Thursday 2021 Spring

Spring 2021 Thursday Seminar

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


Feb 4

Overview of redshift

Mike Hopkins

Feb 11

Introduction to algebraic K-theory

Peter Haine

Overview of the algebraic K-theory functor, including the plus construction, the S construction, and the universal property as a localizing invariant. Notes.
Feb 18

The Land--Tamme theorem

Ian Coley

A discussion of the K-theory of pullbacks. Notes.
Feb 25

Chromatic homotopy theory

Ishan Levy

Basics of chromatic homotopy theory, including telescopic localizations and the Bousfield--Kuhn functor. Notes.
Mar 4

Purity in chromatically localized algebraic K-theory

Shachar Carmeli

Discussion of Section 3 of the Land--Mathew--Meier--Tamme work, ending with the proof of Theorem 3.8. Also, the beginning of our discussion of the Clausen--Mathew--Naumann--Noel work. Notes Part 1. Notes Part 2.
Mar 11

Blueshift and a converse for highly structured ring spectra

Adela 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 K-theory via group actions

Lucy Yang

Discussion of the Clausen--Mathew--Naumann--Noel work, focusing on Section 4. Notes.
Mar 25

THH and TC

Nat Pacheco-Tallaj

The definitions of THH and TC. Statement of the Dundas--Goodwillie--McCarthy theorem. The computations of THH(Fp) and TC(Fp), exhibiting redshift from height -1 to height 0. Notes.
Apr 1

The multiplication on truncated Brown--Peterson spectra

Jeremy Hahn

An overview of the final three talks of the semester, followed by a construction of an E3 algebra structure on BP<n>.
Apr 8

Height shifting and canonical vanishing

Tomer Schlank

A proof that K(BP<n>) has chromatic height n+1. Discussion of the canonical map from S1 homotopy fixed points to Tate fixed points.
Apr 15

Harvard wellness day

No seminar

Apr 22

The Segal and Lichtenbaum--Quillen conjectures

Elden Elmanto

A proof that BP<n> satisfies higher height analogs of both the Segal and Lichtenbaum--Quillen conjectures.


  1. A universal characterization of higher algebraic K-theory.
  2. On the K-theory of pullbacks.
  3. Purity in chromatically localized algebraic K-theory.
  4. A short proof of telescopic Tate vanishing.
  5. Descent and vanishing in chromatic algebraic K-theory via group actions.
  6. On topological cyclic homology.
  7. Redshift and multiplication for truncated Brown-Peterson spectra.
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