MATH 608K: Algebraic K-Theory

Textbook

J. Rosenberg, Algebraic K-Theory and its Applications, Springer, ISBN 0-387-94248-3.

Description

This course will provide an introduction to the subject known as algebraic K-theory. Here is a capsule description of the subject, taken from the preface (written by Dan Grayson and Eric Friedlander) to the just-released Handbook of K-theory:

Informally, K-theory is a tool for probing the structure of a mathematical object such as a ring or topological space in terms of suitably parameterized vector spaces. Thus, in some sense, K-theory can be viewed as a form of higher order linear algebra that has incorporated sophisticated techniques from algebraic geometry and algebraic topology in its formulation. K-theory has its origins in A. Grothendieck's formulation and proof of his celebrated Riemann-Roch Theorem in the mid-1950's. (See for example the classic paper of Borel and Serre explaining Grothendieck's idea.)

The subject of K-theory has applications in number theory, algebraic geometry, geometric topology, symbolic dynamics, functional analysis, and even mathematical physics. Thus the course could go in many different directions, depending on the interests of the audience. I plan to go over the essentials of "classical K-theory" (the definitions and properties of K₀, K₁, K₂), and then decide what to do afterwards depending on the interests of the class. I will assign a limited amount of homework to help people learn the fundamentals.