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:

• Barbara Bolognese: Introduction to Bridgeland stability conditions
• Dylan Allegretti: Introduction to cluster varieties
• Sven Meinhardt: Introduction to Donaldson-Thomas theory and cohomological Hall algebras
• Tom Bridgeland: Stability conditions, Donaldson-Thomas invariants and cluster varieties

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:

• The Harley Hotel: very close and cheap (£30/n), but there is live music downstairs which might affect your sleep
• The Wilson Carlile Centre: also quite close to the Hicks building but slightly more expensive (~40 p/n)
• The Premier Inn Hotel at St Mary's Gate: 10 min walking distance, prices between £30-50/n
• The Jurys Inn: about 15 min walking distance, prices from £50/n

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:

• Students' Union right behind the Hicks Building contains a lot of places on several levels
• Brood Cafe Bar at Roco offers a small but fine selection of food
• University Arms is a pub with typical pub food
• The Doctor's Order is another pub with fancier pub food

Abstracts and titles:

Barbara Bolognese: Introduction to Bridgeland stability conditions

Lecture 1: Triangulated and derived categories

• Motivation
• Triangulated categories
• Basic properties
• Examples: derived categories
• t-structures
• Tilting theory for t-structures

Lecture 2: Stability conditions on abelian and triangulated categories

• Stability functions on abelian categories
• SC on derived categories 1: slicings
• SC on derived categories 2: hearts of t-structures
• Basic properties and relations

Lecture 3: Examples of stability conditions

• Examples (easy): slope stability on curves and stability for quivers
• Why this doesn't work on surfaces
• SC on surfaces
• SC on threefolds (conjecture)

Lecture 4: The space of stability conditions

• Intoduce the space of SC
• As a topological space: the topology
• heorem: structure of a complex manifold
• Example: curves, K3 (what we know)
• The two actions and the connected component of geometric SC

Lecture 5: Stability conditions and MMP

• The moduli space of stable complexes, vague hint of construction
• The wall-and-chamber structure of the space of stability conditions
• The large volume limit and the Gieseker chamber
• Examples: Hilbert scheme of pts of the projective plane (ABCH)

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

• The ring of motives
• Families of representations
• The motivic Hall algebra

Lecture 2: Integration map and Poisson automorphisms

• An example
• The integration map
• Poisson automorphisms

Lecture 3: Donaldson-Thomas invariants

• Lambda-rings
• DT-invariants
• Potentials

Lecture 4: Vanishing cycles

• DT-theory for quiver with potential
• The classical vanishing cycle sheaf

Lecture 5: The cohomological Hall algebra

• Equivariant cohomology
• Moduli spaces and intersection cohomology

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.