is a 'dual Zariski space' associated to a scheme.

A natural question to ask is if it is possible to run Hochster's story in the étale topology. In recent work with Clark Barwick and Saul Glasman, we showed that this is indeed the case by providing an equivalence between a certain category of spectral topoi containing all étale topoi of qcqs schemes and a category of pro-objects in stratified homotopy types. Again, there is an obvious involution on one side of the equivalence, so there is a 'dual étale topos' associated to any qcqs scheme. In this talk we'll give an explicit description of this dual étale topos in terms sheaves on a site of branched covers and (if time) discuss its relationship to the Abhyankar Inertia Conjecture. The latter is joint work with Clark Barwick and Tomer Schlank.

Sarah Kelleher — The Weighted Projective Plane and the Resolution of Singularities

Recently the definition of Geigle-Lenzing Weighted Projective Lines (WPL) has been generalised to higher dimensions. One of the interesting things about WPLs is that we can produce the ADE singularities, and in fact, these are the only instances where we also have an NCCR. We will look at the higher-dimensional analogue, Weighted Projective Planes and whether similar results exist. Under mild conditions we are able to construct a partial resolution of the Veronese subrings. We will look at some examples and obstructions to results which no longer hold. 

Graduate level talk — Introduction to 2-representation theory

I will motivate and explain the ideas of (finitary/fiat) 2-categories and (finitary/fiat) 2-representations. 

Research level talk — Coidempotent subcoalgebras and extensions of finitary 2-representations

I will explain what short exact sequences of 2-representations are and how they are related to coidempotent subcoalgebras of certain coalgebra 1-morphisms in the 2-category. This is closely related to the ideas relating recollements of abelian categories and idempotent quotients of algebras.