Number Theory and Geometry

这个项目涉及数论和几何的研究。这些数学领域可以应用于计算机科学和网络安全，这是该项目的重点领域。除了研究之外，PI还将继续与本科生、研究生和博士后研究人员合作，向他们介绍在他们自己的研究中使用的思想和方法；-为期刊和在线数学网站撰写说明文和笔记；-维护和开发网页，在复杂的研究数学和基本的主题和难题之间进行互动；-目前的说明性数学讲座，一些针对高中生或大学非数学专业的学生，通常以音乐或国际象棋的数学方面为特色。本研究项目涉及算术几何中的基本理论和计算问题：椭圆曲线和更高的变量（特别是K3曲面、Calabi-Yau三倍曲面及其模变量）在有限域、函数域和数域上的行为。探讨了纠错码的广义权数的构造和应用，以及球面码和类似组合结构的相关问题。更好地理解球形和纠错码可以提高依赖于这种结构的模拟和数字通信的效率。

This project concerns research in number theory and geometry. These areas of mathematics have applications to computer science and to cyber security, which are areas of emphasis in the project. In addition to the research the PI will continue to- work with undergraduate and graduate students and postdoctoral researchers, to introduce them to ideas and methods of use in their own research; - write expository articles and notes for journals and for online mathematical sites; - maintain and develop web pages that feature interaction between sophisticated research mathematics and elementary topics and puzzles; - present expository mathematical lectures, some aimed at high school students or non-math majors in college, often featuring mathematical aspects of music or chess. This research project addresses fundamental theoretical and computational questions in arithmetic geometry: the behavior of elliptic curves and higher varieties (especially K3 surfaces, Calabi-Yau threefolds, and their moduli varieties) over finite fields, function fields, and number fields. It also explores the construction and application of generalized weight enumerators of error-correcting codes and related problems on spherical codes and analogous combinatorial structures. Better understanding of spherical and error-correcting codes may increase the efficiency of analog and digital communication that relies on such structures.