Explicit Moduli Spaces and Arithmetic Applications

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Studying polynomial equations and their solutions dates back thousands of years. For certain types of polynomials, understanding these solutions has important applications, ranging from engineering to biology to elliptic curve cryptography. It is still a very difficult question to determine how many solutions an arbitrary polynomial equation might have, especially if one looks specifically for solutions lying in the natural numbers or the rational numbers. A simple example of this phenomenon is Fermat's Last Theorem, which states that there are no positive integer solutions to an equation of the form an n-th power is equal to the sum of two n-th powers if n is an integer larger than 2, so for example a cube of a positive integer cannot be equal to the sum of two cubes of positive integers. While the theorem is easy to state, the method of proof involves many deep ideas in number theory and arithmetic geometry, including the idea of studying many such polynomial equations all at once (the idea of "moduli spaces"). The PI intends to study such spaces and use them to understand properties of certain types of polynomials and associated geometric objects, such as elliptic curves.The PI works in the intersection of three fields: number theory, algebraic geometry, and representation theory. The main theme of this proposal involves the use of representation theory to explicitly construct moduli spaces in algebraic geometry; these constructions also may be applied to solve counting problems in number theory by using geometry-of-numbers techniques. The PI intends to use methods and tools developed in all of these subjects over the last hundred years, including constructions from classical algebraic geometry, ideas from Lie theory, and sieve techniques from analytic number theory.