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Program of Workshop

All lectures will take place at ESI, Boltzmann lecture hall.

First week (21st - 26th June)

Time Monday Tuesday Wednesday Thursday Friday Saturday1
9:15 - 9:30 Opening
9:30 - 10:30 Serguei Barannikov
A_infinity equivariant matrix integrals 1
Serguei Barannikov
A_infinity equivariant matrix integrals 2
Yoannis Vlassopoulos
Weak Calabi-Yau algebras 1
Yoannis Vlassopoulos
Weak Calabi-Yau algebras 2
Yoannis Vlassopoulos
Weak Calabi-Yau algebras 3
Atanas Iliev
Fano manifolds of Calabi-Yau type
10:30 - 11:00 Coffee break Coffee break Coffee break Coffee break Coffee break
11:00 - 12:00 Manfred Herbst
Matrix factorization 1
Ludmil Katzarkov
Mixed Hodge structures and diagrams of Fukaya categories 2
David Favero
Spectra of categories 1
Emanuel Scheidegger
Effective actions, normal functions and matrix factorizations 1
Matt Ballard
Spectra of categories 2
Charles Doran
Modular Invariants for Lattice Polarized K3 Surfaces
2:00 - 3:00 Maxim Kontsevich
Mixed Hodge structures and diagrams of Fukaya categories
Manfred Herbst
Matrix factorization 2
Serguei Barannikov
A_infinity equivariant matrix integrals 3
Albrecht Klemm
Counting Donaldson Thomas invariants with modular forms
Emanuel Scheidegger
Effective actions, normal functions and matrix factorizations 2
Charles Doran
Toric Hypersurface Normal Forms and the Kuga-Satake Hodge Conjecture
3:00 - 3:30 Coffee break Coffee break Coffee break Coffee break Coffee break
3:30 - 4:30 A. Efimov
Formal completion of a category along a subcategory
A. Efimov
HMS for P^1-{\geq 3 points}
Johanna Knapp
Normal functions, effective superpotentials and mirror symmetry
Nils Carqueville
Quiver gauge theory from open topological string theory
Richard Garavuso
Extending Hori-Vafa
4:45 - 5:45 Grigory Mikhalkin
Basic Notion of tropical Geometry 1
Grigory Mikhalkin
Basic Notion of tropical Geometry 2

1Please note: Lecture times on Saturday are 10 a.m., 11:30 a.m. and 1 p.m., respectively.

Second week (28th - 30th June)

Time Monday Tuesday Wednesday
9:30 - 10:30 Mark Gross
TBA
Gregory Pearlstein
The locus of the Hodge classes in admissible variations of mixed Hodge structure 2
Yan Soibelman
Donaldson Thomas invariants and integrable systems 3
10:30 - 11:00 Coffee break Coffee break Coffee break
11:00 - 12:00 Yan Soibelman
Donaldson Thomas invariants and integrable systems 1
Denis Auroux
Mirror symmetry for blowups and hypersurfaces in toric varieties 1
Denis Auroux
Mirror symmetry for blowups and hypersurfaces in toric varieties 2
2:00 - 3:00 Gregory Pearlstein
The locus of the Hodge classes in admissible variations of mixed Hodge structure 1
Yan Soibelman
Donaldson Thomas invariants and integrable systems 2
Mohammed Abouzaid
Functors and Fukaya Categories 2
3:00 - 3:30 Coffee break Coffee break Coffee break
3:30 - 4:30 Mohammed Abouzaid
Functors and Fukaya Categories 1
Mark Gross
TBA
Victor Przhiyalkovskiy
TBA
4:45 - 5:45   Ludmil Katzarkov
Spectra of categories 3

7:30 Dinner at Heurigen

Abstracts
Mohammed Abouzaid (MIT): Functors and Fukaya Categories
I will describe three flavours of the Fukaya category, and define functors between them which are mirror to pullback and restriction functors of sheaves (in very special situations). The technical details are joint work with P. Seidel.

Denis Auroux (MIT,UCB): Mirror symmetry for blowups and hypersurfaces in toric varieties
The goal of these talks is to give a geometric motivation for some recent constructions of mirrors to varieties of general type. In the first part, we will give an overview of the Strominger-Yau-Zaslow approach to mirror symmetry, where mirror pairs are constructed by dualization of real torus fibrations. We will focus especially on the non-Calabi Yau case, where the mirror geometry is modified by a superpotential. The second part will focus on recent joint work with Mohammed Abouzaid and Ludmil Katzarkov, investigating mirror symmetry for blowups of toric varieties. This yields a geometric approach to the construction of mirrors of essentially arbitrary hypersurfaces in toric varieties. The main examples will be pairs of pants and genus 2 curves.

Matt Ballard (UPenn): Spectra of categories
see David Favero

Serguei Barannikov (ENS): A-infinity equivariant matrix integrals
I'll describe the higher dimensional analog of the matrix Airy integral, associated with noncommutative Calabi-Yau varieties, and introduced in my 2006 paper on noncommutative Batalin-Vilkovisky formalism.

Nils Carqueville (Munich): Quiver gauge theory from open topological string theory
TBA

Charles Doran (Alberta): Modular Invariants for Lattice Polarized K3 Surfaces
TBA

Charles Doran (Alberta): Toric Hypersurface Normal Forms and the Kuga-Satake Hodge Conjecture
TBA

A. Efimov (RAS): Formal completion of a category along a subcategory
Following an idea of Kontsevich, I will introduce the notion of formal completion of compactly generated triangulated category along a small subcategory. I will explain that in algebro-geometric setting one obtains in this way the usual formal completion of a Noetherian scheme along a closed subscheme (answering a question of Kontsevich). Moreover, it will be shown that Beilinson-Parshin adeles of Noetherian schemes can be obtained in this way as well.

