CAREER: Undecidability in Number Theory and Applications of Arithmetic Geometry

研究人员将研究与丢番图方程有关的几个问题。其中一个焦点是希尔伯特第十问题的推广。一般情况下，Hilbert关于有理数域和过数域的第十个问题仍然是公开的，而对于有理数的几个子环是已知的不可判定的。一个目标是调查数域的哪些子环的问题是不可判定的。另一种方法是推广目前已知的函数域的不可判定性结果。对希尔伯特第十问题的研究开创了数论、算术几何和逻辑相互作用的新领域。这些都将在本项目中进一步探索。另一个焦点是研究算术几何中应用于密码学的计算问题。一个目标是构造适合于密码学应用的小亏格曲线。另一个目标是将数域的量子算法推广到函数域。研究人员提出了几个研究项目，涉及研究多变量多项式方程的解。在整数或有理数上求这类方程的解是数论的基本问题之一。它有着悠久的历史，可以追溯到古希腊。对于第一个项目，研究人员将研究一个基本问题，即是否有可能找到一个程序来确定一个任意的多变量多项式方程在给定的数系中是否有解。第二个项目集中在某些特殊类型的方程的计算方面，这些方程在密码学中有应用。这一数学领域非常适合激励年轻学生学习数学。这位调查员将向初中和高中的女孩传授密码学及其数学背景。每年将有两次工作坊，其中包括在计算机实验室进行动手实验。此外，亦会为本地数学教师举办专业发展工作坊。

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.