Pacific Institute for the Mathematical Sciences - PIMS

The idea of stable homotopy refinements of Floer homology was first introduced by Cohen, Jones, and Segal in a 1994 paper, but it was only in the last decade that this idea became a key tool in low-dimensional and symplectic topology. The two crowning achievements of these techniques so far are Manolescu's use of his Pin(2)-equivariant Seiberg–Witten Floer homotopy type to resolve the Triangulation Conjecture and Abouzaid-Blumberg's use of Floer homotopy theory and Morava K-theory to prove the general Arnol'd Conjecture in finite characteristic. During this period, a range of related techniques, included under the umbrella of Floer homotopy theory, have also led to important advances, including involutive Heegaard Floer homology, Smith theory for Lagrangian intersections, homotopy coherence, and further connections between string topology and Floer theory. These in turn have sparked developments in algebraic topology, ranging from developments on Lie algebras in derived algebraic geometry to new computations of equivariant Mahowald invariants to new results on topological Hochschild homology.

The goal of the summer school is to provide participants the tools in symplectic geometry and stable homotopy theory required to work on Floer homotopy theory. Students will come away with a basic understanding of some of the key techniques, questions, and challenges in both of these fields. The summer school may be particularly valuable for participants with a solid understanding of one of the two fields who want to learn more about the other and the connections between them.

Speakers and Courses:

• Mohammed Abouzaid, Columbia University: Floer Homotopy
• Nate Bottman, Max Planck: Floer Homology Fundamentals
• Catherine Cannizzo, SCGP: Floer Homology Fundamentals
• Jeff Hicks, University of Edinburgh: Applications
• Cary Malkiewich, Binghamton University: Spectra and Smash Products
• Katherine Poirier, New York City College of Technology: String Topology 
• Hiro Lee Tanaka, Texas State University: Operads and Ring Spectra

School Structure:

The summer school will consist of lecture courses with problem sessions; seminars on recent developments; and two panel discussions about professional development. The lecture courses will be:

Week 1: July 11- 15

Fundamentals of Floer Homology (9 lectures)

Operads and Ring Spectra (3 lectures)

String topology (3 lectures)

Week 2:July 18- 22

Spectra and smash products (4 lectures)

Operads and Ring Spectra continued (4 lectures)

Applications of Floer homology (3 lectures)

Floer homotopy theory (4 lectures)

Most days, an hour and a half will be set aside for problem sessions. There will also be two seminars on recent developments and one panel discussion on professional development each week.

Schedule

Hot Topics

Notes and Exercises 

Lecture Recordings 

Suggested Prerequisites:

Basic algebraic topology (fundamental group, homology, cohomology, Poincaré duality) and basic differential geometry (smooth manifolds, vector fields, flows, transversality, differential forms, deRham cohomology). Some familiarity with generalized cohomology theories (K-theory, bordism, etc.) or spectra would be useful, but not absolutely required.

This event is part of the PIMS CRG on Novel Techniques in Low Dimensions.

COVID-19 Reporting and Precautions for participants attending the summer school in person: 

Precautionary measures:

- Wear you mask at all times indoors or in shared spaces.

- Maintain a safe distance from others (at least 1 metre), even if they don’t appear to be sick.

- Clean your hands often. Use soap and water, or an alcohol-based hand rub.

- Cover your nose and mouth with your bent elbow or a tissue when you cough or sneeze.

Contact:

Should you have any queries about this event, pleasae contact ruth@pims.math.ca.