Babytop Seminar
Fall 2024
This semester Babytop will be about stable equivariant homotopy theory and the Kervaire invariant one problem. We will largely cover HillHopkinsRavenel'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.

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 On the nonexistence of elements of Kervaire invariant one by Hill, Hopkins, and Ravenel.
 The Kervaire invariant problem by Hopkins.
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:
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 Chapter 1 of The Burnside category.
Isabel Longbottom
We give a brief introduction to equivariant homotopy theory. In the nonequivariant 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:
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 Chapter 3 of The Burnside category.
 Section 3 of HHR.
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:
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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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Natalie Stewart
TBA
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Matthew Niemiro
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This seminar is organized by Natalie Stewart.