Workshop on Homological Mirror Symmetry and Related Topics
January 18-23, 2010, University of Miami

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Mohammed Abouzaid (Clay Institute and MIT): 

  1. A functorial point of view on HMS

    Starting with a Fano variety, I will briefly recall how to obtain the conjectural Landau-Ginzburg mirror from the SYZ construction, and in particular how to the mirror of the category of sheaves on the complement of an anticanonical divisor should be the the wrapped Fukaya category of the mirror. Then, I will explain work in progress with Paul Seidel producing mirrors on the A model of the Landau-Ginzburg side for restriction operations on the category of coherent sheaves.

  2. A split-generation criterion for the Wrapped Fukaya category

    In proving homological mirror symmetry, one commonly manages to prove that some subcategory of the Fukaya category is equivalent to the expected category of coherent sheaves. I will explain a geometric condition which guarantees that this subcategory of the Fukaya category is in fact equivalent to the entire Fukaya category, up to the algebraic operations of adding cones and summands.

  3. Floer homology and string topology

    The wrapped Fukaya category of a cotangent bundle Q gives a family of examples where explicit computations may be done: the chains over the based loop space of Q form a differential graded algebra, and I will explain progress on completing the proof that the triangulated closure of the wrapped Fukaya category is equivalent to a certain category of modules over this dga.

Maxim Kontsevich (IHES and U. Miami): 

  1. Singular Lagrangian branes

    Any (Wein)stein manifold X can be contracted to a singular Lagrangian submanifold L, e.g. by the gradient flow of a plurisubharmonic function. I'll argue that L carries a homotopy cosheaf of dg-categories of finite type, whose global section is the wrapped Fukaya category of X. In the case when X is a surface with boundary, we obtain a description of F(X) as a homotopy colimit of a finite diagram of representation categories of quivers of series A. Further examples include Riemann-Hilbert correspondence for irregular holonomic D-modules (via Stokes structures), and dg-algebras associated with Legendrian links.

  2. Mixed non-commutative motives and open Gromov-Witten invariants

    In derived non-commutative algebraic geometry one can define a triangulated category of mixed nc-motives similarly to Voevodsky's theory, but in much more simpler and direct way. The conjecture on the degeneration of Hodge to de Rham spectral sequence extends to the mixed setting. Also, one define a refined Chern class of an object of a saturated dg-category in an analog of Deligne cohomology. I'll present also examples finite diagrams in A-model setting, where one can identify periods of mixed motives with some kind of open GW invariants.

  3. Refined cohomological Hall algebra and topology of holomorphic Chern-Simons functional

    I'll describe a new approach to Donaldson-Thomas invariants of 3-dimensional Calabi-Yau categories with stability structure, via a new type of Hall algebras. An analogy with Landau-Ginzburg models suggests that an extra structure (gluing data) is natural to consider, related in the geometric limit to the counting of a kind of self-dual connections on G_2-manifolds. The refined counting has wall-crossings simultaneously in complex and Kahler moduli spaces.

Bertrand Toën (Univ. Montpellier 2): 

The main purpose of this series of lectures is to present some recent developments in the theory dg-algebras and dg-categories arising in algebraic geometry. In the first lecture, I will present the basic definitions of smooth, proper, and finite type dg-algebras and dg-categories, as well as their properties. The second lecture is concerned with finiteness properties of smooth and proper dg-categories, such as definition over "small" rings, existence of various moduli spaces (of dg-modules, dg-algebras and dg-categories themselves). In the third lecture I will present the notion of topological K-theory of dg-categories, in the complex analytic as well as in the l-adic setting. I will also present a construction of the Gauss-Manin connection associated to a family of smooth and proper dg-categories, and mention its (conjectural) interaction with the induced family of topological K-theory. At the end of this third lecture, several open questions and problems will be mentioned.


Serguei Barannikov (ENS, Paris):  Developments in the noncommutative Batalin-Vilkovisky formalism

Jonathan Block (U. Penn.):  TBA

Tobias Dyckerhoff (U. Penn.):  Isolated hypersurface singularities as noncommutative spaces

Eisenbud's category MF of matrix factrorizations is a 2-periodic differential graded category associated to the local germ X of an algebraic hypersurface. If the hypersurface is regular, then this category is trivial which, in general, allows us to think of MF as an invariant of the singularity of X. We will discuss various properties and invariants of MF in the case where the singularity is isolated.

