Workshop on Homological Mirror Symmetry and Related Topics
January 24-29, 2011, University of Miami

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Mark Gross (UC San Diego):  Theta functions for K3 surfaces and tropical Morse trees.

Maxim Kontsevich (IHES and U. Miami):  Classical and quantum geometry of integrable systems

Naichung Conan Leung (Chinese Univ. of Hong Kong):  SYZ mirror symmetry

In these lectures, I will explain the Strominger-Yau-Zaslow proposal which explains the mirror symmetry as a Fourier transformation. We will discuss the toric cases in more details. In particular, we show that the SYZ map coincides with the mirror map for certain toric Calabi-Yau manifolds.

Andrew Neitzke (U. Texas Austin):  A construction of hyperkahler metrics, with application to Hitchin systems

Pierre Schapira (Institut de Mathématiques de Jussieu):  Microlocal theory of sheaves and applications to non-displaceability

In the first talk we will explain the main notions and results of the microlocal theory of sheaves: the microsupport of sheaves and its behaviour with respect to the 6 operations, with emphasis on the Morse lemma.

In the second talk, inspired by the recent work of Tamarkin but with really different methods and results, we will apply these tools to treat some problems of non-displaceability, including the conservation of Morse inequalities as well as some results specific to positive isotopies. The main tool is a theorem which asserts that any Hamiltonian isotopy admits a unique sheaf quantization (joint work with S. Guillermou and M. Kashiwara).



Mohammed Abouzaid (Clay Institute and MIT):  Homotopy equivalence of nearby Lagrangians

I will explain how enlarging the Fukaya category to admits as objects local systems of arbitrary rank over Lagrangian submanifolds reveals features of their topology going beyond their homology groups. In particular, one can prove that Maslov 0 exact Lagrangians in a cotangent bundle must be homotopy equivalent to the base.

Matthew Ballard (U. Penn.):  Orlov spectra

This talk will cover joint work with D. Favero and L. Katzarkov. Defined by D. Orlov, the Orlov spectrum is a new invariant of a triangulated category. When the triangulated category is of geometric origin, like the bounded derived category of coherent sheaves on a variety or the derived Fukaya category of a symplectic manifold, meaningful geometric information is encoded in the shape of the Orlov spectrum. In this talk, we will begin by reviewing the necessary background for and natural questions concerning Orlov spectra. We will demonstrate an upper bound for the Orlov spectrum of the category of singularities for an isolated hypersurface singularity and, then, use similar methods, plus a theorem of Orlov, to strongly constrain the Orlov spectrum of the bounded derived category of coherent sheaves on a smooth degree n+1 hypersurface in P^n. If time allows or interest demands, we will also discuss the structure of the Orlov spectra of the derived Fukaya categories of symplectic surfaces of higher genus and give upper bounds on the generation time of exceptional collections.

Thomas Kragh (MIT):  Nearby Lagrangians with arbitrary Maslov class.

I will show that ANY closed (compact without boundary) exact Lagrangian in a cotangent bundle is up to a finite covering space lift a homology equivalence. The proof uses several natural properties of a Serre type spectral sequence, which originates by considering the symplectic action functional for a certain family of Hamiltonians as "fibrant" over the base manifold.

Dmitri Orlov (Steklov Institute):  D-branes in LG-models, matrix factorizations, and triangulated categories of singularities: properties and relations.

Tony Pantev (U. Penn.):  Mirror symmetry and mixed Hodge structures

James Pascaleff (MIT):  Floer cohomology in the mirror of CP^2 relative to a conic and a line

In the Strominger-Yau-Zaslow description of mirror symmetry, singularities of the torus fibration lead to difficulties in the construction of mirror spaces and the computation of algebraic structures associated to these spaces. We will discuss one such algebraic structure, the Floer cohomology of Lagrangian sections of the torus fibration, in a space with a simple type of singularity, the Landau-Ginzburg mirror of CP^2 relative to a conic and a line, along with some natural generalizations. Of particular interest will be the canonical basis for the Floer cohomology groups (and hence of sheaf cohomology groups on CP^2) that our construction gives rise to.

Nick Sheridan (MIT):  On the Homological Mirror Symmetry conjecture for pairs of pants

The n-dimensional pair of pants is CPn with n+2 generic hyperplanes removed. We construct an immersed Lagrangian sphere in the pair of pants and compute its A endomorphism algebra in the Fukaya category. On the level of cohomology, the algebra is an exterior algebra with n+2 generators. It is not formal, and we must compute certain higher products to determine it up to quasi-isomorphism. This allows us to give some evidence for the Homological Mirror Symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model (Cn+2,W), where W = z1...zn+2. We show that the endomorphism algebra of our Lagrangian is quasi-isomorphic to the endomorphism (dg) algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.

Bernd Siebert (Univ. Hamburg):  Logarithmic Gromov-Witten invariants

Yan Soibelman (Kansas State U.):  Complex integrable systems, wall-crossing formulas and Calabi-Yau categories

David Treumann (Northwestern):  Plumbings, constructible sheaves, and HMS