The GEARS seminar

João Camarneiro (Edinburgh)

Title: Infinite symplectic staircases and where to find them

Abstract: In 1985, Gromov proved the famous non-squeezing theorem, giving new fundamental insights into the nature of symplectic geometry. Since then, there has been a lot of interest in various symplectic embedding problems. In this talk, I will tell you about the "infinite staircases" that sometimes appear when embedding 4-dimensional symplectic ellipsoids into various targets, and the surprisingly intricate structure governing their existence.

Theresa Ortscheidt (Glasgow)

Title: Combinatorics of Structure Coefficients in the Fusion Ring

Abstract: The fusion or Littlewood-Richardson ring is a quotient of the ring of symmetric functions. While much is known about the Littlewood-Richardson coefficients, which are the structure coefficients of the ring of symmetric functions with respect to the Schur basis, the fusion coefficients are not yet well understood. In particular, one open problem is the lack of a strictly positive combinatorial formula to compute fusion coefficients.

In my talk I will recap some of the combinatorics of Littlewood-Richardson coefficients and then present some of the results of Goodman and Wenzl, who found a (not strictly positive) combinatorial formula to compute fusion coefficients for certain representations of Hecke algebras at roots of unity. I will then briefly discuss some of my progress in trying to find a strictly positive algorithm.

Giorgio Mangioni (Heriot-Watt)

Title: Cyclic splittings of Artin groups

Abstract: Atari founder Nolan Bushnell once said that "every good game is easy to learn but hard to master". As an instance of this principle, it is straightforward to define Artin groups, which arise naturally in representation theory and topology, but most basic questions about this class of groups have been open for over a century. We don't even have a unified solution to the isomorphism problem, which asks to determine when two Artin groups are isomorphic!

In this talk we characterise which Artin groups admit a splitting over Z (that is, an action on a tree with infinite cyclic edge stabilisers), and we use this to shed more light on the isomorphism problem. In the process, we also construct a JSJ splitting for any Artin group, which roughly "refines" every splitting over Z. If time allows, we shall also mention some applications to the study of automorphism groups of Artin groups, and further directions of investigation. 

This talk is based on joint work with Oli Jones and Giovanni Sartori.