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Richard Thomas: Lectures

01/13/2022: Higher rank DT theory from rank 1
01/06/2020: Borisov-Joyce in algebraic geometry
01/08/2018 and 01/09/2018: Introduction to coherent sheaves

January 13, 2022
TITLE: Higher rank DT theory from rank 1

ABSTRACT: DT invariants count stable bundles and sheaves on Calabi-Yau 3-folds X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the Gromov-Witten invariants of X.

Along the way we also show they are also determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

Slides of Lecture

January 6, 2020
TITLE: Borisov-Joyce in algebraic geometry

ABSTRACT: Borisov and Joyce have defined invariants counting sheaves on Calabi-Yau 4-folds. Unfortunately they use real derived differential geometry, which nobody likes. In joint work with Jeongseok Oh we describe a way to define their invariants entirely within the much nicer subject of algebraic geometry. Technically this involves defining a localised “square root Euler class” for isotropic sections – and isotropic cones – in SO(n,C) bundles. We also prove a torus localisation result for our virtual class.

Slides of lecture

 

January 8 and January 10, 2018
TITLE: Introduction to coherent sheaves

ABSTRACT: I will try to give a coherent introduction to sheaf theory.

Coherent sheaves can be thought of as singular holomorphic vector bundles on complex manifolds, and can be used to compactify moduli of bundles. They thus give a way to define higher dimensional gauge theory invariants on projective varieties, and give examples that demonstrate some of the phenomena that can arise on more general manifolds of special holonomy.

After an introductory first lecture I will focus on some of (depending on audience tastes): curve counting via sheaves, stable pairs, the relationship to GW theory (MNOP conjecture) and Gopaukmar-Vafa invariants, the Serre construction relating codimension two subvarieties to rank 2 bundles, smoothing of singularities of reflexive sheaves.

January 10, 2018:

 

January 8, 2018: