Spring 2021

In Spring 2021, Juvitop was about The Galois Action on Symplectic K-Theory.

• • Feb 022021
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  Introduction

  Nat Pacheco-Tallaj

  Notes:

  • Speaker's Notes

  Discussion Section:

  • Handout

  References:

  • The Galois Action on Symplectic K-Theory by T. Feng, S. Galatius, and A. Venkatesh

  • The Galois Action on p-adic Symplectic K-Theory talk by T. Feng

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• • Feb 092021
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  Introduction/Review of Stable Homotopy Theory

  Dylan Wilson

  Notes:

  • Notes by organizers

  • Speaker's Notes

  Overview of (Ring) spectra, stable homotopy groups, infinite loop space machines, Postnikov truncations, Bott elements, S-duality, Moore spectra and homotopy with coefficients.

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• • Feb 162021
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  Three Types of K-Theory

  Lucy Yang

  References:

  • Theorie de Quillen et Homologie du Groupe Orthogonal by M. Karoubi

  Discussion Section:

  • Handout

  Review and introduce the notions of algebraic K-theory, symplectic K-theory, and Hermitian K-theory. Group completion and K-theory spectra. Quillen's devissage and localization. Show how these three thoeries relate to eachother. Bott-inverted K-theory.

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• • Feb 232021
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  Introduction to Class Field Theory

  Niven Achenjang

  Notes:

  • Speaker's Notes

  Discussion Section:

  • Handout

  We start with a crash course on places/completions in algebraic number theory. We then give the statements of the main results of global class field, and look at some of their consequences. In particular, we show the existence of (narrow) Hilbert class fields.

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• • Mar 022021
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  Etale Cohomology

  Daniel Li-Huerta

  Notes:

  • Speaker's Notes

  Discussion Section:

  • Handout

  References:

  • Algebraic K-theory and Etale cohomology by R. W. Thomason

  Etale cohomology is an invariant for schemes that extends singular cohomology for varieties over C. After defining étale cohomology, we'll present several examples and computations to give a feel for the theory. We'll conclude by briefly introducing Thomason's spectral sequence, which relates étale cohomology to algebraic K-theory, and mention how it's used in [FGV].

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• • Mar 092021
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  Hodge Map and Witt Groups

  Morgan Opie

  Notes:

  • Speaker's Notes

  Discussion Section:

  • Handout
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• • Mar 162021
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  CM Abelian Varieties

  Grant Barkley

  Notes:

  • Speaker's Notes

  We'll explain the basics of abelian varieties and the theory of complex multiplication. We use this to construct classes in symplectic K-theory associated to a principally polarized abelian variety, and explain how complex multiplication can be used to parametrize a useful family of principally polarized abelian varieties for this purpose.

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• • Mar 232021
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  CM classes exhaust symplectic K-theory

  Araminta Amabel

  Notes:

  • Speaker's Notes

  We show that the map defined in last week’s talk is surjective. The result says that the so-called CM classes generate symplectic K-theory groups in dimensions 4k-2 for k>0. The proof involves techniques developed throughout the seminar, including a throw back to the Hodge and Betti maps.

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• • Mar 302021
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  Construction of the Galois action on symplectic K-theory

  Yuri Sulyma

  Appendix B.

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• • Apr 062021
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  The Galois action on KSp and on CM abelian varieties

  Zhiyu Zhang
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• • Apr 132021
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  The main theorem and its proof: Part I

  Peter Haine
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• • Apr 202021
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  The main theorem and its proof: Part II

  Vaughan McDonald
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• • Apr 272021
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  Families of abelian varieties and stable homology

  Sanath Devalapurkar
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This seminar was organized by Araminta Amabel, Nat Pacheco-Tallaj, and Lucy Yang.