Workshop on Homological Mirror Symmetry and Related Topics
January 19-24, 2009, University of Miami

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Maxim Kontsevich (IHES and U. Miami):  Geometric and algebraic aspects of wall-crossing

Hiraku Nakajima (Kyoto Univ.):  Instanton counting and wall-crossing in Donaldson invariants

Donaldson invariants of 4-manifolds were originally introduced via the moduli spaces of anti-self-dual connections, but can be approached also by moduli spaces of stable bundles when the underlying manifolds are complex projective surfaces. Based on this approach, we compute the wall-crossing terms of Donaldson invariants, describing what happens when the ample line bundle is changed. This is an old subject, but a recent advance, the `instanton counting', enables us to compute them in the torus equivariant setting. Then the terms are naturally expressed in terms of the so-called Seiberg-Witten curves. Finally, we express the blow-up formula of Donaldson invariants, as a wall-crossing formula via moduli spaces of perverse coherent sheaves on the blow-up.

Talks are based on series of joint works with Kota Yoshioka, Lothar Göttsche, and Takuro Mochizuki.

Richard Thomas (Imperial College):  Curve counting and derived categories

I will describe joint work with Rahul Pandharipande. The celebrated MNOP conjecture relates Gromov-Witten theory to invariants counting embedded curves and subschemes in 3-folds. I will review this theory, and describe another method to count curves that throws away the worst subschemes. This "stable pair" theory lives in the derived category of the 3-fold, and its relation to MNOP theory is via a wall crossing in a space of stability conditions. Via another wall crossing, it gives the conjectural BPS curve counting invariants of Gopakumar-Vafa. I will explain all of these concepts.

Denis Auroux (MIT):  Special Lagrangian fibrations and mirror symmetry

These lectures will focus on the construction of mirror manifolds using special Lagrangian fibrations, with the Strominger-Yau-Zaslow conjecture as a starting point. The first talk will be elementary, and will provide some motivation and basic examples, both in the Calabi-Yau case and in the broader setting of varieties with effective anticanonical divisor. We will in particular explain how Landau-Ginzburg models naturally arise in this setting, viewing the superpotential as a mirror counterpart to a Floer-theoretic obstruction.

The second talk will discuss some simple examples of the wall-crossing phenomena which arise in the non-toric case, as well as some evidence concerning mirror symmetry for pairs consisting of a variety and a Calabi-Yau hypersurface.

Finally, the last talk will discuss joint work in progress with Mohammed Abouzaid and Ludmil Katzarkov regarding the extension of mirror symmetry to arbitrary hypersurfaces in toric varieties, by considering Lagrangian fibrations on blow-ups. The main examples there will be pairs of pants (and their higher-dimensional analogues) and higher-genus curves.


Mohammed Abouzaid (Clay Institute and MIT):  Homological Mirror Symmetry for T^4

I shall explain how the pseudo-holomorphic quilt techniques of Wehrheim and Woodward yield of proof of Homological mirror symmetry for products, assuming mirror symmetry is known for both factors, and stringent conditions are imposed on the behaviour of holomorphic curves. The current conditions are so stringent that the only application is a proof of Homological mirror symmetry for T^4. An application to Lagrangian embeddings in T^4 will be deduced. This work was done jointly with Ivan Smith.

Sabin Cautis (Rice Univ.):  Equivalences from geometric sl2 actions

We explain how certain sl2 actions on derived categories of coherent sheaves can be used to construct new derived equivalences generalizing the spherical twists of Seidel-Thomas. The example we'll discuss in detail is an sl2 action on the cotangent bundle of Grassmannians. More generally we can construct an sl2 action on the derived category of coherent sheaves on Nakajima quiver varieties lifting his sl2 action on their cohomology. (joint with J. Kamnitzer and A. Licata)

Alexandr Efimov (Moscow):  Noncommutative Grassmanians

We will define the so-called noncommutative Grassmanians as examples of noncommutative moduli spaces of objects in derived categories. We will discuss the derived categories of quasi-coherent sheaves on them, the k-points, and the completions of local rings of the k-points. This is a part of the joint preprint with D. Orlov and V. Lunts arxiv:math/0702840.

Kenji Fukaya (Kyoto Univ.):  Mirror symmetry for toric manifolds

This is a report on work with Oh-Ohta-Ono. We are now building rather precise dictionary between theory of quantum cohomology on toric manifold and K. Saito's theory of hypersurface singularity (its generalization to a version over Novikov ring). The bridge is through open-closed Gromov-Witten theory. In the last part of this talk I will (probably) focus on its aspects which identify Poincare duality in GW theory with Residue pairing in Saito's theory. One new feature here is we need to use not only holomorphic disc but also holomorphic annuli to establish this relation.

Alexander Kuznetsov (Steklov Institute):  Fractional Calabi-Yau triangulated categories

A fractional Calabi-Yau is a triangulated category in which a power of the Serre functor is isomorphic to a shift. I will describe a construction which produces such categories and several interesting examples.

Rahul Pandharipande (Princeton):  The tropical vertex group

I will explain the relationship between ordered product formulas for commutators in the tropical vertex group and Gromov-Witten invariants of toric surfaces. Joint work with M. Gross and B. Siebert.

Tony Pantev (U. Penn.):  Geometric Langlands and non-abelian Hodge theory

It is expected that the geometric Langlands correspondence is a combination of homological mirror symmetry and microlocalization. I will explain how this expectation can be made precise via non-abelian Hodge theory and how it can be used to construct Hecke eigensheaves explicitly. I will illustrate the general strategy on the non-trivial explicit example of the projective line with tame ramification at five points. This is a joint work with R. Donagi and C. Simpson.

Yongbin Ruan (Michigan):  Integrable hierarchies and singularity theory

Almost twenty years ago, the celebrated theorem of Witten-Kontsevich asserts that the intersection theory of Deligne-Mumford space is governed by KdV-hierarchies. Around the same time, Witten proposed a sweeping generalization which leads to the representation theory of infinite dimensional Lie algebra and singularity theory. In the talk, we will sketch the recent resolution of Witten's vision by Fan-Jarvis and author and the appearance of new phenomenon such as mirror symmetry in the integrable hierarchy problem.

Paul Seidel (MIT):  Homological mirror symmetry for the genus two curve

Yan Soibelman (Kansas State Univ.):  DT-invariants, quivers and cluster transformations

In a joint work with Maxim Kontsevich we introduced motivic Donaldson-Thomas invariants for 3d Calabi-Yau categories. There is a special class of such categories which are generated by a finite collection of objects, which we call cluster collection. Such categories correspond to quivers with potentials. Motivic DT-invariants gives rise to cluster varieties (both classical and quantum).

Atsushi Takahashi (Osaka Univ.):  HMS for isolated hypersurface singularities

After reviewing some necessary basic definitions, I'll formulate HMS conjecture, give relevant results such as fractional CY property of the categories, the existence of full (strong) exceptional collections and then show HMS for some examples. A relation between Orlov's semi-orthogonal decomposition of categories of singularities and HMS for cusp singularities will also be discussed.

Ilia Zharkov (Kansas State):  Geometry of numbers and tropical curves