Fall 2024

This semester Babytop will be about stable equivariant homotopy theory and the Kervaire invariant one problem. We will largely cover Hill-Hopkins-Ravenel's proof.

We meet at 4:30pm on Tuesdays in Harvard Science Center 232 unless otherwise noted. Click here to add the seminar to your google calendar. Lecture notes will be recorded on Github.

• • Sep 172024
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  Overview

  Mike Hopkins

  In this talk, we will introduce the Kervaire invariant problem and sketch the strategy for its resolution at dimensions away from dimension 126.

  References:

  • On the non-existence of elements of Kervaire invariant one by Hill, Hopkins, and Ravenel.
  • The Kervaire invariant problem by Hopkins.
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• • Sep 242024
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  An intro to equivariant homotopy theory

  Isabel Longbottom

  We give a brief introduction to equivariant homotopy theory. In the non-equivariant setting, homotopy theory is concerned with topological spaces up to weak equivalence. Before we can do equivariant homotopy theory, we need an equivariant notion of weak equivalence. Through a selection of examples, we present and try to motivate the relevant definitions. We then discuss Elmendorf's Theorem and how it gives us a very nice, concrete model for the \(\infty\)-category of \(G\)-spaces as presheaves on the orbit category. We conclude by saying a few words about equivariance in families.

  References:

  • Chapter 1 of The Burnside category.
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• • Oct 012024
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  Mackey functors and equivariant cohomology

  Eunice Sukarto

  We introduce \(\mathbb{Z}\)-graded equivariant cohomology theories on \(G\)-spaces with coefficients in a coefficient system. We will look at an example computation using cellular cochains. We define \(G\)-spectra to be the stabilization of \(G\)-spaces with respect to finite orthogonal \(G\)-representations. They represent \(\mathrm{RO}(G)\)-graded cohomology theories, where Mackey functors now play the role of the coefficients.

  References:

  • Chapter 3 of The Burnside category.
  • Section 3 of HHR.
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• • Oct 082024
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  \(G\)-spectra and the spectral Mackey functor theorem

  Dhilan Lahoti

  We introduce \(G\)-categories and spectral mackey functors, and show that the \(G\)-category of \(G\)-spectra is equivalent to the \(G\)-category of mackey functors valued in spectra. This provides a convenient model for the \(G\)-category of \(G\)-spectra. Along the way, we will define \(G\)-stability and \(G\)-semiadditivity.

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• • Oct 222024
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  The isotropy separation sequence

  Jonathan Buchanan

  TBA

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• • Oct 292024
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  Indexed operations, HHR norms, and a splash of equivariant higher algebra

  Natalie Stewart

  TBA

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• • Nov 032024
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  The slice filtration

  Matthew Niemiro

  TBA

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• • Nov 122024
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  Real bordism I

  TBA

  TBA

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• • Nov 192024
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  Real bordism II

  TBA

  TBA

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• • Dec 032024
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  The detection theorem

  TBA

  TBA

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This seminar is organized by Natalie Stewart.