MATH748R
Selected Topics in Geometry and Topology:
Noncommutative Geometry and Topology

Fall 2007

Course web site:

http://www.math.umd.edu/~jmr/748R/

Meeting times:

MWF, 10:00am-10:50am (MTH 0102). Important note: The course will not meet on Wednesday and Friday, September 5th or 7th, since I will be at an international conference on Noncommutative Geometry, nor on the Fridays of September 14th and 28th and October 5th, because of the Jewish religious holidays. If paticipants are interested, I will arrange makeup times.

Instructor:

Professor Jonathan Rosenberg. His office is room 2114 of the Math Building, phone extension 55166, or you can contact him by email. Office hours are Mondays and Wednesdays after class, or by appointment.

Texts:

The foundation of this subject is the remarkable book Noncommutative Geometry by Alain Connes, Academic Press, 1994, ISBN 0-12-185860-X. From UMCP computers you can view the MathSciNet featured review by John Roe here. More detailed reviews from the AMS Notices are this one by Vaughan Jones and Henri Moscovici and this one by Andrew Lesniewski. Connes has now made the whole book available free on-line at his web page, along with a draft of a book in progress by Connes and Marcolli, and the text of many of his papers, including the key Comptes Rendus note called "C* algebras and differential geometry" (available in both the French original and a new English translation). Other useful books are Joseph Varilly, An introduction to noncommutative geometry, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2006; Matilde Marcolli, Arithmetic noncommutative geometry, University Lecture Series, 36, American Mathematical Society, Providence, RI, 2005; and Nigel Higson and John Roe, editors, Surveys in Noncommutative Geometry, Clay Math. Inst./American Math. Soc. publications, vol. 6, American Mathematical Society, Providence, RI, 2006.

I have two sets of lecture notes on line which may be helpful: Applications of Non-Commutative Geometry to Topology and Applications of noncommutative topology in geometry and string theory.

Prerequisites:

To learn this subject thoroughly you need a background in both geometry/topology and in functional analysis, but for purposes of this course, the equivalents of MATH 730 and of MATH 630 should suffice if you are willing to take some things on faith. We will develop from scratch most of the operator theory we need.

Course Requirements:

There will probably be a few homework assignments, but there will be no exams. At some point participants may be asked to present some material to the rest of the class.

What is noncommutative geometry?

A basic notion in mathematics, going all the way back to Descartes, is that we study a space by means of functions on the space. In fact, the algebra of functions "determines" the space.

Examples of this principle:

Algebraic Geometry: R a commutative ring, Spec R a scheme.
Gelfand-Naimark correspondence: X a locally compact Hausdorff space, C₀(X) a commutative C*-algebra (a Banach algebra that can be realized as a norm-closed *-algebra of bounded operators on a Hilbert space).

Quantum mechanics, however, suggests that some physical systems should be modeled by ``spaces'' on which ``functions'' are not commutative. C*-algebras are natural models for the function algebras, since they have a good structure theory and since quantum mechanics demands that observables be self-adjoint operators on some Hilbert space.

Example: a spinning electron, with two [pure] states: "up" and "down."

Semiclassical model: space S⁰ of two points, possibilities of transitions between them.
Quantum model: space with "functions" M₂(C). Generators are matrix units e_ij with relations e_ije_kl=e_il when j=k.

Second Example from Representation Theory: Take G to be a finite group. The noncommutative space G^ has "algebra of functions" the group ring CG.

Fields Medalist Alain Connes has pursued the idea of noncommutative spaces to a much greater extent, and has developed both "topology" and "differential geometry" on such noncommutative spaces. We will go over the foundations of his theory and then develop a number of very interesting applications to ordinary geometry and topology. There are also many applications to mathematical physics, though we probably won't have time to do much more than mention these in passing. Recently, noncommutative geometry has begun to play a big role in number theory as well, and we will try to explain a bit of this connection.

MATH748R Selected Topics in Geometry and Topology: Noncommutative Geometry and Topology