The proposed research 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. Their result concerns the group of symmetries of closed two-dimensional manifolds or, equivalently, it concerns the moduli space of such manifolds. A major part of Galatius' proposed activity will study the extension of this theory to manifolds of higher dimension. 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.
Galatius' research will concern the study of manifolds and their symmetries. A manifold is a generalized versions 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 real parameters (for example, the surface of the earth is locally parametrized by two parameters, longitude and latitude). A classic 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. A related classical problem is the classification of symmetries of manifolds (for example, rotation about an axis is a symmetry of the sphere). Galatius' research will apply new methods to study these classic questions.