Schafer-Rosenberg RIT on non-commutative geometry and L2-cohomology

Subject:

This RIT will deal with various applications of non-commutative algebra (group rings, operator algebras, etc.) to geometry and topology. This is one of the most active areas in geometry and topology at present, and also accounts for much of the current interest in operator algebras.

References to get started:

Possible Topics for Spring 2003:

  1. The paper of Grigorchuk, Linnell, Schick and Zuk, "On a conjecture of Atiyah", which gives a counterexample to the strong version of the Atiyah conjecture. This can be read in conjunction with the paper by Dicks and Schick: "The spectral measure of certain elements of the complex group ring of a wreath product".
  2. R. Roy's counterexample to the trace conjecture and Lück's proof of a weakened version, given the Baum-Connes conjecture. These papers are more of an index-theory nature than about L2-cohomology, but should be accessible and have obvious possible thesis problems attached. Roy's paper is in K-Theory 17 (1999), pp. 209-213, and Lück's paper is available at the Inventiones Math. site. (Search on "trace conjecture" in the title field.)
  3. The algebraic approach to L2-invariants and the extension of L2-invariants to modules over the group von Neumann algebra. This is available in our notes, papers by Farber, and a paper of Lück in Math. Annalen 309 (1997), pp. 247-285.
  4. A paper by Lück in Topology 33 (1994), pp. 203-214, entitled "L2-Betti numbers of mapping tori and groups". This paper is more of a topological character.
  5. Novikov-Shubin invariants. There are sections in the notes and a paper by Lück, Reich and Schick entitled "Novikov-Shubin invariants for arbitray group actions and their positivity" in Contemporary Math. 231 (1999), pp. 159-176.
  6. The paper by Farber and Weinberger giving a counterexample to the zero-in-the-spectrum conjecture.

Meetings:

For Fall 2003 we will not have regular meetings but will function more as a reading course. We may get together from time to time when someone has something to report on to the group. Those interested in participating should contact one of the two leaders.

Contacts:

James Schafer, room 2112 (phone 55164) Jonathan Rosenberg, room 2114 (phone 55166) or 1106 (55059)

Possible Undergraduate Projects:

See this poster.