Programme. Download as a PDF file.

Chris Brav (3 talks): 'From non-commutative to derived algebraic geometry'.
Abstract:
Differential graded (dg) categories, viewed as being categories of sheaves on putative non-commutative spaces, provide a flexible approach to non-commutative algebraic geometry, with the 'moduli of objects' in a dg category, in the sense of Toën-Vaquié, establishing a bridge from non-commutative to derived algebraic geometry. After reviewing basic notions in the homotopy theory of dg categories and in derived algebraic geometry, we describe how a 'relative Calabi-Yau structure' on a dg functor, in the sense of Brav-Dyckerhoff, induces a Lagrangian structure on the corresponding morphism of moduli spaces.

Tom Bridgeland (2 talks): 'Wall-crossing for Donaldson-Thomas invariants'.
Abstract: The main focus of the talks will be the Kontsevich-Soibelman wall-crossing formula for generalised Donaldson-Thomas invariants. In the first talk I will introduce the relevant context, state the formula and explain how it works in the simplest example of the A2 quiver. In the second talk I will discuss a Riemann-Hilbert problem suggested by the wall-crossing formula. If there is time I will explain how in the case of the resolved conifold the solution to this problem produces the topological string partition function.

You can download the slides here:
Lecture 1 pdf file


Emily Cliff (2 talks):
'Chiral algebras and factorization algebras'.
Abstract: The definition of a vertex algebra was formulated by Borcherds in the 1980s to solve algebraic problems, but these objects turn out to have important applications in mathematical physics, especially related to models of 2d conformal field theory. In the 1990s, Beilinson and Drinfeld gave geometric formulations of the definition, which they called chiral algebras and factorization algebras. These different approaches each have advantages and disadvantages: for example, the definition of a vertex algebra is more concrete and has so far been better studied; on the other hand, the geometric approach of chiral algebras and factorization algebras allows for transfer of knowledge between the fields of geometry, physics, and representation theory, and furthermore admits natural generalizations to higher dimensions. In these talks we will introduce all three of these objects; then we will discuss the relationships between them, especially focusing on how information from any one approach can lead to new understanding in the others.

You can download the slides here:
Lecture 1 pdf file


Adam and Elena Gal (2 talks): 'Hall algebras and higher Hall algebras'.
Abstract: The Hall algebra associated to a category is related to the Waldhausen S-construction in the work of Kapranov and Dyckerhoff. We explain how the higher associativity data can be extracted from this construction in a natural way, thus allowing for various higher categorical versions of Hall algebra. We describe several such constructions. We then discuss a natural and systematic extension of the S-construction construction providing a bi-algebraic structure. We show how it can be used to provide a more transparent proof for the Green's theorem for the Hall algebras of hereditary categories and discuss possible extension to the higher categorical settings.

Dominic Joyce (3 talks):
'Donaldson-Thomas theory of Calabi-Yau 3-folds, and generalizations'.
Abstract: Donaldson-Thomas invariants DT
α(τ) are integers or rational numbers which 'count' τ-semistable coherent sheaves with Chern class α on a Calabi-Yau 3-fold
X, for some stability condition τ. They were introduced by Richard Thomas in 1998 for proper τ-stable moduli spaces, and extended to include strictly τ-semistable sheaves by Joyce and Song in 2008. They have the nice property that they are unchanged by continuous deformations of the Calabi-Yau 3-fold X.
  We discuss 'classical' Donaldson-Thomas theory and its more recent generalizations to categorified and motivic invariants, based on Pantev-Toën-Vaquié-Vezzosi’s theory of shifted symplectic geometry.


You can download the slides here:
Lecture 1 pdf file  Lecture 2 pdf file  Lecture 3 pdf file

Kevin McGerty
and Tom Nevins (3 talks): 'Quantizations and symplectic representation theory'.
Abstract: Much of the representation theory of a reductive algebraic group is captured in the geometry of the cotangent bundle of its flag variety. This is smooth symplectic variety with the unusual property that it is birational to its affinization, and is an example of what is called a symplectic resolution. Recently there has been much interest in extending ideas from classical Lie theory to the setting of arbitrary symplectic resolutions. In these lectures we will begin by casting the classical theory in the context of symplectic geometry, and then discuss some recent advances: localization results for quantized symplectic resolutions which arise as GIT quotients, versions of category O, and categorical decompositions.


You can download the slides here:
Lecture 1 pdf file
Lecture 2 pdf file

Nick Rozenblyum (3 talks): 'Shifted symplectic and Poisson structures in derived algebraic geometry'.
Abstract: Many moduli problems of interest, such as moduli spaces of local systems, come equipped with a natural symplectic or Poisson structure. The theory of shifted symplectic and Poisson structures is a vast generalization of algebraic symplectic geometry which provides a natural framework for studying these symplectic structures. In addition to being a natural setting for the BV approach to Feynman integration, this theory provides a robust framework for various counting problems in geometry and topology, such as the theory of Donaldson-Thomas invariants and its generalizations. In these lectures we will give an overview of derived geometry and the theory of shifted symplectic and Poisson structures with an emphasis on applications to moduli problems.