# Mikala Ørsnes Jansen: Reductive Borel–Serre & Unstable algebraic K-theory

**Time: **
Wed 2022-11-23 13.15

**Location: **
Albano, Cramér room

**Participating: **
Mikala Ørsnes Jansen (Köpenhamn)

**Abstract**

Let *R* by a ring. The term unstable algebraic K-theory will refer to any (family of) anima \(K(R,n)\) built entirely out of linear algebra internal to *R ^{n}* through which the canonical maps \(BGL_n(R) \to K(R)\) factorise. A classical example is Quillen's plus-construction \(BGL_n(R)^+\). Ideally, we want a model for unstable algebraic K-theory to be closer to \(K(R)\) than \(BGL_n(R)\) is in terms of its nature and properties; for example the fundamental group of the plus-construction is closer to \(K_1(R)\) than \(GL_n(R)\) is. The term unstable algebraic K-theory was used in the 1970's by Dennis and Stein in a survey of the functor \(K_2\), and classically unstable algebraic K-theory has been used to derive many important computational results about ''stable'' algebraic K-theory. We introduce a new model for unstable algebraic K-theory inspired by a detailed study of the so-called reductive Borel–Serre compactification of locally symmetric spaces. In this talk I will mention the main results and calculations of this work and also attempt to shed light on the geometric origins of the model, as this is an important and interesting aspect of the story.

This is joint work with Dustin Clausen.