MATH 7350: Topics in Algebra: Algebraic Differential Algebra (Fall 2009)

Instructor: Richard Vale

This course concerns noncommutative algebras (usually over the complex numbers) and their representation theory. Our main aim will be to work through the fascinating 1995 paper of Cuntz and Quillen — Algebra extensions and nonsingularity — in which various notions of algebraic geometry, such as smoothness, are developed in the noncommutative setting, building on some ideas introduced by Connes in the setting of C* algebras. I hope to supplement this by providing background topics and interesting examples, and also results from related papers.

Topics covered will include: Morita theory; Hochschild homology and cohomology; deformations; differential forms; quasi-free algebras; connections on modules; geodesic flow, and probably cyclic (co)homology.

Background material will be taken from various sources including Ginzburg's Lectures on noncommutative geometry (arXiv:math/0506603v1). Prerequisites for the course are familiarity with homological algebra (complexes, exactness, Ext, Tor, etc.) and some basic affine algebraic geometry. Knowledge of spectral sequences will not be assumed.