Spring 2021 Thursday Seminar 

We will discuss redshift in algebraic K theory. We meet on Zoom on Thursdays at 3:30 PM EST. 
Talks 

Feb 4 
Overview of redshiftMike Hopkins 
Feb 11 
Introduction to algebraic KtheoryPeter Haine 
Overview of the algebraic Ktheory functor, including the plus construction, the S_{•} construction, and the universal property as a localizing invariant. Notes. 

Feb 18 
The LandTamme theoremIan Coley 
A discussion of the Ktheory of pullbacks. Notes. 

Feb 25 
Chromatic homotopy theoryIshan Levy 
Basics of chromatic homotopy theory, including telescopic localizations and the BousfieldKuhn functor. Notes. 

Mar 4 
Purity in chromatically localized algebraic KtheoryShachar Carmeli 
Discussion of Section 3 of the LandMathewMeierTamme work, ending with the proof of Theorem 3.8. Also, the beginning of our discussion of the ClausenMathewNaumannNoel work. Notes Part 1. Notes Part 2. 

Mar 11 
Blueshift and a converse for highly structured ring spectraAdela Zhang 
Kuhn's blueshift theorem on telescopic Tate vanishing, as well as the converse theorem for E_{∞} ring spectra. Notes. 

Mar 18 
Descent and vanishing in chromatic algebraic Ktheory via group actionsLucy Yang 
Discussion of the ClausenMathewNaumannNoel work, focusing on Section 4. Notes. 

Mar 25 
THH and TCNat PachecoTallaj 
The definitions of THH and TC. Statement of the DundasGoodwillieMcCarthy theorem. The computations of THH(F_{p}) and TC(F_{p}), exhibiting redshift from height 1 to height 0. Notes. 

Apr 1 
The multiplication on truncated BrownPeterson spectraJeremy Hahn 
An overview of the final three talks of the semester, followed by a construction of an E_{3} algebra structure on BP<n>. 

Apr 8 
Height shifting and canonical vanishingTomer Schlank 
A proof that K(BP<n>) has chromatic height n+1. Discussion of the canonical map from S^{1} homotopy fixed points to Tate fixed points. 

Apr 15 
Harvard wellness dayNo seminar 
Apr 22 
The Segal and LichtenbaumQuillen conjecturesElden Elmanto 
A proof that BP<n> satisfies higher height analogs of both the Segal and LichtenbaumQuillen conjectures. 
