In Spring 2022, I organize the student symplectic topology seminar. The topic is Floer Homotopy theory
Zoom link: https://umn.zoom.us/j/94798206643
Schedule: Friday 9-11AM
Notes and Slides (Latex by Shengzhen Ning)
1.26: Overview (Shuo Zhang)
2.2: Spectrum and stable homotopy theory (Shuo Zhang)
2.9: Spectrum and stable homotopy theory, continued (Shuo Zhang)
2.16: Important examples of Spectra (Shuo Zhang)
2.23: Break
3.2: Realizing chain complexes and Floer homotopy type of cotangent bundles （Jie Min)
3.9: Twisted framed bordism invariant (Tian-jun Li）12-2PM
3.16: Kuranishi Spaces and Kuranishi Flow categories (Shengzhen Ning)
3.23:Kuranishi Spaces and Kuranishi Flow categories, continued (Erkao Bao)
3.30: Break
4.6: Seiberg-Witten-Floer homotopy types (Yi Du) Video
4.12: Real blowup and Three versions of Seiberg-Witten-Floer homologies Video
4.20: Introduction to chromatic homotopy theory (without formal group law). (Shuo Zhang)
4.27: Complex oriented cohomology and formal group law (Shuo Zhang)
5.4 : Break
5.11: Break
5.18: Break due to Covid Notes
5.25: Organizational meeting
6.2: Shengzhen Z-valued Gromov-Witten type invariants Video
6.9: Ke Zhu Nearby Lagrangian I Video
6.16 Ke Zhu Nearby Lagrangian II Video
6.23: Jie HMS for T^4
6.30: Break
7.7: Higher capacities (Shengzhen Ning)
7.14: Localized Mirror functor (Shuo Zhang)
7/21: Break
7.28: HMS for Log CY（Jie Min)
8.4: HMS for Log CY (Jie Min)
8.11: Algebraic capacities(Shengzhen Ning)
8.18: Symplectic homology of Log CY surface (Jie Min)
8.24: Symplectic homology of Log CY surface (Jie Min)
References:
Equivariant homotopy theory: (There are two frameworks: G-spectra and orthognal G-spectra)
Surveys:
EQUIVARIANT STABLE HOMOTOPY THEORY by Greenlees-May. (G-spectra) covered classical equivariant theory like G-CW complexes
LECTURES ON EQUIVARIANT STABLE HOMOTOPY THEORY by Schwede. (Orthognal G-spectra)
Abouzaid-Blumberg's Appendix A. (Orthognal G-spectra)
More comprehensive accounts:
EQUIVARIANT HOMOTOPY AND COHOMOLOGY THEORY by May (orthognal G-spectra). These also covered many classical equivariant theory like G-CW complexes
Equivariant orthogonal spectra and S-module by Mandell-May. These covered general frameworks for G-spectra and orthognl G-spectras and proved they are equivalent.
Equivariant Stable Homotopy Theory by Lewis-May-Steinberger. (G-spectra).
Chromatic homotopy theory:
Chapter 7 of Foundations of Stable Homotopy Theory by Barnes-Roizheim
Luries's lecture notes: https://ncatlab.org/nlab/show/Chromatic+Homotopy+Theory Focus on lecture
(The most detailed and comprehensive one) Complex cobordism and stable homotopy groups of spheres by Doug Ravenel
Morava K theory with an eye toward Floer theory:
https://sanathdevalapurkar.github.io/files/morava-k-thy-floer.pdf
Miscellaneous notes:
Floer Homotopy 2017:
https://sites.google.com/site/floerhomotopy2017/home/summer-school-schedule?authuser=0
Video recordings on youtube
1,Virtual fundamental chains. Abouzaid 19
2,Floer homotopy without spectra. Abouzaid 20
3,Floer homotopy theory, new and old. Cohen 21
4, Abouzaid-Mclean-Smith (by Abouzaid): https://www.youtube.com/watch?v=EoJ73ndrJbU&t=2883s
5, A summer school in ICTS covered some stable homotopy and Floer homotopy:
https://www.youtube.com/watch?v=KeyHuYS4LXU&list=PL04QVxpjcnjg5D51diyf_nzAlhIoRYq75
Spectrum and Stable homotopy theory (Shuo):
1,"Stable homotopy and generalized cohomology , part III' by Adams for motivations
2,"Stable homotopy theory" notes by Rognes. This should satisfy most of our needs
3,"Foundations of Stable homotopy theory" by Barnes and Roitzheim. This should be used as a dictionary.
4,The website nLab is always useful: http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/Introduction+to+Stable+Homotopy+Theory#:~:text=to%20homotopy%20theory.-,1)%20Stable%20homotopy%20theory,%E2%80%9C%E2%88%9E%2Dgroup%E2%80%9D).&text=A%20spectrum%20is%20where%20this,stabilization%E2%86%A6(linearization)Spectra.
5, "STABLE ALGEBRAIC TOPOLOGY" 1945–1966" by May. More motivation.
Realizing chain complexes+Homotopy Viterbo's theorem (Jie).
1,"The Floer homotopy type of the cotangent bundle" by Cohen
2,"Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners" by Cohen.
3,"The Viterbo transfer as a map of spectra '' by Kragh. Also contains a proof for this when M is nonorientable.
Parametrized spectra and Kragh's thesis (Liya）
1,"The Viterbo transfer as a map of spectra" by Kragh
2,"Parametrized Homotopy Theory" by May for backgrounds
Nearby Lagrangian are homotopy equivalent (Ke)
1,"Simple Homotopy Equivalence of Nearby Lagrangians". This is the best result so far.
2,"Parametrized ring-spectra and the nearby Lagrangian conjecture" by Abouzaid-Kragh. This is the proof using spectra.
3,”Nearby lagrangians with vanishing maslov class are homotopy equivalent" by Abouzaid. This didn't use spectra
4,"Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence" by Kragh. A similar result proved by spectral sequence, without spectra.
5,"On the immersion classes of nearby Lagrangians" A related problem.
6,"SHEAVES AND SYMPLECTIC GEOMETRY OF COTANGENT BUNDLES", the sheaf proof of some classical problem, including nonsequeezing, C^0 closedness of symplectomorphism and nearby lagrangian being homotopy equivalence.
Kuranishi Spaces and Kuranishi Flow categories (Erkao, Shengzhen)
1, "Arnold conjecture and Morava K theory". Abouzaid-Blumberg.
1'," Complex cobordism, Hamiltonian loops and global Kuranishi charts", This used one global Kuranishi chart instead of piecing together multiple local charts.
3, [Par16] An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves
4,[MW17] The topology of Kuranishi atlases
5,[FO99] Arnold conjecture and Gromov-Witten invariant
6,[FOOO09] You know which. There's another new book:
7,[FOOO20] Kuranishi Structures and Virtual Fundamental Chains