Chen Group

Geometric Topology of Manifolds

Manifolds are central objects studied in geometry and topology. They are models of the space-time in which we live, and are the foundational concept in many physical theories. On the other hand, manifolds are very useful in the applied sciences: the data cloud in an experiment can form the shape of a manifold, and studying the properties of this manifold will, in turn, provide information on the nature of the data.

The Chen group studies a variety of questions related to manifolds (usually smooth or with more structure) and also general topological spaces. Most of these are concerned with moduli spaces. We are particularly interested in the following simplest example of moduli spaces, called configuration spaces: Given a manifold/space X and a natural number n, we can form a new space Conf(X,n), the points in which are the configurations of n ordered points in X, but the points are not allowed to coincide with each other.

In addition, we study some manifold “invariants” (meaning simpler quantities extracted from a manifold, like numbers or groups) defined by counting points in configuration spaces that satisfy some conditions, as well as the algebraic structures related to configuration spaces (e.g., operads).