**
Tom Bridgeland
**

**Stability conditions on derived
categories, parts 1 and 2.
**

**
Herb Clemens
**

**Special properties of
Noether-Lefschetz loci on Calabi-Yau threefolds: Some examples and
questions.
**

**Abstract
**

We begin with an example, namely the family of lines on the Dwork pencil of quintic threefolds. We use this example as a paradigm for a variational version of the generalized Hodge conjecture for certain families of threefolds.

**
Alastair Craw
**

**On quivers and exceptional
collections for projective toric manifolds.
**

**Abstract
**

I will describe how certain collections of line bundles on a projective toric manifold can be used to construct the manifold as a moduli space of quiver representations. Put another way, I will introduce new quiver gauge theory constructions of these manifolds. The condition on the line bundles is quite weak, and in particular it holds for a complete strong exceptional collection (if one exists). This programme leads to new examples of such collections. This is joint work with Gregory Smith.

**
Emanuel Diaconescu
**

**A vertex formalism for degenerate
torus actions.
**

**Abstract
**

In this talk we will construct a vertex formalism for the residual Gromov-Witten theory of nontoric Calabi-Yau threefolds with degenerate torus action. As an application of this formalism we will present a construction of the Gromov-Witten partition function of local nontoric del Pezzo surfaces.

**
Michael Douglas
**

**What physicists would like to know about the
superpotential, parts 1 and 2.
**

**
Tohru Eguchi
**

**Distribution of flux vacua near
singular loci in Calabi-Yau
moduli space.
**

**
Ezra Getzler
**

**Modular operads revisited.
**

**Abstract
**

Modular operads were introduced, in joint work with Kapranov, for two
reasons: to understand the axiomatization of quantum field theory in two
dimensions, and to give a natural setting for Kontsevich's graph
complexes.
In this talk, I show how to incorporate an open sector into the
definition of modular operads. At the same time, I present a new point
of view on modular operads, in which the role of representations of
the symmetric group *S _{n}* is taken by representations
of the mapping class groups for surfaces with boundary.

**
Sergei Gukov
**

**Topological strings and
categorification of knot invariants, parts 1 and 2.
**

**Abstract
**

We start with a brief introduction into knot homology theories
and categorification of polynomial knot invariants. In particular,
we review the construction and the basic properties of a new homological
knot invariant, recently introduced by Khovanov and Rozansky.
To every knot diagram, it associates a bigraded chain complex
whose graded Euler characteristic is the quantum-group *sl(N)* invariant.
Motivated by the ideas from physics, we then present a reformulation of
the *sl(N)* knot homology in terms of new triply-graded knot invariants.
This leads to new conjectures on the structure of the homological
knot invariants and suggests a larger theory which unifies the *sl(N)*
Khovanov-Rozansky homology (for all *N*) with the knot Floer homology.
We argue that this unification should be accomplished by a triply
graded homology theory which categorifies the HOMFLY polynomial.
We also describe the geometric meaning of the new knot invariants
in terms of the enumerative geometry of Riemann surfaces with
boundaries in a certain Calabi-Yau three-fold.

**
Amihai Hanany
**

**Quiver gauge theories and
Sasaki-Einstein manifolds. Part 1: Tilings, Quviers & Dimers.
**

**Quiver gauge theories and
Sasaki-Einstein manifolds. Part 2: Multiplicities of Toric Diagrams,
Kastelayn matrices & generating functions.
**

**Quiver gauge theories and
Sasaki-Einstein manifolds. Part 3: a maximization & Z
minimization.
**

**Abstract for the series
**

When *D* branes probe a singular CY manifold the gauge theory on the
world volume of the brane gets dramatically changed. The details
depend on the nature of the singularity and in all known cases lead
to a quiver gauge theory. For toric singularities the gauge fields
and matter fields are relatively easy to compute and there exist few
equivalent methods for the computation. It has been a long standing
problem to compute the superpotential which specifies how the matter
fields couple to each other. In these lectures I will present a new
concept - dimers - and will show how they simplify and solve the
problem of computing the superpotential. Along the way we will find a
new set of integers associated with a toric diagram, denoting the
multiplicities of linear sigma model fields.
The quiver gauge theories flow in the infrared to a superconformal
fixed point, the details of it are captured using the AdS/CFT
correspondence - A dual Sasaki-Einstein manifold arises for which
some of its properties are exactly derived using the gauge theory
description.

**
Anton Kapustin
**

*A*-branes and noncommutative
geometry.

**Abstract
**

We describe how noncommutative geometry arises in the study of
topological *D*-branes. In particular, we show that in certain
cases the category of *A*-branes on a Calabi-Yau manifold
*X* can be related to the category of *B*-branes on a
noncommutative deformation of *X*. This duality is distinct from
mirror symmetry and can be regarded as a categorical generalization of
the Seiberg-Witten transform which relates gauge theories on
commutative and noncommutative tori. We sketch an application of this
duality to the geometric Langlands correpondence.

**
Robert Karp
**

*B*-branes on quotient
stacks.

**Abstract
**

In this talk we will discuss three loosely related topics: the
geometric construction of fractional branes in **C**^{d}/**Z**_{n} orbifolds, the
problem of establishing the world-volume theory of a space-filling
D3-brane at a toric singularity, and the proof of the
orbifold/Landau-Ginzburg quantum symmetry in the context of derived
categories. We will see that in all three cases the use of stacks is
indispensable.

**
Ludmil Katzarkov
**

**Homological mirror symmetry for Fano
manifolds and manifolds of general type.
**

**Abstract
**

In this talk we will study HMS for Fano manifolds and manifolds of general type. A disctionary between birational geometry and HMS will be introduced. As an application we will look at the rationality questions for 3 dimensional and 4 dimensional cubics.

**
Conan Leung
**

**Geometry of special holonomy, parts
1 and 2.
**

**
Yi Li
**

**Deformations of generalized complex
structures.
**

**Abstract
**

We discuss the problem of deforming generalized complex structures on generalized Calabi-Yau manifolds, emphasizing the motivation from physics. Relevant mathematical and physical background materials will also be covered.

**
Yuri Manin
**

**Formal deformations with
noncommutative base spaces and Maurer-Cartan equations.
**

**Abstract
**

In this talk I will survey some recent and not-so-recent results on formal deformations of various structures involving algebras, infinity algebras, operads etc stressing the underlying philosophy according to which the type of the object being deformed determines the type of the eventual ``moduli space''.

**
Claude Sabbah
**

**Hermitian metrics on some Frobenius
manifolds.
**

**Abstract
**

I will recall how a canonical Frobenius manifold structure can be
defined on the parameter space of the universal unfolding of any
(convenient and nondegenerate) Laurent polynomial. I will explain how
to construct a *tt ^{*}* structure on this parameter
space. The main theorem is the positive definiteness of the
corresponding Hermitian form. It is obtained through a description of
the Fourier-Laplace transform of a variation of polarized complex
Hodge structure defined on the complex line minus a finite set of
points.

**
Edward Witten
**

**Gauge theory and the Geometric
Langlands Program.
**

**Abstract
**

*N=4* super Yang-Mills theory in four dimensions, compactified to
two dimensions on a Riemann surface *C*, gives, via its
*S*-duality symmetry, a natural framework for understanding the
geometric Langlands program for complex Riemann surfaces. The
correspondence between the tensor algebra of representations of
* ^{L}G* and the Hecke algebra of

**
Kota Yoshioka
**

**Wall crossing formula for the Euler
characteristic
of determinant line bundles.
**

**Abstract
**

Let *M _{H}(c_{1},c_{2})* be the moduli of