Fall 2021

This semester Babytop will be about deforming homotopy theory, synthetic spectra and a modern understanding of the Adams spectral sequence.

We meet at 4:0 on Tuesdays in 2-151 unless otherwise noted.

• • Sep 142021
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  Overview

  rwb

  After a virtual year, I'm excited to announce babytop is back, and in person! This semester we'll be taking a look at synthetic spectra and structured spectral sequences more generally. Today we'll examine several motifs that will recur throughout the semester.

  The seminar will have roughly four phases.

  In the first, we will give a gentle introduction to the Adams spectral sequence and its categorification in the form of the category of synthetic spectra.

  In the second, we will examine the techniques that exist for working with synthetic spectra (with an emphasis on portability).

  In the third, we will examine the myriad of techniques that now exist for constructing deformations that shares many of the formal properties of synthetic spectra.

  In the final phase we will put all the pieces together and look at a couple recent papers. Topics will be chosen based on particpant interests.

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• • Sep 212021
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  Introduction to the Adams spectral sequence

  Piotr Pstragowski

  Computing homotopy classes of maps between even the most simple spectra, such as the stable homotopy groups of spheres, is very difficult. What is usually much easier to access is homology. A powerful insight, due to Adams, is that in the stable context this is already a good start—there's a spectral sequence which starts with homological data and which computes homotopy classes of maps.

  This talk will be a gentle introduction to this remarkable construction, known as the Adams spectral sequence.

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• • Sep 282021
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  Formal properties of synthetic spectra

  Arpon Raksit

  There is a mantra I find helpful for calming my mind: "Synthetic spectra categorify the Adams spectral sequence."

  Without actually constructing or proving anything, I will describe the basic setup of the theory of synthetic spectra, and explain the meaning of this mantra.

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• • Oct 052021
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  How do you calculate with synthetic spectra?

  Ishan Levy

  I will explain how to compute the F_2-synthetic homotopy groups of the sphere in low dimensions using the Adams spectral sequence starting from its E_2 page. In doing so I will explain relevant notions and tools such as Toda brackets and cell diagrams as they come up, and we will see how the synthetic perspective makes for simple arguments.

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• • Oct 122021
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  Calculations (continued)

  David Jongwon Lee

  Continuing from last week, I will compute more stems in the F_2-based Adams spectral sequence with the help of operations in the synthetic homotopy groups.

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• • Oct 192021
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  Goerss–Hopkins theory through synthetic spectra

  Leon Liu

  TBD

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• • Oct 262021
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  Miller's Adams spectral sequence

  Natalie Stewart

  TBD

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• • Nov 022021
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  Synthetic spectra as sheaves

  Dexter Chua

  TBD

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This seminar is organized by Robert Burklund and Piotr Pstragowski.