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:
Interested participants will need to arrange their own accommodation - unfortunately we cannot provide funding for travel or lodging. The following suggestions might help:
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 |
Barbara Bolognese: Introduction to Bridgeland stability conditions
Lecture 1: Triangulated and derived categories
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
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.