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.


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.


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.


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.


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 Sn 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.


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.


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.


In this talk we will discuss three loosely related topics: the geometric construction of fractional branes in Cd/Zn 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.


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.


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.


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.


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.


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 LG and the Hecke algebra of G comes from S-duality between Wilson loops on one side and 't Hooft loops on the other. The correspondence between LG local systems and D-modules on the moduli space of G-bundles depends on some considerations involving branes, or in other words supersymmetric boundary conditions.

Kota Yoshioka

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


Let MH(c1,c2) be the moduli of H-stable sheaves on a surface X. For a divisor D on X, we can associate a determinant line bundle whose first chern class is defined by the μ-map. I consider the Euler characteristic of this determinant line bundle.In particular, I will talk about the wall crossing formula under a suitable assumption on X. This is a joint work with Lothar Goettsche and Hiraku Nakajima.