Arithmetic of configuration spaces

The aim of this project is to explore the wealth of deep connections lying at the interface of three subjects: algebraic topology, quantum algebra, and statistical problems in arithmetic. A meeting point of these subjects is in the study of moduli spaces: these are spaces which parameterize mathematical objects of a given sort. Deep questions in arithmetic may be formulated in terms of properties of these spaces, and those properties may be studied through the lens of algebraic topology. Remarkably, the sort of answers that topological tools provide has a close connection to the study of quantum groups and related objects arising in quantum physics. The main goals of this proposal are two-fold; the first is to more deeply explore how algebro-topological tools can address open questions in quantum algebras, such as the conjecture posed by Andruskiewitsch-Schneider on the generation of pointed Hopf algebras, and the conjecture of Etingof-Ostrik on the cohomology of finite tensor categories. The second is to use these tools, in concert with machinery of the Weil conjectures, to enumerate points on moduli spaces over finite fields, thereby establishing statistical conjectures on the distribution of Galois groups (e.g., those of Malle and Cohen- Lenstra), or moments of L-functions (e.g., those of Conrey-Keating-Farmer-Rubinstein-Snaith). 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.