Third Year Major Oral: How Geometric PDEs and Symplectic Geometry Meet?

Abstract: Morse homology is a way to compute homology of finite-dimensional manifolds M by looking at the downward gradient flow lines of the function on M. In 1988, Andreas Floer adapted the idea of Morse homology to infinite-dimensional setting, and introduced two different approaches to construct the invariants.

One method uses solutions to certain geometric partial differential equations, known as ASD equations, to construct the invariant of the 3-manifolds. The other method uses pseudo-holomorphic curves to provide a powerful tool in studying symplectic geometry, a field that studies geometric structures inspired by classical mechanics.

In the same year, Michael Atiyah proposed that one could apply the latter method to a symplectic manifold naturally associated with a 3-manifold, thereby producing invariants of the 3-manifold. Moreover, Atiyah predicts that these two very different approaches should actually produce the same invariants. This prediction is now known as the Atiyah–Floer conjecture.

In this talk, I will first give an informal overview of these two constructions and explain why they are expected to agree. I will then describe my ongoing projects, a conjecture due to Cielibeck, Giao, and Salamon, which is motivated by the Atiyah–Floer conjecture.

Faculty Advisor: Ali Daemi

Note: This talk is given in partial fulfillment of the Third Year Candidacy Requirement.