Many important questions in number theory concern the statistical behavior of numbers. For example, a classical such question asks what the chance is that a random integer has no repeated prime factors. The PI will investigate several important open questions concerning the statistical behavior of the integers and related objects of interest. The PI has recently made progress toward these questions by bringing in new ideas from several disparate areas of mathematics. The PI plans to continue investigating these connections and deepening our understanding of them. Simultaneously, the PI proposes to support and mentor mathematicians at different levels of their educational career. The project is to work on the main conjectures of arithmetic statistics over function fields. These conjectures include the Cohen-Lenstra conjectures on class groups of quadratic fields, Malle's conjecture on counting global field extensions with specified Galois group, and the Poonen-Rains conjectures on ranks and Selmer groups of elliptic curves. The main idea is to develop tools in topology to compute relevant stable and unstable homology groups. The project will combine ideas in number theory and algebraic geometry with ideas in topology and higher algebra to make progress toward these conjectures. The PI also plans to pursue projects in other areas, such as the Putman-Wieland conjecture. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation