Explicit Moduli Spaces and Arithmetic Applications

对多项式方程及其解的研究可以追溯到几千年前。对于某些类型的多项式，了解这些解具有重要的应用，从工程到生物学再到椭圆曲线密码学。确定一个任意多项式方程可能有多少解仍然是一个非常困难的问题，特别是如果一个人专门寻找位于自然数或有理数中的解的话。这种现象的一个简单例子是费马最后定理，该定理指出，如果n是大于2的整数，则n次方等于两个n次方的和的形式的方程没有正整数解，因此，例如，正整数的立方体不能等于两个正整数立方体的和。虽然这个定理很容易表述，但证明方法涉及数论和算术几何中的许多深层思想，包括同时研究许多这样的多项式方程的思想(“模空间”的思想)。PI的目的是研究这种空间，并利用它们来理解某些类型的多项式和相关几何对象的性质，如椭圆曲线。PI的工作涉及三个领域：数论、代数几何和表示论。该方案的主要内容是利用表示论来显式地构造代数几何中的模空间，这些构造也可以用于利用数的几何技巧来解决数论中的计数问题。PI打算使用过去一百年来在所有这些学科中发展起来的方法和工具，包括来自经典代数几何的构造，来自李论的想法，以及来自解析数论的筛选技术。

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.