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Quantum Cohomology and Mirror Symmetry / Cohomologie quantique et symétrie miroir
Org: Kai Behrend (UBC)

IONUT CIOCAN-FONTANINE, University of Minnesota, School of Mathematics, Minneapolis, Minnesota
Gromov-Witten invariants of GIT quotients

I will discuss some conjectures about the relationship between the genus zero Gromov-Witten invariants of a non-abelian GIT quotient X//G, and those of the associated abelian quotient X//T, for T a maximal torus in G. The conjectures were inspired by explicit computations of Gromov-Witten invariants on Grassmannians, and may be viewed as "quantum" generalizations of results (due to Ellingsrud-Strømme, and Martin) relating the cohomology rings of X//G and X//T. I will then sketch proofs for some of the conjectures in the cases when localization methods apply, in particular for all type A flag manifolds. This is joint work with Aaron Bertram and Bumsig Kim.

Towards the McKay correspondence in higher dimensions

For a finite subgroup G of SL(n,C), if the quotient singularity Cn/G admits a crepant resolution Y then the McKay correspondence principle asserts that the geometry of Y should be equivalent to the G-equivariant geometry of Cn. This is always true for the euler number of Y, a weak geometric invariant.

With additional assumptions (which hold in dimension two or three), Bridgeland, King and Reid established the McKay correspondence as an equivalence of derived categories for a special choice of a crepant resolution. This talk will discuss a programme to generalise this result to higher dimensions.

CHUCK DORAN, Columbia University, New York, New York, USA
Integral structures, toric geometry, and homological mirror symmetry

We establish the isomorphisms over Z of cohomology/K-theory, global monodromy, and invariant symplectic forms predicted by Kontsevich's Homological Mirror Symmetry Conjecture for certain one dimensional families of Calabi-Yau threefolds with h2,1 = 1. These families arise as hypersurfaces or complete intersections in Gorenstein toric Fano varieties, and their mirrors are described by the Batyrev-Borisov construction. Our method involves (1)  classifying all rank four integral variations of Hodge structure over P1\{0,1,¥} with maximal unipotent local monodromy about 0 and local monodromy about 1 unipotent of rank 1, and (2)  checking, using properties of nef partitions of reflexive polytopes, that the Z-VHS of our Calabi-Yau families match those picked out by the K-theory of their mirrors via the HMS Conjecture. This is joint work with John Morgan.

BARBARA FANTECHI, SISSA, Via Beirut 4, 34014  Trieste, Italy
Quantum cohomology for orbifiolds

In this mostly expository talk we will recall the definition of quantum cohomology (and Gromov Witten invariants) for orbifolds, i.e. smooth proper complex Deligne Mumford stacks with projective coarse moduli space. We will stress the main differences with the quantum cohomology for smooth varieties: the need to allow for stacky points in the nodal curves which are the domain of stable maps, and as a consequence the fact that the evaluation maps have the inertia stack as target (the definition of inertia stack will also be recalled). Special attention will be devoted to the degree zero case, defining what is called orbifold or Chen Ruan cohomology. We will then present an overview of the (not very many) computational results available so far: indeed, perhaps the most striking difference between quantum cohomology for varities and for orbifolds is that for varieties many computations were carried out even before a general definition was complete, while for orbifolds the definition (both in the symplectic and in the algebraic language) is already a few years old but explicit results are still few and far apart. The talk should be understandable to mathematicians working in algebraic or symplectic geometry: previous familiarity with either quantum cohomology or orbifolds/stacks (or both) will be useful but not necessary.


SÁNDOR KOVÁCS, University of Washington, Department of Mathematics, Seattle, Washington  98195, USA
Zero sets of holomorphic one forms on varieties of general type

This is joint work with Christopher Hacon (Utah).

It has been conjectured that a global holomorphic one form on a variety of general type always has a non-empty sero set. One of several applications of this statement is that a variety with this condition does not admit a smooth morphism to an abelian variety.

This conjecture has been confirmed by Zhang for canonically polarized varieties and by Luo and Zhang for threefolds. We confirm the conjecture for several other cases, including all smooth minimal models of general type of arbitrary dimension.

RAVI VAKIL, Stanford University, Stanford, California, USA
Schubert induction

I'll describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the geometric Littlewood-Richardson rule.

Schubert problems are among the most classical problems in enumerative geometry of continuing interest. As an application of Schubert induction, I will address several natural questions related to Schubert problems, including: the "reality" of solutions; effective numerical methods; solutions over algebraically closed fields of positive characteristic; solutions over finite fields; a generic smoothness (Kleiman-Bertini) theorem; and monodromy groups of Schubert problems.

I may spend some time at the end discussing the geometry behind the geometric Littlewood-Richardson rule.


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