## Spring 2023

This semester Babytop will be about telescopically localized stable homotopy theory.

We meet at 4:00 on Tuesdays in 2-361 unless otherwise noted.

• Feb 072023
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### Ishan

The goal of the seminar is to learn about the T(n)-localizations of the stable homotopy category.

In the first part of the seminar, we will first go over some of the basics of chromatic homotopy theory and introduce the T(n)-local and K(n)-local categories. We will then focus on understanding ambidexterity and its consequences, which is one of the fundamental features of these categories.

In the second part of the seminar, we will read works surrounding the telescope conjecture. In particular, we will learn about the original proofs of the telescope conjecture, attempted disproofs and related conjectures at higher heights, and implications of these conjectures on the size of the stable homotopy groups of spheres.

Below is a rough list of the talks for the first half of the semester.

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• Feb 142023
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### Eunice Sukarto

This will be a crash course on chromatic homotopy theory. We will first look at the connection between stable homotopy theory and formal groups. Then, we will define the Morava E-theories E(n) and the Morava K-theories K(n) and state the nilpotence and periodicity theorems. Finally, we will look at K(n) vs T(n) localizations and state the telescope conjecture.

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• Feb 212021
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### Rushil Mallarapu

Reference: https://people.math.harvard.edu/~lurie/papers/Ambidexterity.pdf

We will introduce and give motivating examples of the categorical phenomena of ambidexterity. The main goal is to outline the proof that pi-finite Kan complexes are ambidextrous in K(n)-local spectra, and reduce this to questions about the behavior of K(n)-local En-modules. Along the way, we’ll introduce some interesting ideas about alternating powers of formal groups to contextualize these computations.

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• Feb 282021
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### David Jongwon Lee

Reference: https://arxiv.org/abs/1811.02057

It is a theorem of Kuhn that the category of T(n)-local spectra is 1-semiadditive. Taking this as the base case, I will explain the inductive argument by Carmeli-Schlank-Yanovski for proving that this category is ∞-semiadditive. It will be based on a new power operation that can be constructed in symmetric monoidal 1-semiadditive categories.

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• Mar 072021
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### Natalie Stewart

Reference: https://arxiv.org/abs/2103.02471

This talk concerns galois extensions in the K(n)-local category and lifts of Abelian such extensions to the T(n)-local category.

We begin by introducing the notion of Galois extensions in higher algebra. We introduce higher cyclotomic extensions and use nil-conservativity to prove that height n p^r'th cyclotomic extensions are T(n)-locally Galois.

Then, we introduce separable closures and Rognes' 'Brave New Galois Correspondence.' We will see that E_n is the separable closure of the K(n)-local sphere, with absolute galois group given by the big morava stabilizer group. By an explicit computation, we then show that all K(n)-local abelian galois extensions lift to the T(n)-local setting.

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• Mar 282021
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### None

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• Apr 042021
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### Tristan Yang

Reference: https://arxiv.org/abs/2210.12822

We define the chromatic fourier transform and explain how it relates to semiadditive height and higher roots of unity. we then apply this to the T(n)-local setting

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• Apr 112021
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### Tomer Schlank

This talk will explain a proof of the height 1 telescope conjecture via checking descent with respect to the maximal cyclotomic extension of the T(1)-local sphere.

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• Apr 182021
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### Ishan Levy

Reference: https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-92/issue-2/brm-o-resolutions/pjm/1102736799.full

This talk will explain the approach due to Mahowald to understanding the bo-based Adams spectral sequence at the prime 2. In particular, it gives a proof of the height 1 telescope conjecture at the prime 2.

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This seminar is organized by Ishan Levy.