The proposer aims to study homotopy theoretical properties of various moduli spaces. These are moduli spaces of manifolds with a geometric structure, for example a complex structure. The case of complex dimension one gives the moduli space of Riemann surfaces, which by now is relatively well understood. In contrast, much less is understood about moduli spaces of higher dimensional complex manifolds. This is project will investigate what can be said about this and related spaces using methods of homotopy theory. Another goal of this project is to study moduli spaces of manifolds with singularities. Here, the most basic case is the Deligne-Mumford compactification of Riemann's moduli space. For many application this space is more important than the uncompactified moduli space, and a homotopy theoretic understanding should be fruitful. In particular, the proposer will study applications to Gromov-Witten theory.
A manifold is a general notion of space. Manifolds are fundamental in geometry, topology, analysis, and other areas of mathematics. In physics, they are the underlying geometric structure in Einstein's general theory of relativity, and play fundamental roles in mechanics, quantum field theory, and probably many other areas. Studying manifolds from a mathematical perspective can be done in two ways: You can study the geometric properties each manifold, one at a time, or you can study geometric properties of the collection of all manifolds at once. The "collection of all manifolds" can itself be thought of as a kind of space, and each individual manifold is a point in this space. This space is an example of a moduli space, and the current project aims at understanding geometric and topological properties of this moduli space and its variations.
