Basic Notions Seminar talk on October 18, 2002

Speaker:   Dev Sinha (University of Oregon)

Title:   Geometry and Topology of Configuration Spaces

Abstract: 
    Given a space X, the space of configurations of n points in X, denoted C_n(X), is the subspace of the product X^n of all n-tupes (x_1, ..., x_n) where x_i = x_j only when i=j. Configuration spaces are particulary interesting when X is a manifold.

    Configuration spaces arise and touch on a number of fields within matematics. In this talk I will share some of my favorite facts about configuration spaces. These will fall roughly into three parts.
    1) The computation of H_*(C_n(R^k)) and its connection with Lie algebras. Here I will share my definition of the "Jacobi manifold", which gives a geometric incarnation (in terms of celestial mechanics, of sorts) of the Jacobi identity for a Lie algebra.
    2) The construction of a map from C_n(R^k) to the space of maps from S^k to itself, which places configuration spaces at the heart of algebraic topology.
    3) My elementary construction of compactifications of C_n(M) due to Fulton-MacPherson, Kontsevich, and Axelrod-Singer (who used these compactifications to gain insight into the topology of configuration spaces themselves, to define integrals which solved problems in deformation quantization and in knot theory, and to rigorously treat Chern-Simons theory in dimension three, respectively). These compactifications have many remarkable properties, and it is fun to understand them geometrically.

If I am running low on time, I will focus more on 1) and 3) than 2).