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Regular version of the site

Introduction to cobordism theory

Weekly seminar on Tuesdays 13.00-16.10 (organised by Semyon Abramyan and Alexey Gorinov)

Dear all,

the seminar on basic cobordism theory (organised by Semyon Abramyan and Alexey Gorinov) starts on Tuesday 7 September. The tentative syllabus is below. The first meeting will take place on Tuesday 1pm-4.10 pm in the HSE math department (6 Usacheva), and in Zoom. The room number will be posted on the door of room 326 before the class.

Youtube channel of the math department: https://youtube.com/channel/UCASlwNxf7mHBUEPr1s6fsDg

All subsequent meetings will tentatively be on Tuesday, same room, same time, but this is subject to change depending on the participants' wishes and the covid situation. All information about the seminar will appear on the seminar web page

https://sites.google.com/view/intro-cobordisms-hse-2021-2022/home

Also, if you are officially registered for the seminar or would like to be on the seminar mailing list, please fill out this form: https://forms.gle/UXUqY9wTGBVAtavy6

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PREREQUISITES: Smooth manifolds as covered in the compulsory course; homology and cohomology as covered in Algebraic topology 1 or the first three chapters of Hatcher’s Alegbraic topology.

SYLLABUS:

1. Examples of bordisms: oriented, non-oriented, complex and framed bordisms.

2. The Pontrjagin – Thom theorem.

3. Spectra and their homotopy groups: a reminder. The Thom spectra.

4. The Adams spectral sequence.

5. Applications of the Adams spectral sequence to the calculation of bordism groups.

6. The Hurewicz homomorphism.

7. Orientations of vector bundles with respect to multiplicative cohomology theories. Complex oriented theories.

8. Formal group laws and Quillen’s theorem.

9. Cohomology operations and the Landweber-Novikov theorem.

10. (*) Brown – Peterson spectra.

11. (*) Landweber’s exact functor theorem.

12. (*) Elliptic cohomology. Topological modular forms.

13. (*) Chromatic spectral sequence and Morava’s K-theories.

TEXTBOOKS: Haynes Miller, Vector fields on spheres etc. (online notes).

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Best wishes,

Semyon Abramyan, Alexey Gorinov

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