Math 8390  
                 Topics in Mathematical Physics
		      String Topology

Time: 2:30pm-3:20 p.m. MWF
Room: VinH 207
Instructor: Sasha Voronov

Topics: String Topology, including intersection theory on loop spaces,
the Pontryagin-Thom construction, BV algebras, Hochschild cohomology,
PROPs and operads, Topological and Conformal Field Theories, Morse
theory on loop spaces, Symplectic Field Theory

In this one-semester course, I plan to give an introduction to the new
subject of String Topology, which started from a 1999 paper by M. Chas
and D. Sullivan, who introduced new invariants of manifolds, defined
in the topological setting by analogy with String Theory and in
particular Gromov-Witten invariants in symplectic and algebraic
geometry. String Topology was quickly recognized as a fascinating and
powerful subject and has been developing quite actively since then. I
will be following the outline of a survey paper "Notes on String
Topology" of R. Cohen and myself, see

http://front.math.ucdavis.edu/math.GT/0503625,

which we are currently expanding to a book. I do not know if the
lectures will be following the future book or the book will be
following the lectures, but the element of freshness and novelty will
definitely be present in the course.

No exams will be given or homework assigned, as the material is new
and keeping up with the course will already require a certain amount
of work.

Prerequisites: You need to be familiar with some topology, including,
manifolds, homotopy, and homology. No knowledge of physics or advanced
topics in topology is required - all what is needed will be introduced
along the way.

Outline of the course: intersection theory in loop spaces; BV
algebras; the Pontryagin-Thom construction; the Chas-Sullivan loop
product and bracket; Hochschild cohomology and Chen integrals;
introduction to PROPs and operads; the little disks operad and BV
algebras; the cactus operad and its action on a loop space via
correspondences; 2d Quantum Field Theories, including Topological,
Conformal, and Motivic ones; open-closed string topology; introduction
to Morse theory; the energy functional and the symplectic action,
leading to Morse theory on the loop space; higher-dimensional theory:
"brane" topology, sphere spaces, and higher Hochschild homology;
Symplectic Field Theory.

If you contemplate taking the course, please, make sure to REGISTER -
otherwise, the course may not run, and you will have no choice. You
may register on line by going to "Schedule Fall 06" on our
departmental web page. Note that although Math 8601 is listed as a
prerequisite, it is not supposed to be one, and you may simply ignore
it.