Summary

The eCHT Foundations of Hermitian K-Theory Reading Seminar is an online reading seminar in Fall 2021. The target audience is graduate students and postdoctoral faculty, although more senior faculty are also welcome.

In its original form, Hermitian K-theory is the study of unimodular quadratic and symmetric forms and their isometry groups using the techniques of algebraic K-theory as introduced by Quillen. So far, it has been most accessible for rings in which 2 is invertible. To obtain a satisfactory theory in general it is advantageous to study derived generalisations of quadratic and symmetric forms, as formalised by Lurie’s notion of Poincaré categories.

The seminar will consist of 4 thematic blocs:

• We will start by introducing and studying Poincaré categories. Associated to such is a Grothendieck-Witt spectrum and an L-spectrum, which we will study in detail.
• We will study Ranicki’s algebraic surgery and how it can be used to analyse Grothendieck-Witt and L-theory of rings.
• We prove the universal properties of Grothendieck-Witt theory and L-theory, viewed as invariants of Poincaré categories, and establish a fundamental fibre sequence relating algebraic K-theory, Grothendieck-Witt theory and L-theory.
• Finally, we apply the results to determine the Grothendieck-Witt theory of the integers.

Prerequisites

Familiarity with (stable) infinity-categories, or at least the willingness to take the machinery for granted, will be required. In addition, familiarity with (stable) homotopy theory covering spectra and E-infinity-spaces would be very helpful. Previous exposure to algebraic K-theory might be helpful as well.

Student Expectations

Students will take turns giving presentations and are expected to attend all presentations.

Over the course of the semester, each student will give a 60 minute lecture via Zoom. For examples of possible formats, see recordings of recent eCHT presentations. In addition, we will meet for another 30 minutes after each talk, where the organisers will have prepared additional examples and addenda to the lectures.

Students are encouraged to skim the source material being presented each week and post a response in the private seminar workspace prior to that week’s talk. These responses can be in the form of comments or questions related to the material. The shared online workspace will also serve as a forum where students can ask questions as they prepare for their presentations.

Students are encouraged to obtain independent study credit from their home universities for their participation in the eCHT Seminar, but this is not required.

Organizers

The seminar is run by Fabian Hebestreit (Bonn) and Markus Land (Copenhagen), with the assistance of Danika Van Niel (Michigan State University). Contact any of us for more information.

Application

Please submit the application form to apply. Applications received by August 25 will receive full consideration.  

Schedule

There will be an organizing meeting on Monday, September 6 at 10:00am Eastern. After that, the seminar meets regularly at 10:00am Eastern on Thursdays for talks, starting September 9 and ending December 16.