Symplectic Geometry Seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
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Thursday
February 15, 2018

1:00 PM
Math Tower 5-127
Brice Loustau, Rutgers University
Bi-Lagrangian structures and Teichmüller theory

A Bi-Lagrangian structure in a smooth manifold consists of a symplectic form and a pair of transverse Lagrangian foliations. Equivalently, it can be defined as a para-Kähler structure, that is the para-complex equivalent of a Kähler structure. Bi-Lagrangian manifolds have interesting features that I will discuss in both the real and complex settings. I will proceed to show that the complexification of a real-analytic Kähler manifold has a natural complex bi-Lagrangianstructure, and showcase its properties. I will then specialize this discussion to moduli spaces of geometric structures on surfaces, which have a rich symplectic geometry. I will show that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory, while revealing other new geometric features; as well as deriving several well-known results of Teichmüller theory by pure differential geometric machinery. Time permits, I will also mention the construction of an almost hyper-Hermitian structure in the complexification of any real-analytic Kähler manifold, and compare it to the Feix-Kaledin hyper-Kähler structure.


Thursday
February 22, 2018

1:00 PM
Math Tower 5-127
Yuhan Sun, Stony Brook University
Finding nondisplaceable Lagrangian tori via toric degeneration.

Starting with a semi-Fano toric surface, I will explain how to find families of Lagrangian tori with nontrivial deformed Floer cohomology. The proof uses toric degeneration method to compute the potential functions of these tori.


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