Workshop

We will give an overview of the Strominger-Yau-Zaslow (SYZ) approach to mirror symmetry in the setting of non-compact Calabi-Yau varieties given by the complement U of an anticanonical divisor D in a projective variety X. Namely, U is expected to carry a Lagrangian torus fibration, and a mirror Calabi-Yau variety U' can then be

constructed as a (suitably corrected) moduli space of Lagrangian torus fibers equipped with local systems. (Partial) compactifications of U deform the symplectic geometry of these Lagrangian tori by introducing holomorphic discs; counting these discs yields distinguished regular functions on the mirror U'. The goal of the talk will be to illustrate these concepts on simple examples, such as the complement of a conic in C^2.

If time permits we will also try to explain the relation of this story to the symplectic cohomology of U and its product structure

I will give a self-contained introduction to cluster algebras and cluster varieties.

In my two talks I'll explain the general features of my conjecture, joint with Gross, Hacking and Siebert, that the algebra of regular functions on an affine log CY (with maximal boundary), comes with a natural vector space basis, generalizing the characters of an algebraic torus, for which the structure constants for the multiplication rule are positive integers, counting holomorphic discs (on the mirror).

In particular, I'll explain how using these "theta functions" one can generalize the basic constructions of toric geometry. E.g. a single choice of anti-canonical normal crossing divisor on a Fano Y (conjecturally) canonically determines for each line bundle L on Y a basis of sections parameterized by the integer points of a "polytope" (more precisely, a piecewise integer affine  manifold with boundary).  The talks will be at a very general level -- in particular I won't assume any familiarity with cluster varieties or mirror symmetry

These talks will be part of the continuing exposition of the ideas of GHKK. I will focus in particular on the actual mechanics of how scattering diagrams for cluster algebras are constructed and give rise to theta functions via sums over broken lines.

Stability conditions on triangulated categories allow us to define moduli spaces of objects in the category which undergo wall-crossing as the stability condition varies.   We will consider certain Calabi-Yau-3 triangulated categories whose space of stability conditions has a wall-and-chamber structure which "categorifies" mutation in a corresponding cluster algebra.  Following a recent article of Bridgeland I will show how to enhance this wall-and-chamber structure to a scattering diagram which in certain cases coincides with the scattering diagram associated to the cluster algebra by Gross-Hacking-Keel-Kontsevich

Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category equipped with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation.

Let S be an oriented surface with punctures, and a finite number of special points on the boundary considered modulo isotopy. It give rise to a moduli space X(m, S), closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. For each puncture of S, there is a birational Weyl group action on the space X(m, S). 
We calculate the DT-transformation of the moduli space X(m,S), with few exceptions. 

We discuss the relationship between DT-transformations and duality conjectures in general.

In particular, our main result, combined with the work of Gross, Hacking, Keel and Kontsevich,
gives rise to a canonical basis in the space of regular functions on the cluster variety X(m,S), and in the upper cluster algebra with principal coefficients related to the pair (SL(m), S)

In the talk I will try to explain the appearance and interpretation of wall crossing structures in mirror symmetry

In this talk I will explain a construction by K. Rietsch of the mirrors of homogeneous spaces. The latter include Grassmannians, quadrics and flag varieties, and their mirrors can be expressed using Lie theory. More precisely the mirrors of homogeneous spaces live on so-called `Richardson varieties', which possess a cluster structure, and the mirror superpotential is defined on these Richardson varieties.

I will start by detailing Rietsch's general construction, then I will present some recent results by Marsh-Rietsch on Grassmannians, as well as joint work with Rietsch (resp. Rietsch and Williams) on Lagrangian Grassmannians (resp. quadrics). In particular I will show how the restriction of the superpotential on each cluster chart is a Laurent polynomial, which changes as we change cluster charts

This is a report on joint work with B. Leclerc and J. Schröer.

Recall that Derksen, Weyman and Zelvinsky managed to prove many combinatorial conjectures about cluster algebras with symmetric exchange matrix by studying certain representations of the corresponding quiver together with a non-degenerate potential.

We proposed in [GLS2], that in this situation the generic values of Palu's cluster character on the strongly reduced components of the corresponding representation varieties should be a good candidate for a basis of the cluster algebra. It is not hard to see that these elements have the following properties:

(a) contain all cluster monomials,

(b) they are independent of the mutation class,

(c) belong to the upper cluster algebra.

On the other hand, we showed in [GLS2] that the dual of Lusztig's semicanonical basis (which is defined for the enveloping algebra of the positive  part of any Kac-Moody Lie algebra with symmetric Cartan matrix) restricts to a basis of the coordinate ring of corresponding unipotent cell associated to an Weyl group element.

This dual semicanonical basis contains all cluster monomials.

In [GLS2] we showed, that in  the above mentioned candidate for a "generic basis" and the dual semicanonical basis coincide.

In this talk I will try to explain this development in the special case of a symmetric Cartan matrix of finite type and the longest element in the Weyl group.

References

[GLS1] Ch. Geiss, B Leclerc, J Schröer:

Kac–Moody groups and cluster algebras

Advances in Mathematics 228 (2011), 329-433

[GLS2] Ch. Geiss, B Leclerc, J Schröer:

Generic bases for cluster algebras and the Chamber Ansatz Journal of the American Mathematical Society 25 (2012), 21-76

We will study the geometry of cluster varieties from the perspective of stability conditions on the associated Calabi-Yau-3 triangulated category.  I will focus on ideas introduced by Gaiotto-Moore-Neitzke which suggest how to produce cluster coordinates from stability conditions.   In particular we will consider the class of examples associated to triangulations of marked bordered surfaces for which the cluster variety is a moduli space of rank 2 local systems and the space of stability conditions is a space of quadratic differentials with prescribed singularities on an associated closed surface