School of Mathematics

A converse to a theorem of Gross--Zagier, Kolyvagin and Rubin, II

Ashay Burungale
Universite Paris 13; Member, School of Mathematics
May 1, 2018

Let $E$ be a CM elliptic curve over a totally real number field $F$ and $p$ an odd ordinary prime. If the ${p^{\infty}\mbox{-}\mathrm{Selmer}}$ group of $E$ over $F$ has ${\mathbb{Z}_{p}\mbox{-}\mathrm{corank}}$ one, we show that the analytic rank of $E$ over $F$ is also one (joint with Chris Skinner and Ye Tian). We plan to discuss the setup and strategy.

Birational Calabi-Yau manifolds have the same small quantum products.

Mark McLean
Stony Brook University
April 30, 2018

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.

Algorithms for the topology of arithmetic groups and Hecke actions II

Michael Lipnowski
Member, School of Mathematics
April 24, 2018

At the November workshop, I described a new algorithm to cover compact, congruence locally symmetric spaces by balls. I’ll discuss how to compute the nerve of such a covering and Hecke actions on its cohomology. Joint work with Aurel Page.At the November workshop, I described a new algorithm to cover compact, congruence locally symmetric spaces by balls. I’ll discuss how to compute the nerve of such a covering and Hecke actions on its cohomology.
 
Joint work with Aurel Page.

Mirror spaces from formal deformation of Lagrangians and their gluing.

Hansol Hong
Harvard University
April 23, 2018

For given a Lagrangian in a symplectic manifold, one can consider deformation of A-infinity algebra structures on its Floer complex by degree 1 elements satisfying the Maurer-Cartan equation. The space of such degree 1 elements can be thought of as giving a local chart of the mirror. In this talk, I will explain how to glue local charts from different Lagrangians using isomorphisms between Lagrangians in the Fukaya category.
 
As an application, we will discuss the mirror construction for Gr(2,4) that recovers its Lie-theoretical mirror.