The GEARS seminar

13:30-14:20 Noah Dizep (Heriot-Watt)

Title: Difference Equations, Relative Gromov-Witten theory and the Geometry of the Nekrasov-Shatashvili limit

Abstract: Recently there has been an increased interest in the geometry of the refined topological string on a Calabi-Yau threefold in the Nekrasov-Shatashvili limit. In this talk, building up on the work by Bousseau and Brini-Schueler, we will discuss the enumerative meaning of the quantum corrected B-period of the resolved Conifold. This serves to showcase the curious interplay between difference equations arising from the quantization of mirror curves, relative Gromov-Witten theory and refined sheaf counts.

15:00-15:50 Mikhail Vasilev (Glasgow)

Title: Calogero-Moser systems, integrability and Cherednik algebras

Abstract: I will give a gentle introduction to the phenomenon of integrability in general and in the theory of Calogero-Moser systems. I will introduce both classical and quantum integrable Calogero-Moser systems, including scalar and spin(matrix) versions of this model. On the algebraic side one of the structures, which controls the integrability of the Calogero-Moser systems are Cherednik algebras. I will show how the representation theory of Cherednik algebras can be utilised to prove the integrability of the known Calogero-Moser systems. If time permits, I will briefly talk about our recent work with Misha Feigin and Martin Vrabec about the derivation of the spin deformed Calogero-Moser systems from the Cherednik algebras. 

16:30-17:20 Emanuel Roth (Edinburgh)

Title: The structure of instability of moduli of bundles

Abstract: In the moduli theory of bundles, we impose stability to obtain varieties or schemes parametrizing bundles with nice properties (smoothness, separatedness, etc.). When bundles are unstable, we have less of a grasp on how to classify them. Harder and Narasimhan constructed a filtration of bundles that measures how close a bundle is to semistability, which works for principal bundles (by Ramanathan), as well as for Higgs bundles and parahoric torsors. Similarly, Jordan-Hölder filtrations measure how close a semistable bundle is to being stable. I will talk about constructing complementary polyhedra (from Kai Behrend's PhD thesis), a root-theoretic object that records the (semi)-stability of bundles and provides a unified approach to Harder-Narasimhan/Jordan-Hölder filtrations. I will explain how I hope to use this to prove the normality of the stack of parahoric Higgs torsors in a future project.