Explicit Moduli Problems and Arithmetic Statistics

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One way to study complicated objects in mathematics is to compute simpler "invariants" of the object, which express some (but not complete) information about the object. This project studies invariants arising from objects associated to sets of polynomial equations, and how certain invariants are distributed when considering large families of these objects. For example, if the invariant is an integer, one may ask how likely the integer is, say, zero for a "random" object. Such results then translate into a better understanding of the initial equations and their behavior.This project involves problems using explicit constructions of moduli spaces in algebraic geometry, as well as applications of these parametrizations to obtain statistical information about arithmetic invariants, using methods ranging from classical algebraic geometry constructions to Lie theory and sieve techniques from analytic number theory. The PI proposes to work on finding correspondences between orbits of representations of algebraic groups and moduli spaces of curves, abelian varieties, and other varieties. The PI also plans to use these explicit descriptions of the moduli spaces, along with techniques from the geometry of numbers and analytic number theory, to study the asymptotic growth of arithmetic data related to the parametrizations. Such applications include studying the distributions of the ideal class groups of number fields, bounding the average rank of elliptic curves, and finding densities of curves with a given number of points.