The study of the (co)homology of various moduli spaces, mapping class groups and arithemtic groups is a central topic in topology, with connections to algebraic geometry, number theory, combinatorial group theory and more. The problems on which the PI proposes to work include the following. 1. Confirming a broad conjectural picture of the cohomology of pure braid groups, Torelli-type groups, and congruence subgroups. Such conjectures are phrased in the language of {em representation stability}, a theory (recently discovered by the PI and T. Church) that imports representation theory as a powerful new tool into the study of homological stability phenomena. 2. Applying cohomological computations to computing arithmetic statistics for algebraic varieties, for polynomials, and for maximal tori in algebraic groups over finite fields. 3. Giving a deeper geometric understanding of the Morita-Mumford-Miller classes via a remarkable coincidence between certain characteristic numbers, as discovered by the PI and T. Church. 4. Constructing a large number of linearly independent unstable cohomology classes in mapping class groups and arithmetic groups. While it has been indirectly deduced that super-exponentially many such dimensions of such cohomology must exist, almost no such classes are known. The technique proposed here is a new one, using torsion groups to detect rational homology classes. 5. Constructing $p$-torsion in the homology of level $p$ congruence subgroups of arithmetic groups, mapping class groups, and other groups. Again the technique here is new, and has already been applied successfully by the PI and T. Church.
Moduli spaces, or the spaces of shapes, are fundamental objects in mathematics. Understanding their structure and describing their basic topological properties is an important problem. Such descriptions are needed if one wants to understand the evolution of shapes over time, or if one wants to find the most efficient shape needed to solve a problem. The problem is the topological structure of moduli spaces is extremely complicated to describe. The purpose of this proposal is to apply the powerful machinery of representation theory in order to give a simpler, easier-to-work-with encoding of these complicated structures. The PI and T. Church discovered that such a language is applicable to structures all over mathematics, allowing for new descriptions and new insights into the structure of complicated objects. The PI proposes to apply these ideas to a variety of problems, with applications to topology, Lie algebras, and counting problems in number theory.
