Workshop in Sheffield


From the 9th until the 13th of January the University of Sheffield will host a mini workshop on

Stability conditions, Donaldson-Thomas invariants and cluster varieties

aimed at graduate students or postdocs looking for a gentle introduction to these subjects. If you are interested in any of the topics, you are warmly invited to join the workshop. It will consist of 4 mini lecture series, each of 5 lectures. The speakers are:

The lectures take place in room 20 on floor F in the Hicks Building and start at 10am, except for the first lecture which starts at 11:30am. If you would like to participate in the workshop, please send a short email to Sven allowing us to estimate the amount of coffee needed and so on.



Accommodation:

Interested participants will need to arrange their own accommodation - unfortunately we cannot provide funding for travel or lodging. The following suggestions might help:

If you have any further questions, please, don't hesitate to ask Sven or Tom.



Schedule:

Time Monday Tuesday Wednesday Thursday Friday
10:00 - 11:00 Dylan Sven Tom Barbara
11:30 - 12:30 Barbara Sven Tom Barbara Dylan
Lunch
14:00 - 15:00 Dylan Barbara Barbara Dylan Sven
15:30 - 16:30 Sven Tom Dylan Sven Tom
17:00 - 18:00 Tom
19:00 - open Pub Social Dinner



Where to go for lunch:



Abstracts and titles:

Barbara Bolognese: Introduction to Bridgeland stability conditions

Lecture 1: Triangulated and derived categories

Lecture 2: Stability conditions on abelian and triangulated categories Lecture 3: Examples of stability conditions Lecture 4: The space of stability conditions Lecture 5: Stability conditions and MMP



Dylan Allegretti: Introduction to cluster varieties

Lecture 1: Cluster varieties and quantization

Abstract: I will discuss the general definition of the cluster Poisson variety. I will begin by defining the notion of quiver mutation and the compact quantum dilogarithm function. I will then define automorphisms of the noncommutative fraction field of a quantum torus. In the classical limit, these automorphisms give rise to rational maps known as cluster transformations. Cluster varieties are schemes obtained by gluing together algebraic tori using these cluster transformations.

Lecture 2: Moduli spaces of local systems

Abstract: I will discuss examples of cluster varieties arising from marked bordered surfaces. These are defined as compact oriented surfaces with boundary together with finitely many marked points. Associated to such a surface, there is a moduli space parametrizing $PGL_2(\mathbb{C})$-local systems with an additional datum called a framing. I will define rational coordinates on this moduli space and show that it is birational to the cluster Poisson variety.

Lecture 3: Teichmuller and lamination spaces

Abstract: I will explain how cluster varieties appear in low-dimensional geometry and Teichm\"uller theory. I will define the enhanced Teichm\"uller space of a punctured surface and show that it is identified with the set of points of a cluster variety defined over the positive real numbers. I will define a tropicalization of this cluster variety which parametrizes measured laminations on the surface.

Lecture 4: Canonical bases for coordinate rings

Abstract: I will describe a canonical vector space basis for the algebra of regular functions on a cluster variety associated to a punctured surface. This canonical basis is parametrized by certain laminations on the surface. I will explain how this canonical basis can be deformed to a canonical set of elements of the quantized algebra of regular functions. In preparation for the final lecture, I will introduce a construction known as the quantum symplectic double.

Lecture 5: Representations of quantized cluster varieties

Abstract: I will explain how the quantized algebra of functions on a cluster variety can be represented by densely defined operators on an infinite-dimensional Hilbert space. The crucial ingredient in this construction is the non-compact quantum dilogarithm function.




Sven Meinhardt: Introduction to Donaldson-Thomas theory and cohomological Hall algebras

Lecture 1: The motivic Hall algebra

Lecture 2: Integration map and Poisson automorphisms Lecture 3: Donaldson-Thomas invariants Lecture 4: Vanishing cycles Lecture 5: The cohomological Hall algebra



Tom Bridgeland: Stability conditions, Donaldson-Thomas invariants and cluster varieties

Lecture 1: Overall aims. Simple tilts in CY3 categories.

Lecture 2: Exchange graph. Space of stability conditions. Cluster variety.

Lecture 3: Examples from triangulated surfaces: triangulations, quadratic differentials.

Lecture 4: Wall-crossing formula for DT invariants.

Lecture 5: Riemann-Hilbert problems.





Last update: 22/12/2016