January 18-23, 2010, University of Miami

**Mohammed Abouzaid** (Clay Institute and
MIT):

- A functorial point of view on HMS
Starting with a Fano variety, I will briefly recall how to obtain the conjectural Landau-Ginzburg mirror from the SYZ construction, and in particular how to the mirror of the category of sheaves on the complement of an anticanonical divisor should be the the wrapped Fukaya category of the mirror. Then, I will explain work in progress with Paul Seidel producing mirrors on the A model of the Landau-Ginzburg side for restriction operations on the category of coherent sheaves.

- A split-generation criterion for the Wrapped Fukaya
category
In proving homological mirror symmetry, one commonly manages to prove that some subcategory of the Fukaya category is equivalent to the expected category of coherent sheaves. I will explain a geometric condition which guarantees that this subcategory of the Fukaya category is in fact equivalent to the entire Fukaya category, up to the algebraic operations of adding cones and summands.

- Floer homology and string topology
The wrapped Fukaya category of a cotangent bundle Q gives a family of examples where explicit computations may be done: the chains over the based loop space of Q form a differential graded algebra, and I will explain progress on completing the proof that the triangulated closure of the wrapped Fukaya category is equivalent to a certain category of modules over this dga.

**Maxim Kontsevich** (IHES and U. Miami):

- Singular Lagrangian branes
Any (Wein)stein manifold X can be contracted to a singular Lagrangian submanifold L, e.g. by the gradient flow of a plurisubharmonic function. I'll argue that L carries a homotopy cosheaf of dg-categories of finite type, whose global section is the wrapped Fukaya category of X. In the case when X is a surface with boundary, we obtain a description of F(X) as a homotopy colimit of a finite diagram of representation categories of quivers of series A. Further examples include Riemann-Hilbert correspondence for irregular holonomic D-modules (via Stokes structures), and dg-algebras associated with Legendrian links.

- Mixed non-commutative motives and open
Gromov-Witten invariants
In derived non-commutative algebraic geometry one can define a triangulated category of mixed nc-motives similarly to Voevodsky's theory, but in much more simpler and direct way. The conjecture on the degeneration of Hodge to de Rham spectral sequence extends to the mixed setting. Also, one define a refined Chern class of an object of a saturated dg-category in an analog of Deligne cohomology. I'll present also examples finite diagrams in A-model setting, where one can identify periods of mixed motives with some kind of open GW invariants.

- Refined cohomological Hall algebra and topology
of holomorphic Chern-Simons functional
I'll describe a new approach to Donaldson-Thomas invariants of 3-dimensional Calabi-Yau categories with stability structure, via a new type of Hall algebras. An analogy with Landau-Ginzburg models suggests that an extra structure (gluing data) is natural to consider, related in the geometric limit to the counting of a kind of self-dual connections on G_2-manifolds. The refined counting has wall-crossings simultaneously in complex and Kahler moduli spaces.

**Bertrand Toën** (Univ. Montpellier 2):

- Saturated dg-categories I: definitions and basic properties
- Saturated dg-categories II: finiteness and moduli spaces
- Saturated dg-categories III: topological invariants

**Serguei Barannikov** (ENS, Paris):
Developments in the noncommutative Batalin-Vilkovisky formalism

**Jonathan Block** (U. Penn.):
TBA

**Tobias Dyckerhoff** (U. Penn.):
Isolated hypersurface singularities as noncommutative spaces

**David Favero** (U. Penn.):
The dimension spectrum of a triangulated category; spherical and
exceptional objects

**Kenji Fukaya** (Univ. Kyoto):
TBA

**Victor Ginzburg** (Univ. of Chicago):
Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections

Let *Y, Z * be a pair of smooth coisotropic subvarieties in a smooth
algebraic Poisson variety *X*. We show that any data of first order
deformation of the structure sheaf *O _{X}* to a sheaf of noncommutative
algebras and of the sheaves

Our construction is motivated by, and is closely related to, a result of Behrend-Fantechi, who considered the case of Lagrangian submanifolds in a symplectic manifold.

**Kentaro Hori** (IPMU):
Mirror symmetry and reality

**Dmitry Kaledin** (Steklov Institute):
Hochschild cohomology as a factorization algebra

**Bernhard Keller** (Univ. Paris 7):
The periodicity conjecture via Calabi-Yau categories

**Alexander Kuznetsov** (Steklov Institute):
Hochschild homology and cohomology of admissible subcategories

**Valery Lunts** (Indiana):
Uniqueness of enhancement for triangulated categories I

**Dmitri Orlov** (Steklov Institute):
Uniqueness of enhancement for triangulated categories II

**Grigory Mikhalkin** (Univ. Genève):
TBA

**Yan Soibelman** (Kansas State U.):
Cohomological Hall algebra