Princeton University

Construction of the Kuranishi Structure on the Moduli Space of Pseudo-Holomorphic Curves

Kenji Fukaya
Simons Center for Geometry and Physics
April 12, 2013

To apply the technique of virtual fundamental cycle (chain) in the study of pseudo-holomorphic curve, we need to construct certain structure, which we call Kuranishi strucuture, on its moduli space. In this talk I want to review certain points of its construction.

A Converse to a Theorem of Gross-Zaqier-Kolyvagin

Christopher Skinner
Princeton University; Member, School of Mathematics
April 4, 2013

The theorem of the title is that if the L-function L(E,s) of an elliptic curve E over the rationals vanishes to order r=0 or 1 at s=1 then the rank of the group of rational rational points of E equals r and the Tate-Shafarevich group of E is finite. This talk will describe an approach to the converse. The methods are mostly p-adic.

Dimers and Integrability

Richard Kenyon
Brown University
March 29, 2013

This is joint work with A. B. Goncharov. To any convex integer polygon we associate a Poisson variety, which is essentially the moduli space of connections on line bundles on (certain) bipartite graphs on a torus. There is an underlying integrable Hamiltonian system whose Hamiltonians are weighted sums of dimer covers.

Resonance for Loop Homology on Spheres

Nancy Hingston
The College of New Jersey; Member, School of Mathemtics
March 15, 2013

Fix a metric (Riemannian or Finsler) on a compact manifold M. The critical points of the length function on the free loop space LM of M are the closed geodesics on M. Filtration by the length function gives a link between the geometry of closed geodesics, and the algebraic structure given by the Chas-Sullivan product on the homology of LM and the “dual” loop cohomology product.

An Analogue of the Ichino-Ikeda Conjecture for Whittaker Coefficients of the Metaplectic Group

Erez Lapid
Hebrew University of Jerusalem and Weizmann Institute of Science
March 14, 2013

A few years ago Ichino-Ikeda formulated a quantitative version of the Gross-Prasad conjecture, modeled after the classical work of Waldspurger. This is a powerful local-to-global principle which is very suitable for analytic and arithmetic applications. One can formulate a Whittaker analogue of the Ichino-Ikeda conjecture. We use the descent method of Ginzburg-Rallis-Soudry to reduce the Whittaker version to a purely local identity which we prove in the p-adic case under some mild hypotheses. Joint work with Zhengyu Mao

"Intermediate Symplectic Capacities"

Alvaro Pelayo
Washington University; Member, School of Mathematics
March 1, 2013

In 1985 Misha Gromov proved his Nonsqueezing Theorem, and hence constructed the first symplectic 1-capacity. In 1989 Helmut Hofer asked whether symplectic d-capacities exist if 1 < d < n. I will discuss the answer to this question and its relevance in symplectic geometry. This is joint work with San Vu Ngoc.

Contact Non-Squeezing and Rabinowitz Floer Homologhy

Peter Albers
Universitat Munster
February 15, 2013

We will present joint work with Will Merry. Using spectral invariants in Rabinowitz Floer homology we present an abstract contact non-squeezing theorem for periodic contact manifolds. We then exemplify this in concrete examples. Finally we explain connections to the existence of a biinvariant metric on contactomorphism groups. All this is connected and generalizes work by Eliashberg-Polterovich and Sandon.