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

  We define the geometric fixed points of a $G$-spectrum from the spectral Mackey functor point of view. We discuss the basic properties of this construction, including the isotropy separation sequence. If time permits, we will discuss the tom Dieck splitting, which allows us to decompose the ordinary fixed points of suspension spectra in terms of geometric fixed points, and also possibly a characterization of $G$-spectra for $G = C_p$

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• • Nov 052024
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  Indexed operations, HHR norms, and a splash of equivariant higher algebra

  Natalie Stewart

  We introduce indexed (co)products, indexed tensor products, \(G\)-symmetric monoidal categories, and \(G\)-commutative algebras; in particular, we will introduce Hill-Hopkins-Ravanel norms and their basic properties.

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• • Nov 122024
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  Real bordism I

  Keita Allen

  We give an introduction to the Real bordism spectrum \(MU_{\mathbb{R}}\), from which we will eventually construct the periodic spectrum helping us to solve the Kervaire invariant one problem. This is a \(C_2\)-equivariant enrichment of the complex bordism spectrum \(MU\), and enjoys an important role in the theory of Real-oriented cohomology theories, analogous to the close relation between \(MU\) and complex orientations.

  Today, we will highlight some nice "cellularity" properties of \(MU_{\mathbb{R}}\) and its images under the norm functors, and how these are both yielded from and yield equivariant refinements of classical computations with MU.

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• • Nov 192024
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  The slice filtration

  Matthew Niemiro

  TBA

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• • Dec 032024
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  Real bordism II

  TBA

  TBA

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• • Dec 102024
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  The detection theorem

  TBA

  TBA

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This seminar is organized by Natalie Stewart.