Parts of the research conducted under this grant will concern the study of manifolds and their moduli. Manifolds are a generalized concepts of space, appearing all over mathematics and science. The defining property of a manifold is that it is locally parametrized by a finite number of parameters (for example, the surface of the earth can locally be parametrized by two parameters, namely longitude and latitude). A classical topic in algebraic topology, Galatius' field of research, is the classification of manifolds: How does one decide whether two manifolds are isomorphic, and how does one write a list of all possible manifolds? The research conducted here concerns the related question of classifying how manifolds may vary in families: what are their symmetries and what are their deformations. The research conducted under this grant will apply new methods to study these classical and important questions. Another part of the research conducted under this grant concerns deformations of objects arising from number theory, studied using methods from homotopy theory.
The proposed activity will continue the development of a relatively new area of mathematics in which homotopy theory is used to study various moduli spaces. An important early result this area is the solution, by Madsen and Weiss, of a conjecture of Mumford. Significant new developments have happened since then, many of which, such as the solution to a 1995 conjecture of Hatcher, are due to the principal investigator. This subject lies in the overlap between algebraic topology and other branches of mathematics, with expected applications in algebraic geometry, symplectic geometry, and possibly theoretical physics. Another part of the project concerns applications of homotopy theoretic ideas in number theory. Working in a derived setting is a very familiar idea in homotopy theory, and has often been successfully imported into algebra and other fields. Recently, the use of a derived setting for the study of deformations has received much attention. The project will develop and apply these techniques to Galois representations, objects of fundamental importance in number theory.