Seminars Spring 2008

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Fall 2009

Thursday, April 3, 2008

Yan Soibelman, Kansas State University

Donaldson-Thomas invariants, motivic integration and cluster transformations

I am going to discuss the joint work with Maxim Kontsevich which is devoted to the general approach to the count of stable objects in 3d Calabi-Yau categories (BPS states in the language of physics, DT-invariants in the language of mathematics). The approach is based on the ideas of motivic integration. In particular it leads to new wall-crossing formulas. If time permits, I am going to discuss the relationship of our approach to quantum dilogarithm and cluster transformations.

Friday, April 4, 2008

Dmytro Shklyarov, Kansas State University

Hirzebruch-Riemann-Roch Theorem for DG algebras

Due to results of A. Bondal, M. Van den Bergh, B. Keller, any reasonable scheme is affine in the dg (differential graded) sense. This means that the derived category of perfect complexes on the scheme is equivalent to the derived category of perfect modules over a dg algebra. The most popular example of such an equivalence is the celebrated result of A. Beilinson that describes the derived categories of the projective spaces in terms of certain quiver algebras with relations. These results have prompted the idea that Noncommutative Algebraic Geometry should incorporate the study of dg algebras, dg categories and various homological invariants thereof.
The talk is devoted to a Hirzebruch-Riemann-Roch type formula for proper dg algebras (such algebras are though of as proper schemes in the dg world). Namely, for any proper dg algebra A, I'll describe an explicit pairing on the Hochschild homology of A and for an arbitrary perfect A-module I'll present an explicit formula for its Chern class with the value in the Hochschild homology of A. The Hirzebruch-Riemann-Roch formula in this setting expresses the Euler characteristic of the Hom-complex between any two perfect A-modules in terms of the pairing of their Chern classes.

Wednesday, April 09, 2008

Stavros Garoufalidis, Georgia Institute of Technology

Arithmetic Resurgence in Quantum Topology

Quantum Topology assigns numerical invariants (such as the Jones polynomial) to knotted 3-dimensional objects. The asymptotic expansions of these invariants are conjecturally related to riemannian (and mostly hyperbolic) geometry in 3-dimensions. In our talk, we will repackage the quantum invariants to two power series, and formulate a precise Arithmetic Resurgence Conjecture regarding the position of the singularities and the structure of the local and global monodromy. Finally, we will give some proofs of our conjecture in some test cases, that include a series of Kontsevich-Zagier, and the Kashaev series of the simplest hyperbolic $4_1$ knot. The second part is joint work with O. Costin.

Thursday, April 10, 2008

Takuya Okuda, UCSB

Bubbling Calabi-Yau geometry

A saddle point of an integral beck-reacts when the integrand is multiplied by a large function. In gauge/gravity duality, the space-time (in gravity) representing a saddle point gets deformed when an operator is inserted in gauge theory path integral. The new space-time develops many new cycles carrying flux quantum numbers, and is called the bubbling geometry.
First, basic examples from AdS/CFT will be intuitively explained, where the anti-de Sitter space back-reacts. Second, the idea of bubbling will be applied to the gauge/gravity (large N) duality of the topological string. In Chern-Simons gauge theory, I will consider the Wilson loop along a knot in three-sphere. I will argue that the conifold back-reacts to the Wilson loop and produces a new bubbling geometry. So bubbling conjecturally associates a six-manifold to a knot. I will present explicit bubbling Calabi-Yau geometries that are dual to the unknot in three-sphere and lens spaces. Third, if there is time, I will show that the BPS invariants of any Lagrangian brane determine the BPS invariants of the six-fold related by geometric transition.

Monday, April 14, 2008

Paul Bressler, Indiana University

Atiyah Class and Hochschild Cohomology

The Atiyah class of the tangent bundle $T$ endows the (shifted) tangent bundle $T[-1]$ with a canonical structure of a Lie algebra object in the derived category of coherent sheaves. This structure was lifted by Kapranov to an $L$-infinity structure on $T[-1]$. I will sketch an alternative construction of an $L$-infinity structure and its relationship to the DGA of Hochschild cochains on the structure sheaf.

Thursday, April 17, 2008

Daniel Sternheimer, Keio University, Japan

The deformation philosophy, quantization and noncommutative
space-time structures

The role of deformations in physics and mathematics lead to the deformation philosophy promoted in mathematical physics by Flato since the 70's, exemplified by deformation quantization and its manifold avatars, including quantum groups and the "dual" aspect of quantum spaces. Deforming Minkowski space-time and its symmetry to anti de Sitter has significant physical consequences that we sketch (e.g. singleton physics). We end by presenting an ongoing program in which anti de Sitter would be quantized in some regions, speculating that this might explain baryogenesis in a universe in accelerated expansion.

Thursday, April 24, 2008

Benjamin Young, UBC

Counting colored 3D Young diagrams with vertex operators

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 also turn out to be orbifold Donaldson-Thomas partition functions for $\mathbb{C}^3/G$, where $G$ is a finite abelian subgroup of $\text{SO}(3)$. This talk should also serve as an introduction to the vertex operator calculus of Okounkov/Reshetikhin.

Thursday, May 1, 2008

Jake P. Solomon, Princeton

Differential equations for the open Gromov-Witten potential

I will describe a system of differential equations for the genus 0 open Gromov-Witten potential of a Lagrangian submanifold of a symplectic 4-manifold fixed by an anti-symplectic involution. These equations involve both the open Gromov-Witten potential and the closed Gromov-Witten potential. They are sufficiently restrictive that in significant examples they completely determine both the open and closed Gromov-Witten potentials up to a finite number of constants. The proof relies on an open-closed generalization of the topological conformal field theory behind the WDVV equation. If time permits, I will discuss target spaces of dimension 0 and 6 as well.

Thursday, May 15, 2008

Elizabeth Gasparim, University of Edinburgh

Nekrasov conjecture for toric varieties

Nekrasov conjecture predicts a relation between the N=2 SUSY gauge theory partition function and the SW prepotential. This conjecture was proven for instantons on $\mathbb{R}^4$ by Nekrasov-Okounkov, Nakajima-Yoshioka, and Braverman-Etingof. I will report on joint work with Melissa Liu, where we prove the conjecture for instantons on toric surfaces.

Thursday, May 22, 2008

Alina Marian, Institute for Advanced Study

Theta dualities on moduli spaces of sheaves on K3 surfaces

I will consider pairs of topologically complementary moduli spaces of sheaves on a K3 surface. I will discuss conjectures and results concerning natural maps between spaces of sections of determinant line bundles on them. These maps are induced by Brill-Noether-type divisors. I will also consider briefly the case of sheaves on abelian surfaces. The talk will partly report on work in progress with Dragos Oprea.

Thursday, May 29, 2008

Dagan Karp, Berkeley

Quantum geometry of local $\mathbb{C}P^1$ and its cyclic quotients

In this talk I hope to introduce a conjectural triangle of equivalences, relating (1) the (orbifold) Gromov-Witten theory of (cyclic quotients of) local $P^1$, (2) the Gromov-Witten theory of their resolutions and (3) the Chern-Simons theory of their large-N duals.

The equivalence of (2) and (3) is a conjecture of Aganagic-Klemm-Marino-Vafa and I will report on work in progress with S. Koshkin. The equivalence of (1) and (2) has been announced (in genus zero at least) by various subsets of Coates-Corti-Iritani-Tseng. The relationship between (1) and (3) would follow from the above, and represents a new CS/OrbGW duality.

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