Lectures Abstracts
- Mohammed Abouzaid (M.I.T.): "Homological Mirror Symmetry for T^4 and applications to Lagrangian embeddings".
- Kwok Wai Chan (Harvard): "SYZ transformations and mirror symmetry". Abstract: In joint work with Conan Leung, we propose a program which is aimed at understanding mirror symmetry by using Fourier-type transformations (SYZ transformations). In this talk, we will discuss the applications of SYZ transformations to mirror symmetry for toric Fano manifolds. In particular, we will see how quantum cohomology is transformed to Jacobian ring and how Lagrangian torus fibers are transformed to matrix factorizations.
- Charles Doran (U of Alberta): "Algebraic Cycles, Regulator Periods, and Local Mirror Symmetry".
- Kenji Fukaya (Kyoto University): "Singularity theory over Novikov ring and Mirror symmetry" (joint with Oh-Ohta-Ono). Abstract: Generating function of open-closed Gromov-Witten invariant that is potential function with bulk in our sense, is a formal function which is some kinds of formal power series converging in appropriate adic topology, over universal Novikov ring. In the case of toric manifold and its Lagrangian fiber this function provides a nice example of universal family of hypersurface singularities, and becomes a `rigid analytic analogue' of K. Saito's theory of isolated hypersurface singularity. This is actually a global theory and so is different from classical Saito's theory. Saito's theory is an imortant source of so called Frobenius manifold structure (= Saito's flat structure). Another important source of Frobenius manifold structure is (big) quatum cohomology. We find that they coincides in the case of arbitrary toric manifold.
- Ilia Itenberg (U of Strasbourg): "Welschinger invariants of toric Del Pezzo surfaces" (joint work with V. Kharlamov and E. Shustin) Abstract: The Welschinger invariants are designed to bound from below the number of real rational curves passing through a given generic real collection of points on a real rational surface. In some cases these invariants can be calculated using G. Mikhalkin's approach which deals with a corresponding count of tropical curves. Using the tropical approach we establish a logarithmic equivalence of Welschinger and Gromov-Witten invariants in the case of generic collections of real points on a toric Del Pezzo surface equipped with an arbitrary real structure (with non-empty real part).
- Ludmi Katzarkov (U of Miami): "Conic Bundles Old and New". Abstract: We will formulate a HMS approach to a classical question of rationality of conic bundles.
- David Morrison (University of California, SB): "SYZ and the moduli of Calabi--Yau threefolds."
- Anvar Mavlyutov (Oklahoma State U): "Deformation of toric varieties and Calabi-Yau hypersurfaces". Abstract: In the 90's, Klaus Altmann studied deformations of affine toric varieties. He constructed families of deformations of affine toric varieties as complete intersections in another toric variety using Minkowski sums of polyhedra. We found a generalization of this construction for arbitrary toric varities. In a particular important case of complete simplicial toric varieties which are partial crepant resolutions of the projective toric varieties corresponding to reflexive polytopes, this new construction coincides with our previous construction of deformations of such toric varieties obtained by a different method via homogeneous coordinates. These deformations are important as they induce deformations of Calabi-Yau hypersurfaces.
- Yong-Geun Oh (U of Wisconsin-Madison): "Seidel's exact sequence for closed Calabi-Yau manifolds". In this talk, we will explain how construction of Seidel's long exact sequence of Floer cohomology under the symplectic Dehn twists can be extended to general, especially closed, Calabi-Yau manifolds. The highlight of the talk is our usage of the notion of `anchored Lagrangian submanifolds' and some study of compactness issue of the moduli space of pseudo-holomorphic sections in the setting of symplectc Lefschetz fibrations.
- Tony Pantev (U of Pennsylvania): "Mirror symmetry for del Pezzo surfaces". Abstract: I will discuss the general mirror symmetry question for del Pezzo surfaces in a setup that goes beyond the Hori-Vafa ansatz. I will describe the mirror map explicitly and will describe non-trivial tests for homological mirror symmetry. This is a joint work with Auroux, Katzarkov and Orlov.
- Bernd Siebert (U of Hamburg): "The tropical vertex". Abstract: One insight of mirror symmetry is the fact that the enumerative geometry of rational curves is related to the deformation theory of a ``mirror variety''. Now there is a pro-nilpotent group of automorphisms of the algebraic 2-torus ruling the constructions of maximal degenerations (Kontsevich/Soibelman, Gross/S.). On the mirror side this group should have some enumerative geometry meaning. In the talk I will present joint work with M. Gross (UCSD) and R. Pandharipande (Princeton) showing that this group indeed organizes a class of natural enumerative geometry problems on toric surfaces into an algebraic structure. The correspondence runs via tropical geometry.
- Yan Soibelman (Kansas State U): "Complex integrable systems,stability structures and invariants of Donaldson-Thomas type". Abstract: In a recent joint work with Maxim Kontsevich we offered a general approach to Donaldson-Thomas type invariants ("counting of BPS states" in the language of physicists). I am going to discuss an application of our approach to complex integrable systems of Seiberg-Witten type. In particular, I will explain how the wall-crossing formulas in Seiberg-Witten theory are related to the wall-crossing formulas of our DT-invariants, and how tropical geometry appears in the description of the spectrum of SW-model.
- Benjamin Young (McGill University): "Counting colored 3D Young diagrams with vertex operators". Abstract: I will show how to compute some multivariate generating functions for 3D Young diagrams (otherwise known as "plane partitions"). Each box in a 3D Young diagram gets assigned a "color" according to a certain pattern; the variables keep track of how many boxes of each color there are. My generating functions turn out to be orbifold Donaldson-Thomas partition functions for C^3/ G, where G is a finite abelian subgroup of SO(3). If time permits, I will discuss recent work on the more general problem of the orbifold topological vertex and the combinatorial crepant resolution conjecture.