Reading seminar on Floer homotopy I
(Summer 2024)

Meeting place:

Zoom (link will be sent via mailing list; please email if you’d like to be put on the mailing list)

Meeting time:

Tuesdays and Thursdays, 1:00-2:30pm EST

Overview:

The goal of this reading seminar is to understand the recent work of Abouzaid-Blumberg on the foundations of Floer homotopy theory via flow categories. We will focus on 1) building the stable infinity-category of (un)structured flow categories (where “structure” refers to a choice of bordism theory) and 2) showing the choice of framed bordism recovers the familiar stable infinity-category of spectra. To motivate things, we will also review the geometry inspiring the algebraic constructions (Morse/Floer theory, bordism, etc.) and the “classical” framed flow categories of Cohen-Jones-Segal.

References:
Non-exhaustive, will be added to as we go

Abouzaid-Blumberg, Foundations of Floer homotopy theory I: Flow categories
Atiyah, K-theory
Audian-Damian, Morse theory and Floer homology
Auroux, A beginner’s introduction to Fukaya categories
Cohen-Jones-Segal, Floer’s infinite dimensional Morse theory and homotopy theory
Conner-Floyd, Differentiable periodic maps
Côté-Kartal, Equivariant Floer homotopy theory via Morse-Bott theory
Freed, Bordism: old and new
Large, Spectral Fukaya categories for Liouville manifolds
McDuff-Salamon, Introduction to symplectic topology
Pedroza, A quick view of Lagrangian Floer homology
Schwarz, Morse homology
Seidel, Graded Lagrangian submanifolds
Wehrheim, Smooth structures on Morse trajectory spaces, featuring finite ends and associative gluing

Schedule:
May 28: Logistics, overview, Morse homology, original CJS construction
May 30: Symplectic geometry, broad overview of (Lagrangian) Floer homology
June 4: Floer homotopy theory on Liouville manifolds
June 7: Overview of infinity-categories (Joseph Hlavinka)
June 13: Overview of stable infinity-categories (Joseph Hlavinka)
June 15: Bordism
June 18: Derived stratified orbifolds with corners
July 9: Structured derived stratified orbifolds with corners
July 23: (Un)structured flow categories (Mira Wattal)
July 25: (Un)structured flow simplices and outline of rest of paper