Babytop Seminar
Fall 2023
In Fall 2023, Babytop was about topological automorphic forms.-
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Andy
The goal of this seminar is to understand TAF and related constructions. This talk will give an overview of the topic. Below is a rough schedule of the rest of the semester.
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Keita Allen
In my talk, I’ll be trying to motivate some of the big ideas in stable and chromatic homotopy theory and introduce p-divisible groups, a class of objects with connections to the theory of algebraic groups and formal groups. The plan is to keep the talk relatively beginner-friendly, so all are welcome!
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Leonard Tokic
In this talk we will discuss Lurie's sheaf of E_infty rings on a site of p-divisible groups (in a formulation due to Jack Davies), focusing on applications. To this end I hope to cover how to use it to construct p-complete (and integral) KU, KO, and Lubin-Tate E_n’s.
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Tristan Yang
We will explain the formulation of TMF as a product of Lurie's Theorem from the previous talk, and discuss various properties. To spotlight the computability of things, we will compute TMF[1/6].
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Natalie Stewart
Following work of Mark Behrens, we define a spectrum Q(ℓ) in terms of the structure of isogenies of elliptic curves via the Goerss-Hopkins sheaf. For (p,ℓ) = (3,2), we construct the K(2)-local sphere as an extension of Q(ℓ) and its K(2)-local Spanier-Whitehead dual, reinterpreting a resolution due to Goerss, Henn, Mahowald, and Rezk.
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Rushil Mallarapu
The theory of abelian varieties is a natural source for many 𝑝-divisible groups, giving an algebro-geometric inroad to TAF. The goal of this talk is to briskly introduce the main concepts in this theory and then discuss the Serre-Tate theorem, which states that the deformation theory of abelian varieties are controlled by the deformation theory of their associated 𝑝-divisible group. We will first define abelian varieties and isogenies, discuss the construction of the dual of an abelian variety, introduce some aspects of the theory of 𝑝-divisible groups, and end with an overview of the proof of the Serre-Tate theorem.
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Dhilan Lahoti
Give a sketch of the construction of TAF, discuss PEL structures.
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David Lee
on Monday 14:00 in 2-449 instead of usual time!
Given a quadratic imaginary field F, I will describe the moduli stack of polarized O_F-linear abelian surfaces with certain PEL structure in terms of the moduli stack of elliptic curves, which will give us a description of TAF in terms of TMF in this case.
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Ishan
I will explain how there is extra functoriality of the TAF construction, which in particular gives rise to the action of Hecke operators. I will then explain how to construct Q_GU, an analog of Q_l, and how it can be described as the limit of a semisimplicial E_infty ring constructed from the Bruhat-Tits building of GU. I may also describe what happens K(n)-locally and make things explicit at height 1.
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Andy
In room 4-265 at 14:45!
In this talk, I will describe some interesting families of p-divisible groups which arise from families of curves, and how to check that they give rise to E_infty-ring spectra. Some of these families are related to topological automorphic forms, but most are not.
Along the way, we'll touch upon Grothendieck-Messing deformation theory, how to compute infinitesimal variations of Hodge structure, and the relationship between Dieudonne modules and de Rham cohomology.
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This seminar was organized by Ishan Levy and Andy Senger.