Cyclic Cohomlogy and Noncommutative Geometry

A brief history of the subject:

The ideas of non commutative space , noncommutative geometry and accompanying invariants like K-theory, K-homology and cyclic cohomology grew out of attempts to solve several outstanding problems in topology ( Novikov conjecture ) and analysis ( extensions of Weyl-Von Neumann theorem , extensions of index theorem to foliated spaces ) . Cyclic cohomology replaces de Rham homology in noncommutative setting . interestingly enough , very recently noncommutative geometry has found some applications and interactions with particle physics. This course is an introduction to cyclic cohomology and noncommutative geometry of Alain Connes and should be of interest to graduate students ( and faculty members ) in mathematics and elementary particle physics . The lectures will be self-contained and I will cover the necessary background material as it may be required.

Outlines and Lecture notes:

• Fundamental duality between spaces and commutative algebras rudiments of topological algebras and topological K-theory
• cyclic cohomology and its basic properties ( Morita invariance , homotopy invariance , operations )
• computations ( smooth functions - relations with de Rham cohomology , stable matrices , relations with Lie algebra cohomology ) .

Marking Policy:  Students are expected to prepare and present reports on selected topics of current interest . I will choose these topics in consultation with each student .

References:

[1] Noncommutative Geometry by A. Connes published by Academic Press, 1994 .

[2]Cyclic Homology , second edition by J. L. Loday published by Springer Verlag, 1998.