The Kervaire invariant is an invariant of a framed (4k+2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. The question of in which dimensions n there are n-dimensional framed manifolds of nonzero Kervaire invariant is called the Kervaire invariant problem. In 1969 Browder reformulated Kervaire invariant problem in terms of the stable homotopy groups of spheres which translated the problem into the world of stable homotopy theory. In 2016 Hill, Hopkins and Ravenel answered the problem in all the dimensions except n=126. Their solution extensively uses equivariant algebraic topology and chromatic homotopy theory along with some fascinating calculating techniques. In our course we plan to give a survey of equivariant stable homotopy theory and sketch main parts of their proof.
* May be skipped, if everyone are familiar with the topic.
** If we would have enough time.
|Introduction and Arf invariant||Artem Prikhodko||September, 13||Talks 0 and 1|
|Presentable and stable categories||Grigory Kondyrev||September, 20||References for the talk: 1, 2, 3|
|Adams-Novikov spectral sequence||Grigory Kondyrev||September, 27||Talk 3|
|Kervaire invariant||Nikolai Konovalov||October, 4||Talk 4|
|Browder's theorem||Nikolai Konovalov||October, 11||Talk 5|
|Unstable equivariant homotopy theory, Elmendorf's theorem||Artem Prikhodko||October, 18||Talk 6|
|Stable equivariant homotopy category, duality theory||Artem Prikhodko||October, 25||Talk 7|
|Mackey functors, tom Dieck splitting||Artem Prikhodko||November, 1|
|Thom spectra||Grigory Kondyrev||November, 8||Talk 9|
|The slice filtration, Gap theorem||Artem Prikhodko||November, 15||Talk 10|
|Norms, Homotopy Fixed Points theorem||Artem Prikhodko||November, 22|
|Periodicity theorem||Artem Prikhodko||November, 29||References: Talks 15 and 17 of  and section 9 of .|
|Reduction and Slice theorems||Nikolai Konovalov||December, 6||References: Talk 16 of  and sections 6-7 of .|
|Aside: Segal's Burnside ring conjectrure||Vladimir Shajdurov||December, 13|
|Detection theorem||Grigory Kondyrev||December, 20||References: Talk 18 of  and section 11 of .|
|Bonus: Parametrized higher category theory||Grigory Kondyrev||December, 27|
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