Noncommutative Geometry and Cyclic Homology

General Inforamtion:

This course is an introduction to current ideas and methods of noncommutative geometry and cyclic homology as has been developed in Alain Connes' program. since 1980.

Outlines and Lecture notes:

• Introduction . Euclid, Riemann and Connes: A lightning introduction to geometric thinking through history. A brief review of algebraic methods á la Descartes, Gelfand, Grothendieck and Connes: Noncommutative space.
• Associative and Lie algebra theory by examples.
• Hochschild and cyclic homology.
• Computations.
• The return of infinitesimal methods: Dixmier Trace.

Note:  While a mature underestanding and knowledge of graduate level geometry, algebra , topology and analysis would be helpful , I will do my best to make this course as self contained as possible.

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] A. Connes, Noncommutative geometry, Academic Press, 1994.

[2]J. Cuntz and M. Khalkhali , Noncommutative geometry and cyclic cohomology, Fields Institute Communications 17, American Mathematical Society, 1997.

[3] J. L. Loday, Cyclic homology, Springer Verlag, 1993.