Stabilization and destabilization of moduli spaces

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In my work I apply the methods of algebraic topology to problems in differential topology, low-dimensional topology, and number theory. I do so by studying the homotopy type of moduli spaces. Moduli spaces contain information about mathematical objects of a given type, and families thereof. By applying topological invariants to moduli spaces you learn a lot about the objects in question. Many moduli spaces depend on a parameter, and it is a surprising phenomenon that they become easier to understand as we let the parameter go to infinity. The theme of my research is that you can recover information about the individual moduli spaces from this limiting value, and vice versa; "destabilization" and "stabilization". My first goal is to find rational models of moduli spaces of high-dimensional manifolds, or equivalently their diffeomorphism groups. To do so I will build on methods developed to understand diffeomorphisms of disks in joint work with Oscar Randal-Williams. Diffeomorphisms of disks are the universal correction term between diffeomorphisms and self-embeddings of a given manifold. We may thus "stabilize" the disk by adding handles, study the diffeomorphisms and self-embeddings of the resulting high-dimensional analogues of surfaces, and then "destabilize" by comparing these. Once we understand the correction term, we use rational models for spaces of self-embeddings to give similar models for diffeomorphisms. Through a remarkable algebraic coincidence, this has deep applications to Torelli groups of surfaces. Together with my collaborators, postdoctoral fellows, and students, I will work out this program and explore its applications to the study of manifolds and other fields such as symplectic and Riemannian geometry. My second goal is to understand the relationship between the homology of general linear groups of a ring and its algebraic K-theory. For fields, I want to relate several filtrations on algebraic K-theory. On the one hand, the "motivic filtration" contains deep number-theoretic information. On the other hand, the "rank filtration" is related to homological stability for general linear groups. In joint work with Soren Galatius and Oscar Randal-Williams I proved that the rank filtration has a vanishing line conjectured by Soule and Beilinson for the motivic filtration. Clarifying the relationship between these two filtrations is thus of great importance, and I propose two lines of investigation to do so. I will work on this with my collaborators and postdoctoral fellows. For rings of integers, I am interested in the (co)homology of arithmetic groups with coefficients in a representation, such as the Steinberg module. These are conjectured to vanish in a range, and if so, this gives new results on homological stability and Vandiver's conjecture. I want to relate these questions to Voronoi complexes and perfect forms. Extensions to other rings make good projects for students.