David Favero (U. Penn.):  The dimension spectrum of a triangulated category; spherical and exceptional objects

The dimension spectrum is a set of natural numbers which is an invariant of any classically finitely generated triangulated category. For categories appearing in algebraic and symplectic geometry, it seems to retain information about monodromy, rationality, and degenerations. We will give an overview of the known results in this area and discuss some aspects of a joint work with L. Katzarkov on elements of the spectrum which are obtained from spherical and exceptional objects.

Kenji Fukaya (Univ. Kyoto):  TBA

Victor Ginzburg (Univ. of Chicago):  Gerstenhaber-Batalin-Vilkoviski structures on coisotropic intersections

Let Y, Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf OX to a sheaf of noncommutative algebras and of the sheaves OY and OZ to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf TorOX (OY, OZ ). The induced Gerstenhaber bracket on the Tor-sheaf turns out to be canonically defined; it is independent of the choices of deformations involved. There are similar results for Ext-sheaves as well.

Our construction is motivated by, and is closely related to, a result of Behrend-Fantechi, who considered the case of Lagrangian submanifolds in a symplectic manifold.

Kentaro Hori (IPMU):  Mirror symmetry and reality

Dmitry Kaledin (Steklov Institute):  Hochschild cohomology as a factorization algebra

Deligne conjecture claims that Hochschild cohomology of an associative algebra has the structure of a "2-algebra", by which people usually understand "an algebra over the chain operad of the operad of small discs". Unfortunately, the language of operads is not very well suited to the problem, so that all the proofs are cumbersome and involve irrelevant combinatorics. I will try to convince the audience that it is much more convenient to treat 2-algebras as certain collections of constructible sheaves living on the spaces of points on a disc. The resulting notion is in fact a special case of the chiral algebras of Beilinson and Drinfeld. I will also discuss possible generalizations.

Bernhard Keller (Univ. Paris 7):  The periodicity conjecture via Calabi-Yau categories

The periodicity conjecture was formulated in mathematical physics at the beginning of the 1990s, in the work of Zamolodchikov, Kuniba-Nakanishi and Ravanini-Valleriani-Tateo. It asserts that a certain discrete dynamical system associated with a pair of Dynkin diagrams is periodic and that its period divides the double of the sum of the Coxeter numbers of the two diagrams. The conjecture was proved by Frenkel-Szenes and Gliozzi-Tateo for the pairs (An, A1), by Fomin-Zelevinsky in the case where one of the diagrams is A1 and by Volkov and independently Szenes when both diagrams are of type A. We will sketch a proof of the general case which is based on Fomin-Zelevinsky's work on cluster algebras and on the theory relating cluster algebras to triangulated Calabi-Yau categories.

Alexander Kuznetsov (Steklov Institute):  Hochschild homology and cohomology of admissible subcategories

I will explain how one can compute the Hochschild homology and cohomology of admissible subcategories (i.e. components of semiorthogonal decompositions) of bounded derived categories of coherent sheaves on smooth projective varieties. Some interesting examples will be discussed.

Valery Lunts (Indiana):  Uniqueness of enhancement for triangulated categories I
Dmitri Orlov (Steklov Institute):  Uniqueness of enhancement for triangulated categories II

We will present general results on the uniqueness of a DG enhancement for triangulated categories. As a consequence we obtain such uniqueness for the unbounded categories of quasi-coherent sheaves, for the triangulated categories of perfect complexes, and for the bounded derived categories of coherent sheaves on quasi-projective schemes. If a scheme is projective then we also get a strong uniqueness for the triangulated category of perfect complexes and for the bounded derived categories of coherent sheaves. These results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product.

Grigory Mikhalkin (Univ. Genève):  TBA

Yan Soibelman (Kansas State U.):  Cohomological Hall algebra

This is a report on a joint work with Maxim Kontsevich. Starting with a quiver with potential (more generally, a 3d Calabi-Yau category) we construct an associative algebra in the category of exponential mixed Hodge structures,called Cohomological Hall algebra. Its Hilbert-Poincare series gives motivic Donaldson-Thomas invariants of the category. The series enjoys interesting integrality properties. A choice of stability condition gives rise to a PBW-type basis of the algebra. Cohomological Hall algebra is a mathematical incarnation of the algebra of BPS states in N=2 supersymmetric gauge theories. It also gives an appropriate theory of cycles for matrix integrals, algebraic description of line operators recently introduced by Gaiotto and is closely related to (holomorphic) Chern-Simons theory and the WKB theory for Schroedinger-type operators in the complex domain. If time permits,I plan to discuss some of those relationships.