Content: Algebraic K-Theory

Algebraic K-theory associates to a ring a graded abelian group. This invariant has a rich history and numerous applications. For example, the algebraic K-theory of group rings has applications to geometric topology, the algebraic K-theory of rings of integers in number fields has applications to number theory, and algebraic K-theory of the coordinate ring of an affine variety has applications to algebraic geometry. Following the insight of Quillen, Segal, and Waldhausen it is beneficial to generalize the input to a category and the output to a space whose homotopy groups are the algebraic K-theory groups. Therefore category theory and topology in addition to algebra are fundamental to the subject.

In this course, we will survey different constructions of algebraic K-theory, describe some fundamental properties of algebraic K-theory, and discuss some important applications. The prerequisites are Topology I-III or equivalent. The course will be targeted at second year masters students who have finished the Topology sequence.