Talks

Here are titles and abstracts of some of the talks that will be taking place during the conference. For more information, participants should look in their conference package.

Colloquia

Representations of p-adic groups and why "success is not in never failing, but rising every time you fall"

Prof Jessica Fintzen

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups.

In my talk I will introduce p-adic groups and provide an overview of our understanding of their representations, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.

I will finish with some remarks about my career path and explain why I wrote "success is not in never failing, but rising every time you fall" on my pinboard.

Infinite staircases, reflexive polygons, and quadratic surds (A good enough mathematician.)

Dr. Ana Rita Pires

A classic result due to McDuff and Schlenk asserts that the function that encodes when a four-dimensional symplectic ellipsoid can be embedded into a four-dimensional ball has a remarkable structure: the function has infinitely many corners, determined by the odd-index Fibonacci numbers, that fit together to form an infinite staircase.

In this talk we will discuss a general framework for analyzing the question of when the ellipsoid embedding function for other symplectic 4-manifolds is partly described by an infinite staircase, in particular by giving an obstruction to the existence of an infinite staircase that experimentally seems strong.

We will then look at the special case of rational convex toric domains / closed symplectic toric manifolds: there are six families of targets with infinite staircases, and they are distinguished by the fact that their moment polygon is reflexive. We conjecture that in this case, these six families constitute a complete answer to the question in the paragraph above, and we will examine this conjecture by translating it to a number theoretical conjecture related to quadratic surds and Ehrhart functions.

This is based on joint work of Dan Cristofaro-Gardiner, Tara Holm, Alessia Mandini, and Ana Rita Pires.

Short Talks

Balanced metrics on 6-dimensional cohomogeneity one manifolds.

Izar Alonso Lorenzo

In the context of Hermitian geometry, the Strominger system is a system of non-linear PDEs on heterotic string theory. In this talk we will describe cohomogeneity one manifolds and then look for solutions to the Strominger system in three dimensions in the cohomogeneity one setting. This leads us to consider simply connected three dimensional complex manifolds endowed with an invariant nowhere-vanishing holomorphic (3,0)-form which are of cohomogeneity one under the almost effective action of a connected Lie group G. We show that one of such M would have to be compact and have a certain principal orbit type, up to G-equivariant diffeomorphism.

Distributions of Character Sums

Ayesha Hussain

Over the past few decades, there has been a lot of interest in partial sums of Dirichlet characters. Montgomery and Vaughan showed that these character sums remain a constant size on average and, as a result, a lot of work has been done on the distribution of the maximum. In this talk, we will investigate the distribution of these character sums themselves, with the main goal being to describe the limiting distribution as the prime modulus approaches infinity. This is motivated by Kowalski and Sawin’s work on Kloosterman paths.

A Mathematical Life Told in Three Projects

Sue Sierra

I will talk about the 3 projects which have shaped my career. Two are mathematical: the classification of noncommutative projective surfaces and chain conditions in enveloping algebras of infinite-dimensional Lie algebras. One is human: the "Sane Mathematician Project", without which the other two wouldn't be possible. I'm interested in encouraging more mathematicians to work on all three!

On the Galois-Gauss sums of weakly ramified characters

Yu Kuang

In an attempt of proving a natural extension of the Galois module theory developed by Fröhlich and M. Taylor, Bley, Burns and Hahn have recently introduced techniques of relative algebraic K-theory to formulate a conjecture of wildly ramified Galois-Gauss sums. We adapt results from Agboola and Caputo, and obtain some new evidence in support of Bley, Burns and Hahn's conjecture on arithmetic properties of Galois-Gauss sums associated to characters that are 'weakly ramified' in the sense of Erez.

Organisations and networks supporting the careers of women in mathematics

Prof Gwyneth Stallard

This talk will give an overview of organisations that support the careers of women in mathematics and, more generally, women in STEM. It will give links to sources of useful information, inspiring role models, details of relevant fellowships and networks.