Joshua Graham - A Brief Introduction to Algebraic K-theory and why it matters

Talk given by Joshua Graham (University of New South Wales) at the Australian Postgraduate Algebra Colloquium on 12th May 2022.

Abstract: Algebraic K-theory arguably had its first appearance, albeit rudimentary, in the work of Grothendieck whilst he was studying a reformulation of the Riemann-Roch theorem. He did this by placing a group structure on the category of locally free coherent sheaves on a given scheme/variety (now called K_0). It was later found that the techniques behind his ideas had implications in other areas, particularly in vector bundles over compact Hausdorff topological spaces and also classifying finitely generated projective modules.

The study of Whitehead torsion in topology later motivated the definition of a different functor (now called K_1) which was found to fit an exact sequence that related the K_0 and K_1 functor, and it was this idea that lead mathematicians (particularly Daniel Quillen) to extend these K functors to all n \ge 0. We will look at the details of this construction as well as some important applications of these groups.

[john-the-cook]

I'm assuming this is not the "Joshua Graham" from "fallout 4"... (the game).
🤔😉🙂

[stevelam]

What are we even talking about here? Is that representation theory? Topology? Group theory? All of them combined?