Mathematics 790STC - CharacteristicClasses&K-Theory

This class is a second course in algebraic topology, taking the next steps after (co)homology and homotopy. The central objects of study are vector bundles, which are families of vector spaces parametrized by a topological space. Associated to a vector bundle are cohomology classes called characteristic classes, which provide a measure of how ``twisted" the bundle is. The first part of the course will focus on motivating examples, the axiomatic approach, and culminate with the notion of a classifying space, which provides a concrete source for characteristic classes. The second part of the course will develop the natural context for studying vector bundles: K-theory, as a generalized cohomology theory. We will also develop special features of K-theory, notably Bott periodicity and the Chern character, which relates K-theory to characteristic classes. Additional topics such as cobordism, the Atiyah-Singer index theorem, or Adams operations and the Hopf invariant one problem may be discussed subject to time and student interest.

MATH 672 Prerequisites: Math 672 or the consent of the instructor.

Familiarity with the basics of manifolds will be helpful.