CAREER: Undecidability in Number Theory and Applications of Arithmetic Geometry

研究者将研究与丢番图方程有关的几个问题。其中一个重点是希尔伯特第十问题的推广。在有理数和数域上的希尔伯特第十问题仍然是开放的，而有理数的几个子环的不可判定性是已知的。一个目标是调查的数字字段的子环的问题是不可判定的。另一种方法是推广现有的函数域不可判定性结果。希尔伯特第十问题的研究为数论、算术几何和逻辑之间的相互作用开辟了新的领域。本项目将进一步探讨这些问题。另一个重点是研究算术几何中的计算问题，这些问题在密码学中有应用。一个目标是构造适合于密码学应用的小亏格的曲线。另一个目标是将数域的量子算法推广到函数域。研究者提出了几个研究项目，涉及研究多变量多项式方程的解。在整数或有理数上寻找这类方程的解是数论中的基本问题之一。它有着悠久的历史，可以追溯到古希腊。对于第一个项目，调查员将研究的基本问题，是否有可能找到一个程序，确定是否任意多变量多项式方程有一个解决方案，在一个给定的数字系统。第二个项目的重点是计算方面的某些特殊类别的方程，有密码学的应用。这一数学领域非常适合激励年轻学生学习数学。调查员将教授初中和高中女生密码学及其数学背景。将有两个年度研讨会，其中将涉及在计算机实验室动手实验。还将为当地数学教师举办专业发展讲习班。

The investigator will study several questions related to diophantine equations. One focus is on generalizations of Hilbert's Tenth Problem. Hilbert's Tenth Problem over the rational numbers and over number fields in general is still open while undecidability is known for several subrings of the rational numbers. One goal is to investigate for which subrings of number fields the problem is undecidable. Another is to extend the currently known undecidability results for function fields. Research on Hilbert's Tenth Problem has led to new areas of interaction between number theory, arithmetic geometry and logic. These will be further explored in this project. Another focus is the study of computational problems in arithmetic geometry that have applications to cryptography. One goal is to construct curves of small genus that are suitable for cryptographic applications. Another goal is to generalize the quantum algorithms for number fields to function fields.The investigator proposes several research projects that involve studying the solutions to multivariable polynomial equations. Looking for solutions to such equations over the integers or rational numbers is one of the fundamental problems in number theory. It has a long history that goes back to ancient Greece. For the first project the investigator will study the fundamental question of whether it is possible to find a procedure that determines whether an arbitrary multivariable polynomial equation has a solution in a given number system. The second project focuses on computational aspects of certain special classes of equations that have applications to cryptography. This area of mathematics is very well-suited for motivating young students to study mathematics. The investigator will teach middle school and high school girls about cryptography and its mathematical background. There will be two yearly workshops which will involve hands-on experiments in the computer lab. There will also be professional development workshops for local mathematics teachers.