January 19-24, 2009, University of Miami

**Maxim Kontsevich** (IHES and U. Miami):
Geometric and algebraic aspects of wall-crossing

**Hiraku Nakajima** (Kyoto Univ.):
Instanton counting and wall-crossing in Donaldson invariants

Donaldson invariants of 4-manifolds were originally introduced via the moduli spaces of anti-self-dual connections, but can be approached also by moduli spaces of stable bundles when the underlying manifolds are complex projective surfaces. Based on this approach, we compute the wall-crossing terms of Donaldson invariants, describing what happens when the ample line bundle is changed. This is an old subject, but a recent advance, the `instanton counting', enables us to compute them in the torus equivariant setting. Then the terms are naturally expressed in terms of the so-called Seiberg-Witten curves. Finally, we express the blow-up formula of Donaldson invariants, as a wall-crossing formula via moduli spaces of perverse coherent sheaves on the blow-up.

Talks are based on series of joint works with Kota Yoshioka, Lothar Göttsche, and Takuro Mochizuki.

**Richard Thomas** (Imperial College):
Curve counting and derived categories

I will describe joint work with Rahul Pandharipande. The celebrated MNOP conjecture relates Gromov-Witten theory to invariants counting embedded curves and subschemes in 3-folds. I will review this theory, and describe another method to count curves that throws away the worst subschemes. This "stable pair" theory lives in the derived category of the 3-fold, and its relation to MNOP theory is via a wall crossing in a space of stability conditions. Via another wall crossing, it gives the conjectural BPS curve counting invariants of Gopakumar-Vafa. I will explain all of these concepts.

**Denis Auroux** (MIT):
Special Lagrangian fibrations and mirror symmetry

These lectures will focus on the construction of mirror manifolds using special Lagrangian fibrations, with the Strominger-Yau-Zaslow conjecture as a starting point. The first talk will be elementary, and will provide some motivation and basic examples, both in the Calabi-Yau case and in the broader setting of varieties with effective anticanonical divisor. We will in particular explain how Landau-Ginzburg models naturally arise in this setting, viewing the superpotential as a mirror counterpart to a Floer-theoretic obstruction.

The second talk will discuss some simple examples of the wall-crossing phenomena which arise in the non-toric case, as well as some evidence concerning mirror symmetry for pairs consisting of a variety and a Calabi-Yau hypersurface.

Finally, the last talk will discuss joint work in progress with Mohammed Abouzaid and Ludmil Katzarkov regarding the extension of mirror symmetry to arbitrary hypersurfaces in toric varieties, by considering Lagrangian fibrations on blow-ups. The main examples there will be pairs of pants (and their higher-dimensional analogues) and higher-genus curves.

**Mohammed Abouzaid** (Clay Institute and MIT):
Homological Mirror Symmetry for T^4

**Sabin Cautis** (Rice Univ.):
Equivalences from geometric sl_{2} actions

**Alexandr Efimov** (Moscow):
Noncommutative Grassmanians

**Kenji Fukaya** (Kyoto Univ.):
Mirror symmetry for toric manifolds

**Alexander Kuznetsov** (Steklov Institute):
Fractional Calabi-Yau triangulated categories

**Rahul Pandharipande** (Princeton):
The tropical vertex group

**Tony Pantev** (U. Penn.):
Geometric Langlands and non-abelian Hodge theory

**Yongbin Ruan** (Michigan):
Integrable hierarchies and singularity theory

**Paul Seidel** (MIT):
Homological mirror symmetry for the genus two curve

**Yan Soibelman** (Kansas State Univ.):
DT-invariants, quivers and cluster transformations

**Atsushi Takahashi** (Osaka Univ.):
HMS for isolated hypersurface singularities

**Ilia Zharkov** (Kansas State):
Geometry of numbers and tropical curves