Canonical orientations in Heegaard Floer theory / Quantitative closing lemmas

SPECIAL DOUBLE SEMINAR

Speakers: Ciprian Manolescu (Stanford) and Michael Hutchings (UC Berkeley)

Ciprian Manolescu

will speak 2:15 – 3:15 pm on: Canonical orientations in Heegaard Floer theory

Heegaard Floer homology was originally defined over the integers by Ozsvath and Szabo using choices of coherent orientations on the moduli spaces. In this talk I will explain how to construct orientations in a more canonical way, by using a coupled Spin structure on the Lagrangian tori. This allows us to prove naturality of Heegaard Floer homology over the integers. The talk is based on joint work with Mohammed Abouzaid.

Michael Hutchings 

will speak 3:30 – 4:30 pm on: Quantitative closing lemmas

We consider the dynamics of Reeb vector fields on three-manifolds. Irie proved that for a C^\infty generic contact form, the periodic orbits of the Reeb vector field (“Reeb orbits”) are dense. The key step is a closing lemma asserting that one can modify a contact form by a C^\infty-small perturbation to create a Reeb orbit passing through a given neighborhood. In this talk we discuss a quantitative refinement of Irie’s closing lemma. Roughly speaking this asserts that one can create a Reeb orbit of period at most L via a perturbation of size O(L^{-1}). The proof uses spectral invariants related to embedded contact homology, and a key ingredient is a Weyl law for these invariants arising from Seiberg-Witten theory.