THROUGH THE LOOKING-GLASS:
Mirror Symmetry and Quantum Cohomology
A conference celebrating the mathematical influences of Alexander Givental
on his 60th birthday
Titles and Abstracts
Jim Bryan: Banana Manifolds, Donaldson-Thomas theory, and modular forms
Abstract: Banana Manifolds are a class of compact Calabi-Yau threefolds which are fibered by Abelian surfaces and have singular fibers with banana configurations: three genus zero curves meeting each of the other in two points. We construct the basic banana manifold as well as some exotic ones which are rigid and have very small Hodge numbers. We compute in closed form the Donaldson-Thomas/Gromov-Witten partition function of Banana manifolds for fiberwise classes. We show that the genus g Gromov-Witten potential is an explicit genus 2 Siegel modular form of weight 2g-2.
Ron Donagi: Non-Abelian Hodge Theory, Mirror Symmetry, and Geometric Langlands
Abstract: We will review the Geometric Langlands Conjecture, a non-abelian
generalization of the theory of curves and their Jacobians. We will
compare it to its arithmetic variants and discuss its overlap with
homological mirror symmetry. We will then outline our program for
proving GLC using non Abelian Hodge theory and Hitchin's system.
Finally, we will describe some recent results on the construction of
automorphic sheaves in several specific cases.
Abstract: It is known that there is no reasonable smooth classification of singularities of smooth mappings up to a generic $C^\infty$-perturbation. On the other hand, there are h-principle type results which allows to get rid of most of singularities via a $C^0$-perturbation provided that some homotopical conditions are met. It turns out that even without any pre-conditions by a $C^0$-perturbation one can reduce singularities to an observable list. This result is an essential step in the proof of Nadler’s arborealization conjecture, which is our joint project with David Alvarez-Gavela, David Nadler and Laura Starkston.
Pavel Etingof: Cyclotomic Double affine Hecke algebras and multiplicative quiver varieties
Abstract: I'll show that the partially spherical cyclotomic rational
Cherednik algebra (obtained from the full rational Cherednik algebra by
averaging out the cyclotomic part of the underlying reflection group)
has four other descriptions: (1) as a subalgebra of the degenerate DAHA
of type A given by generators; (2) as an algebra given by generators and
relations; (3) as an algebra of differential-reflection operators
preserving some spaces of functions; (4) as equivariant Borel-Moore
homology of a certain variety. Also, I'll define a q-deformation of this
algebra, called cyclotomic DAHA. Namely, I'll give a q-deformation of
each of the above four descriptions of the partially spherical rational
Cherednik algebra, replacing differential operators with difference
operators, degenerate DAHA with DAHA, and homology with K-theory. In
addition, I'll explain that spherical cyclotomic DAHA are quantizations
of certain multiplicative quiver and bow varieties, which may be
interpreted as K-theoretic Coulomb branches of a framed quiver gauge
theory. Finally, I'll apply cyclotomic DAHA to prove new flatness
results for various kinds of spaces of q-deformed quasiinvariants. This
is joint work with A. Braverman and M. Finkelberg.
Ezra Getzler: The Batalin-Vilkovisky formalism for field theory and general covariance
Abstract: AKSZ field theories, developed at the University of California, are a class of field theories generalizing the Chern-Simons model. They exhibit general covariance under diffeomorphisms of spacetime. In this talk, I will recast general covariance as a Maurer-Cartan equation for a certain curved differential graded Lie algebra.
In joint work with Sean Pohorence, I have shown that the superparticle, a toy model of the (Green-Schwarz) superstring, while not an AKSZ model, does exhibit general covariance in the sense that the curved Maurer-Cartan equation mentioned above may be solved. The proof requires the use of the Thom-Whitney normalization of curved differential graded Lie algebras, introduced by Sullivan in his work on rational homotopy theory.
Alexander Givental: The adelic Hirzebruch-RR in higher genus quantum K-theory
Abstract: I will explain how the problem of expressing K-theoretic Gromov-Witten invariants in terms of cohomological ones leads to an elegant quantum-mechanical formula.
Dusa McDuff: How to count curves in symplectic geometry
Abstract : Coulomb branch of a 3d gauge theory is defined (after
Braverman-Finkelberg-N) as the spectrum of a certain commutative ring,
defined as a convolution algebra of a certain infinite dimensional
variety. A variant of its definition gives a (partial) resolution, when
we have the so-called flavor symmetry in the theory. We identify the
resolution with smooth Cherkis bow variety for a quiver gauge theory of
an affine type A. (Joint work with Braverman and Finkelberg).
Abstract: We present a conjecture on the partition function counting the N-colored solid partitions (i.e. four dimensional Young diagrams, attached to N complex numbers, the so-called Coulomb parameters), with the complex weights depending on N additional complex parameters, called masses, and an element q of the maximal torus of the SU(4) group. Based on a joint work with Nicolo' Piazzalunga.
Andrei Okounkov: Enumerative geometry and geometric representation theory
Abstract: This will be an introductory discussion of the connections between the two fields in the title.
