Explicit Moduli Spaces and Arithmetic Applications

显式模空间和算术应用

• 批准号: 1406066
• 负责人: Wei Ho
• 金额: $ 13.46万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406066&HistoricalAwards=false
• 关键词: Explicit; Moduli; Spaces; Arithmetic; Applications

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.

研究多项式方程及其解决方案的历史可以追溯到数千年。 对于某些类型的多项式，了解这些解决方案具有重要的应用，从工程到生物学到椭圆曲线密码学。 确定任意多项式方程可能具有多少解决方案仍然是一个非常困难的问题，尤其是当人们专门寻找位于自然数或理性数字中的解决方案时。该现象的一个简单例子是费马特的最后一个定理，该定理指出，如果n的整数大于2，则没有正面的形式方程的正整数解决方案等于两个n-the的总和。 虽然定理易于说明，但证明方法涉及数字理论和算术几何形状中的许多深层思想，包括一次研究许多这样的多项式方程的想法（“ Moduli Space”的想法）。 PI打算研究此类空间并使用它们来理解某些类型的多项式和相关几何对象（例如椭圆曲线）的性质。PI在三个领域的交集中起作用：数字理论，代数几何学和表示理论。 该提案的主要主题涉及使用表示理论在代数几何形状中明确构建模量空间。这些结构还可以通过使用数量的技术技术来解决数字理论中的计数问题。 PI打算在过去一百年中使用所有这些主题中开发的方法和工具，包括来自经典代数几何形状的结构，谎言理论的思想以及分析数理论的筛分技术。