Localization in Floer Theories Seminar (LIFTS)

Fall 2018

Critical Information

Time: Wednesdays, 12 - 1 pm
Location: Mathematics Building Room 528
Organizer / Contact: Melissa Zhang,

Goals and References

Our first goal is to read and understand Seidel-Smith's Localization for Involutions in Floer Cohomology [published version] [arXiv version].

The paper is well-organized, so we'll essentially follow the sections outlined by the paper. Roughly, we'd first carefully study the classical Smith inequality in the context of Morse theory, without relying on the 'easier' proof in the singular case. The recipe is always to study an equivariant version of the theory first, and then to study the localization map. After understanding the Morse case, we'll port our knowledge to Floer homology. By my optimistic estimates, this should take about a month. Afterwards, if people are interested, we can discuss the definition of symplectic Khovanov homology and apply the localization theorem to that case. We may, at this point, already have many more ideas about what to learn next.

Schedule of Talks

Date Speaker Topic
9/19 Melissa Zhang Intro to localization and Smith-type inequalities
I'll start with an introduction to the idea of localization and Smith-type inequalities, discuss the motivation behind studying this phenomenon in various theories, and describe the current landscape in this line of work. Afterwards, we'll set up a rough schedule of the talks to follow.

Here are my notes for the talk.
9/24 Alex Pieloch Equivariant Morse cohomology
 
10/3 Alex Pieloch Invariant Morse functions
 
10/10 Alex Zhang The localization map, part 1
 
10/17 Alex Zhang The localization map, part 2
 
10/24 Melissa Zhang Equivariant Floer cohomology
I'll describe the equivariant theory for Floer cohomology.  
10/31 Ali Daemi Obstructions to defining the equivariant cohomology theory
 
11/7 (no seminar)  
 
11/14 Ali Daemi The normal polarization and equivariant transversality
 
11/21 No seminar. Thanksgiving week.
 
11/28 No seminar.
 
12/5 Melissa Zhang Example: Hendricks's localization results for HFK