CAREER: Theory, Heuristics, and Data for Arithmetic Invariants

One way to study complicated objects in mathematics is to compute simpler invariants of the object, which provide information about the object. Objects associated to sets of polynomial equations, called algebraic varieties, are fundamental in many areas of mathematics and model numerous real-world phenomena. This project studies invariants arising from algebraic varieties, especially how certain invariants are distributed when considering large families of them. For example, if the invariant is an integer, one may ask how likely the invariant is, say, zero for a "random" object. Such results then translate into a better understanding of the solutions of the given polynomial equations. Along with the research proposed, the PI will organize a range of outreach activities, including after-school and weekend activities for school-age girls, workshops for graduate students, and regional workshops for students and postdocs. The research in this project is at the intersection of algebraic and analytic number theory, algebraic geometry, and representation theory, and focuses on distributions of arithmetic statistics. One may ask for not only theoretical results but also heuristic predictions, as well as computational data to predict or verify conjectures. The PI will study all three aspects--theoretical results, heuristics, and data--concerning questions about class groups of number fields, ranks of elliptic curves, and other invariants of algebro-geometric objects. The PI intends to use methods from classical algebraic geometry, Lie theory, random matrix theory, and sieve techniques from analytic number theory to pursue the proposed research. 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.