Institute for Geometry and Physics Miami-Cinvestav-Campinas
Opening Conference — Miami, January 2012 — Abstracts
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Talk titles and abstracts

Mohammed Abouzaid (Clay Math Institute and MIT):  On the Fukaya categories of plumbings

Valery Alexeev (Univ. Georgia):  Extending Torelli map to different compactifications of the Siegel space

Garrett Alston (Kansas State Univ.):  Real Lagrangians in the quintic

Denis Auroux (UC Berkeley):  Lagrangian fibrations on conic bundles and mirror symmetry for hypersurfaces
Denis Auroux (UC Berkeley):  Homological mirror symmetry for punctured spheres

Matthew Ballard (Wisconsin):  Phases of the B-model: Variation of GIT for toric LG-models

Via a well-known construction of Cox, semi-projective toric varieties can be described as GIT quotients of the spectrum of the Cox ring. Choosing a degree zero element of the Cox ring, gives a function on all GIT quotients, hence each quotient can by thought of as a toric Landau-Ginzburg model. In the Calabi-Yau case, physicists Herbst, Hori, and Page, have related these LG-models as different phases of the same gauged linear sigma model, predicting equivalent categories of matrix factorizations. These predictions were proven mathematically in work of Herbst and Walcher. The general (non Calabi-Yau) case is part of joint work with D. Favero and L. Katzarkov. Here, one obtains semi-orthogonal decompositions relating the varying GIT quotients. Combined with categorical renormalization group flow, as in the thesis of Isik, this recovers a well known result of Orlov.

Kevin Costello (Northwestern):  The quantum BCOV theory and higher genus mirror symmetry

Colin Diemer (Miami):  Secondary Stacks and Toric Hypersurface Degenerations

Alexander Efimov (Moscow):  Quantum cluster variables via Donaldson-Thomas theory, I
Alexander Efimov (Moscow):  Quantum cluster variables via Donaldson-Thomas theory, II
Alexander Efimov (Moscow):  Cyclic homology of matrix factorizations

Yakov Eliashberg (Stanford):  Rigid and flexible Weinstein manifolds

David Favero (Vienna):  Phases of the B-model: Applications, including Homological Projective Duality

Kenji Fukaya (Kyoto):  Smoothing moduli spaces and Lagrangian Floer theory of arbitrary genus.
Kenji Fukaya (Kyoto):  Applications of Lagrangian Floer theory of arbitrary genus.

Alexander Goncharov (Yale):  Ideal webs and moduli spaces of local systems on surfaces

Mark Gross (UC San Diego):  Towards Mirror Symmetry for Varieties of General Type I

Assuming the natural compactification X of a hypersurface in (C*)n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection X in a toric variety which works for any Kodaira dimension of X. The mirror dual might be a reducible and is equipped with a sheaf of vanishing cycles. We give evidence for the duality by proving the symmetry of the Hodge numbers when X is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, we will explain relations to homological mirror symmetry and the Gross-Siebert construction.

Dmitry Kaledin (Steklov):  Cyclic K-theory

Gabriel Kerr (Miami):  Symplectomorphism Group Relations and Landau-Ginzburg Degenerations

Maxim Kontsevich (IHES):  Stability I
Maxim Kontsevich (IHES):  Stability II
Maxim Kontsevich (IHES):  Geometry of Wall-Crossing Formulas
Maxim Kontsevich (IHES):  Hodge conjecture for abelian varieties - pro and contra

Grigory Mikhalkin (Univ. Genève):  Quantized enumeration of planar curves

Tony Pantev (U.Penn.):  Shifted symplectic structures and quantization
Tony Pantev (U.Penn.):  Constructions of generalized monopoles

Alexander Polishchuk (Univ. Oregon):  Phases of Lagrangian-invariant objects in the derived category of an abelian variety

Bjorn Poonen (MIT):  Néron-Severi groups under specialization

Helge Ruddat (Univ. Mainz):  Towards Mirror Symmetry for Varieties of General Type II

Assuming the natural compactification X of a hypersurface in (C*)n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection X in a toric variety which works for any Kodaira dimension of X. The mirror dual might be a reducible and is equipped with a sheaf of vanishing cycles. We give evidence for the duality by proving the symmetry of the Hodge numbers when X is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, we will explain relations to homological mirror symmetry and the Gross-Siebert construction.

Bernd Siebert (Univ. Hamburg):  Generalized theta functions

The degeneration approach to mirror symmetry provides a canonical basis of sections of the polarizing line bundle. In the case of abelian varieties they give the classical theta functions. In the talk I will explain some aspects of these generalized theta functions in mirror symmetry (joint work with Mark Gross, Paul Hacking and Sean Keel).

Carlos Simpson (Univ. Nice):  Topological structure at infinity of moduli spaces of flat connections

We discuss a few basically easy things one can say about the topological structure of the various moduli spaces of flat connections on a curve, essentially in the case of rank two connections on P1 - 4 points.

Yan Soibelman (Kansas State U.):  Motivic knot invariants

I am going to discuss some ideas about the relationship of motivic Donaldson-Thomas invariants and wall-crossing formulas with knot invariants.

Ravi Vakil (Stanford):  Stabilization of discriminants in the Grothendieck ring

We consider the "limiting behavior" of discriminants, by which we mean informally the closure of the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected — we use the first to understand the second. We describe their classes in the "ring of motives", as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization can be described in terms of the motivic zeta values. The results extend parallel results in both arithmetic and topology. I will also present a conjecture (on "motivic stabilization of symmetric powers") suggested by our work. Although it is true in important cases, Daniel Litt has shown that it contradicts other hoped-for statements. This is joint work with Melanie Wood.

Ilia Zharkov (Kansas State U.):  Tropical Homology