A. Efimov (RAS): HMS for P^1-{\geq 3 points}
We will show how to prove homological mirror symmetry for P^1-{\geq 3 points}. On the A side we have perfect complexes over (wrapped) Fukaya category of $\P^1\setminus\{\geq 3 points\},$ and on the B side we have the Karoubian completion of $D_{sg}(W^{-1}(0)),$ where $(X,W)$ is a mirror (toric open CY) Landau-Ginzburg model with monomial potential.

David Favero (Wien): Spectra of categories
The dimension spectrum is a certain invariant that one can attach to any strongly generated triangulated category. We will give an overview of subject with regards to the study of the derived category of coherent sheaves on an algebraic variety. We also describe some recent developments from joint work with M. Ballard and L. Katzarkov.

Richard Garavuso (Alberta): Extending Hori-Vafa
TBA

Mark Gross (UCSD): TBA
TBA

Manfred Herbst (Augsburg): Matrix Factorization
TBA

Atanas Iliev (Wien,BAS): Fano manifolds of Calabi-Yau type
A Fano manifold of Calabi-Yau type(or a FCY manifold) is a Fano manifold of odd dimension with Hodge structure similar to this of a Calabi-Yau threefold. The FCY manifolds share some basic common properties with CY threefolds, inherited by the similar VHS - similar type of gauged moduli spaces, over which the relative intermediate Jacobian is an integrable system. We give various examples of FCY manifolds, and for some of them we show the infinite generatedness of their Griffiths groups - a property known to take place for all non-rigid CY threefolds.

Ludmil Katzarkov (Wien): Mixed Hodge structures and diagrams of Fukaya categories
see Maxim Kontsevich

Ludmil Katzarkov (Wien): Spectra and categories
see David Favero

Maxim Kontsevich (IHES): Mixed Hodge structures and diagrams of Fukaya categories
It is a joint work in progress with L.Katzarkov and T.Pantev, on the extension of mirror symmetry to singular cases. We propose mirror duals to Fano varieties with anticanonical divisors consisting of several irreducible components, and also to non-maximal degenerations of Calabi-Yau varieties.

Albrecht Klemm (Bonn): Counting Donaldson Thomas invariants with modular forms
In this talk we discuss how Donaldson Thomas invariants on Calabi-Yau threefolds are encoded in terms of modular forms. Emphasis is laid on the interplay between modularity and non-holomorphicity. In particular the holomorphic anomaly lead to the construction of almost holomophic forms for counting of D6-D2-D0 BPS states, which are related to DT and Gromov-Witten invariants. We report on recent observation that link the problem of counting D4-D2-D0 BPS states.

Johanna Knapp (IMPU): Normal functions, effective superpotentials and mirror symmetry
TBA

Grigory Mikhalkin (Geneva): Basic Notion of tropical Geometry
TBA

Tony Pantev (UPenn): TBA
TBA

Gregory Pearlstein (UMich): The locus of the Hodge classes in admissible variations of mixed Hodge structure
Let S' be a Zariski-open subset of a complex manifold S, and let V be a variation of mixed Hodge structure on S'. Suppose that V is defined over the integers, graded polarizable, and admissible with respect to S. Let Hdg(V ) denote the locus of Hodge classes in V . Then each component of Hdg(V ) extends to an analytic space, finite and proper over S.

Victor Przhiyalkovskiy (RAS,Wien): TBA
TBA

Emanuel Scheidegger (Augsburg): Effective actions, normal functions and matrix factorizations
I will describe aspects of the relation between effective actions for B-type D-branes on Calabi-Yau threefolds and normal functions. By the Calabi-Yau/Landau-Ginzburg correspondence, the D-branes are described in terms of matrix factorizations and their deformations. We will give explicit examples.

Yan Soibelman (KSU): Donaldson Thomas invariants and integrable systems
In a series of papers with Maxim Kontsevich we introduced and studied properties of a generalization of the classical Donaldson-Thomas invariants (BPS invariants in the language of physics) of 3d Calabi-Yau categories. I am going to discuss the relationship of these invariants to complex integrable systems and tropical geometry.

Yoannis Vlassopoulos (IHES): Weak Calabi-Yau algebras
It is well known (Kontsevich-Soibelman, Costello) that the Hochschild chain complex of a Calabi-Yau (CY) algebra (a compact $A_\infty$ algebra with scalar product) has the structure of a Topological Quantum Field theory. The compactness means that the algebra has finite dimensional cohomology and it is desirable to extend such a result beyond such algebras. We will show how to do that for what we shall call weak CY algebras: Let $A$ be a vector space and $A^*$ its dual. A weak CY algebra structure on $A$ is an $A_\infty$ algebra structure on $A \oplus A^*$ compatible with the natural scalar product and such that $A$ is an $A_\infty$ subalgebra of $A \oplus A^*$. We will show that the Hochschild chain complex of $A$ has the structure of an algebra over a dg-PROP of chains in the moduli space $\mathcal{M}_{g,m,n}$ of stable curves with $m$ incoming and $n$ outgoing marked points, where $m,n\geq 1$. The chains we construct correspond to directed ribbon graphs (ribbon quivers). As an example we shall consider the case of the algebra defined by the Pontryagin product on chains in $\Omega X$ the space of based loops in an oriented compact manifold $X$. These results are joint work with Maxim Kontsevich.

As of June 24, 2010