Homological Mirror Symmetry

Opening Ceremony for Fuqua

Maxim Kontsevich, Institut des Hautes Études Scientifiques  Video

Hiroshi Iritani, Kyoto University: Fourier analysis of equivariant quantum cohomology I

We discuss a D-module version of Teleman's conjecture relating the equivariant quantum cohomology of a Hamiltonian T-space X and the quantum cohomology of a symplectic reduction X//T. We explain a "global Kaehler moduli space" picture arising from this conjecture, and explain how a (formal) decomposition of quantum cohomology D-modules for projective bundles or blowups follows from the Fourier spectral analysis.  Video

Daniel Pomerleanu, University of Massachussetts  Video

Tony Yue Yu, CalTech  Video

Mohammed Abouzaid, Stanford University  Video

John Pardon, Stony Brook  Video

Kenji Fukaya, Stony Brook  Video

Maxim Kontsevich, Institut des Hautes Études Scientifiques  Video

Hiroshi Iritani, Kyoto University: Fourier analysis of equivariant quantum cohomology II

We discuss a D-module version of Teleman's conjecture relating the equivariant quantum cohomology of a Hamiltonian T-space X and the quantum cohomology of a symplectic reduction X//T. We explain a "global Kaehler moduli space" picture arising from this conjecture, and explain how a (formal) decomposition of quantum cohomology D-modules for projective bundles or blowups follows from the Fourier spectral analysis.  Video

Daniel Halpern-Leistner, Cornell University: Dispatches from the ends of the stability manifold I

The manifold of Bridgeland stability conditions parameterizes a homological structure on a triangulated category that is analogous to a Kaehler structure on a projective variety. Recently, I have proposed a "noncommutative minimal model program" in which the quantum differential equation of a projective variety determines paths toward infinity in the stability manifold of that variety, and that these paths can be used to define canonical (semiorthogonal)decompositions of its derived category. It is natural to ask if these paths actually converge in some partial compactification of the stability manifold.

I will discuss a partial compactification of the stability manifold with a nice modular interpretation, the space of augmented stability conditions. To do this, I will introduce a structure on a triangulated category that we call a multi-scale decomposition, which generalizes a semiorthogonal decomposition, and a new moduli space of multi-scale lines that is closely related to the moduli spaces of multi-scale differentials which are of recent interest in dynamics. The main conjecture about the space of augmented stability conditions is that it is a manifold with corners (in a specific way that I will explain).

One consequence: If this conjecture holds for any smooth and proper dg-category, then any stability condition on a smooth and proper dg-category admits proper moduli spaces of semistable objects.  Video

Constantin Teleman, UC Berkeley  Video

Daniel Pomerleanu, University of Massachusetts  Video

Denis Auroux, Harvard University: Holomorphic discs of negative Maslov index and extended deformations in mirror symmetry

Given a Lagrangian torus fibration on the complement of an anticanonical divisor in a Kahler manifold, one usually constructs a mirror space by gluing local charts (moduli spaces of local systems on generic torus fibers) via wall-crossing transformations determined by counts of Maslov index 0 holomorphic discs; this mirror also comes equipped with a regular function (the superpotential) which enumerates Maslov index 2 holomorphic discs. Holomorphic discs of negative Maslov index deform this picture by introducing inconsistencies in the wall-crossing transformations; the geometric features of the resulting mirror can be understood in the language of extended deformations of Landau-Ginzburg models. We illustrate this phenomenon on an explicit example (a 4-fold obtained by blowing up a Calabi-Yau toric variety), and discuss a family Floer approach to the geometry of the corrected mirror in this setting.  Video

Maxim Kontsevich, Institut des Hautes Études Scientifiques  Video

Hiroshi Iritani, Kyoto University: Fourier analysis of equivariant quantum cohomology III

We discuss a D-module version of Teleman's conjecture relating the equivariant quantum cohomology of a Hamiltonian T-space X and the quantum cohomology of a symplectic reduction X//T. We explain a "global Kaehler moduli space" picture arising from this conjecture, and explain how a (formal) decomposition of quantum cohomology D-modules for projective bundles or blowups follows from the Fourier spectral analysis.  Video

Constantin Teleman, UC Berkeley  Video

Denis Auroux, Harvard University  Video

Hülya Argüz, University of Georgia  Video

Daniel Halpern-Leistner, Cornell University: Dispatches from the ends of the stability manifold II

The manifold of Bridgeland stability conditions parameterizes a homological structure on a triangulated category that is analogous to a Kaehler structure on a projective variety. Recently, I have proposed a "noncommutative minimal model program" in which the quantum differential equation of a projective variety determines paths toward infinity in the stability manifold of that variety, and that these paths can be used to define canonical (semiorthogonal)decompositions of its derived category. It is natural to ask if these paths actually converge in some partial compactification of the stability manifold.

I will discuss a partial compactification of the stability manifold with a nice modular interpretation, the space of augmented stability conditions. To do this, I will introduce a structure on a triangulated category that we call a multi-scale decomposition, which generalizes a semiorthogonal decomposition, and a new moduli space of multi-scale lines that is closely related to the moduli spaces of multi-scale differentials which are of recent interest in dynamics. The main conjecture about the space of augmented stability conditions is that it is a manifold with corners (in a specific way that I will explain).

One consequence: If this conjecture holds for any smooth and proper dg-category, then any stability condition on a smooth and proper dg-category admits proper moduli spaces of semistable objects.  Video