Cornell Math - MATH 751, Fall 2000

MATH 751 — Fall 2000
Berstein Seminar in Topology

Two short papers by Maxim Kontsevich, written in the early 1990's, relate invariants of certain topologically important groups, including mapping class groups and outer automorphism groups of free groups, to invariants of certain infinite-dimensional Lie algebras. The connection is made via various complexes of finite graphs, and a variation of these complexes leads to Kontsevich's graph cohomology, which has important applications to quantum field theory. Applications are also indicated to other topological and geometric invariants such as Vasiliev invariants of knots and Chern-Simons invariants of 3-manifolds, as well as to other topics in mathematical physics.

These papers are difficult to understand, partly because they involve so many different areas of mathematics, and partly because few proofs are given, and those that are given are only sketched. The goal of this seminar is to make progress towards understanding the theory described in these papers, starting from ground zero. We will draw on the experience of students in various fields for basic definitions and theory, as well as working on understanding the definitions, constructions and proofs in Kontsevich's papers. The papers are

• Formal (non)commutative symplectic geometry.  The Gelfand Mathematical Seminars, 1990--1992, 173--187, Birkhäuser Boston, Boston, MA, 1993.
  
• Feynman diagrams and low-dimensional topology.  First European Congress of Mathematics, Vol. II (Paris, 1992), 97--121, Progr. Math., 120, Birkhäuser, Basel, 1994.

I will begin with an introduction to these papers, and we will proceed as usual in the Berstein seminar with student lectures.