MATH 740 in Srping 2022:

Stable Homotopy Theory

This reading group aims to go through some basic material about stable homotopy theory, which is increasingly important for modern geometry and topology, and is related to various fields of research. Participants can choose either to give a talk or just sit down and be an audience, but we recommend everyone to stand in the front and give a talk.

Schedule

We'll meet every Friday from 4:00pm to 5:20pm at KAP 146. Here's the calendar.

Time	Contents	Speaker	References	Video Recordings
01/21	Introduction I	Joseph Helfer	[Ada], [SpWh] and [Ati]	introduction
01/26	Introduction II	Joseph Helfer		Introduction, 2
01/28	Basics of Model Categories	Suraj Yadav	[BaRo], Appendix and possibly [MaPo], chapter 14	Basics of Model Categories
02/04	Basics of Homotopy Theory	Tianle Liu	[Hat], Section 4.1 through 4.3 and 4.J	Basics of Homotopy Theory
02/11	Basics of Stable Homotopy Theory	Haoyang Liu	[BaRo], Chapter 1	Basics of Stable Homotopy Theory
02/18	K-Theory and Bott Periodicity	Haosen Wu	[Hat03] and [Mag]	Bott Periodicity
02/25	Sequential Spectra	Siyang Liu	[BaRo], Section 2.1-2.4	Sequential Spectra
03/04	spectra in a general model category	Jishunu Bose	[BaRo], Chapter 3	Loops and Suspensions
03/11	triangulated structure	Siyang Liu	[BaRo], Chapter 4 and [GeMa], Chapter 4	Triangulated Structures
03/13-03/20	Spring Recess
03/25	the Steenrod algebra and the Adams spectral sequence	Fan Yang	[BaRo], Section 2.5 and 2.6	Recording available here.
04/01	three modern categories of spectra	Qiyu Zhang	[BaRo], Section 5.1-5.4
04/08	other categories of spectra	Jonathan Michala	[BaRo], Section 5.5-5.7	Recording available here.
04/15	monoidal structures on spectra	Haosen Wu	[BaRo], Section 6.1-6.5	Recording available here.
04/22	monoidal structures, II	Haosen Wu	[BaRo], Section 6.1-6.5	Recordings available here.
04/29	Bousfield localization	Qiyu Zhang	[BaRo], Section 7.1-7.3	Recordings available here.

Notes and Recordings

Here's the texed notes by the organizer. Email to the organizer if you find any errors in this note.

There'll be recordings once the seminar is held online.

References

[Ada] Adams, J. F., & Adams, J. F. (1974). Stable Homotopy and Generalised Homology. University of Chicago Press.
[Ati] Atiyah, M. F. (1961). Thom Complexes. Proc. London Math. Soc., s3-11(1), 291–310. doi: 10.1112/plms/s3-11.1.291
[BaRo] Barnes, D., & Roitzheim, C. (2020). Foundations of Stable Homotopy Theory. Cambridge University Press. doi: 10.1017/9781108636575
[Hat] Hatcher, A. (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN: 0-521-79160-X; 0-521-79540-0
[Hat03] Hatcher, A. (2003). Vector Bundles and K-Theory.
[GeMa] Gelfand, S. I., & Manin, Y. I. (2003). Methods of Homological Algebra. Springer. doi: 10.1007/978-3-662-12492-5
[Mag] Magill, M. (2017). Topological K-theory and Bott Periodicity (Dissertation). Retrieved from http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-322927
[MaPo] May, J. P., & Ponto, K. (2012). More Concise Algebraic Topology: Localization, Completion, and Model Categories. University of Chicago Press.
[SpWh] Spanier, E. H., & Whitehead, J. H. C. (1955). Duality in homotopy theory. Mathematika, 2(1), 56–80. doi: 10.1112/S002557930000070X