Babytop Seminar
Spring 2024
This semester Babytop will be about the K(n)-local sphere and the Two (p-adic) Towers. The goal of the seminar is to learn about https://math.bu.edu/people/jsweinst/chromatic.pdf and some of the tools that go into its proof.
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We meet at 4:00 on Tuesdays in 2-151 unless otherwise noted.
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Jared Weinstein
In this talk I'll present the main theorem and discuss the objects involved in its proof, including Morava K-theory and E-theory, Faltings' isomorphism between the two towers, and the pro-etale topology.
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Maxime Ramzi
Talk notes: https://sites.google.com/view/maxime-ramzi-en/notes/adic Further references: https://math.bu.edu/people/jsweinst/AWS2017.pdf S 1, https://people.math.rochester.edu/faculty/doug/otherpapers/scholze-berkeley.pdf S 1-5, https://math.stanford.edu/~conrad/Perfseminar/
This talk will be a very brief introduction to adic spaces. I will introduce valuations, Huber rings and Huber pairs, and explain how formal geometry and rigid analytic geometry fit into this picture. Finally, I will define adic spaces and describe a key example: the open unit disk. Time permitting, I will give more examples and draw pictures.
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Rushil Mallarapu
Notes: https://sudo-rushil.github.io/assets/papers/condense.pdf Reference: https://www.math.uni-bonn.de/people/scholze/Condensed.pdf
Condensed mathematics is a formalism for constructing abelian categories of “topological” objects, and shows up in the computation of [BSSW24] as a means of accessing continuous cohomology. In this talk, we will provide an overview of the information from condensed math necessary for the rest of the seminar, covering condensed sets/abelian groups, solidity and its connection to continuous cohomology, and 𝑝-complete and 𝑝-adic complexes of condensed abelian groups.
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Dylan Pentland
Notes: https://dpentland.github.io/files/proet.pdf https://people.math.rochester.edu/faculty/doug/otherpapers/scholze-berkeley.pdf S 8.2, https://www.math.uni-bonn.de/people/scholze/pAdicHodgeTheory.pdf S 3 & 4
In this talk, I will introduce the étale and pro-étale sites of a rigid analytic space. I will then motivate and define perfectoid spaces and show that perfectoid objects in the pro-étale site of a rigid analytic space form a basis for the topology (when the base field contains Q_p). This result will then be applied to prove some statements about condensed pro-étale cohomology we will need later in the seminar.
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The two towers
Andy
Reference: https://www.math.uni-bonn.de/people/scholze/Moduli.pdf
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p-adic Lie groups S 3.7
Piotr
http://www.numdam.org/item/PMIHES_1965__26__5_0/ Théorème V.2.4.10, or 'Cohomology of p-adic Analytic Groups' in New horizons in pro-p groups, Theorem 5.2.4
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Galois cohomology of $O_C$, S 4
Keita Allen/Tristan Yang
https://www.math.purdue.edu/~tongliu/teaching/598/p-divisible.pdf Section 3
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Galois cohomology of ${O}_{C}$, S 4 pt 2
Keita Allen/Tristan Yang
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Pro-etale cohomology of rigid analytic spaces, S 5
Ishan/David
https://www.math.uni-bonn.de/people/scholze/CDM.pdf, https://people.mpim-bonn.mpg.de/scholze/integralpadicHodge.pdf S 6 & 8, https://arxiv.org/abs/1710.06145 Thm 4.11
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Pro-etale cohomology of rigid analytic spaces, S 5 pt 2
Ishan/David
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The final proof, S 6
Isabel Longbottom
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This seminar is organized by Ishan Levy, Piotr Pstragowski, and Andy Senger.