Stability Phenomena in Topology and Number Theory

Algebraic topology is the study of geometric objects using certain tools called algebraic invariants. One of the most powerful such tools in algebraic topology is the theory of homology. Homology is a way of measuring holes in a geometric object called a topological space. It has had many applications both within pure mathematics and in the physical and social sciences. For example, many phenomena in electricity and magnetism, like the Aharonov-Bohm effect, are homological in nature. Homology can also be used to show the existence of Nash equilibria in economics. The aim of this project is to explore patterns in the homology of sequences of spaces and to use these patterns to better understand other areas of mathematics such as number theory. Training graduate students and postdocs is also key to the mission of the grant. Some specific goals include establishing homological stability and its variants for a wide range of moduli spaces and classifying spaces. These include classifying spaces of arithmetic groups, moduli spaces of graphs, and Hurwitz spaces. Stability for these spaces is important due to connections with analytic number theory over function fields, algebraic K-theory, and the theory of multiple polylogarithms. Techniques from homological algebra, operads, Hopf algebras, locally symmetric spaces, Étale cohomology, graph complexes, and combinatorial simplicial complexes will be employed to establish and apply homological stability. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.