Hermitian K-theory and derived categories

Lead Research Organisation: University of Warwick
Department Name: Mathematics

Abstract

Dylan's project aims to generalize a result of mine and Hornbostel to DVRs with residue field characteristic 2. One hopes to relate the Hermitian K-theory of the DVR with the hermitian K-theory of residue and fraction fields. This requires understanding how derived equivalences influence spaces of quadratic forms. The longterm goal is to compute the symplectic K-theory of the integers which would have applications to the homology of the stable mapping class group.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509796/1 01/10/2016 30/09/2021
1789390 Studentship EP/N509796/1 03/10/2016 30/09/2020 Dylan Madden
 
Description Our goal since last year has been to calculate 'Hermitian K-groups of Burnside form rings'. The 'classical' Hermitian K-theory groups encode information about symmetric bilinear or quadratic forms over rings with involution. However, the interaction between these groups and derived category methods motivate taking a viewpoint which can encompass anything which deserves the name 'form'; this viewpoint has recently been developed by Schlichting.
Instead of rings with involution, we take the more general viewpoint of 'form rings'. Burnside form rings are a particularly natural class of form rings, and we aim to calculate their Hermitian K-groups. We have done this in various generalities and for various examples.
Exploitation Route The specific goal of our current programme of research was to calculate the 0th Hermitian K-group of the integer Burnside form ring, which is of fundamental importance, being the initial object in the category of form rings. This has not been achieved yet. If it is not done by the end of the available time, my thesis may be of help to others who wish to attempt to calculate it.
Sectors Education,Other