Babytop Seminar
Spring 2023
This semester Babytop will be about telescopically localized stable homotopy theory.
We meet at 4:00 on Tuesdays in 2361 unless otherwise noted.

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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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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 Etheories E(n) and the Morava Ktheories 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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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 pifinite Kan complexes are ambidextrous in K(n)local spectra, and reduce this to questions about the behavior of K(n)local Enmodules. Along the way, we’ll introduce some interesting ideas about alternating powers of formal groups to contextualize these computations.
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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 1semiadditive. Taking this as the base case, I will explain the inductive argument by CarmeliSchlankYanovski for proving that this category is ∞semiadditive. It will be based on a new power operation that can be constructed in symmetric monoidal 1semiadditive categories.
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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 nilconservativity 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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MIT spring break
None
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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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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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Ishan Levy
Reference: https://projecteuclid.org/journals/pacificjournalofmathematics/volume92/issue2/brmoresolutions/pjm/1102736799.full
This talk will explain the approach due to Mahowald to understanding the bobased 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.