RUI: Arithmetic of Modular Forms and of Diagonal Hypersurfaces

该奖项资助Fernando Gouvea教授在本科院校研究项目下的研究。Gouvea教授将研究有限域上模形式的算法和对角超曲面的算法。在工作的第一部分的中心主题是两个算子介绍了阿特金的研究。在第二部分，研究者将研究关于阿贝尔变的最简单的算法。这项研究属于数论的总标题。数论是对整数性质的研究，是数学中最古老的分支。从一开始，数论中的问题就为这门学科几乎所有部分的新数学的创造提供了动力。现代数论的哲学之一是，许多关于数字的事实可以用微积分和几何来研究。这种方法在这个项目的工作中非常明显，因为模形式是用微积分研究的，而阿贝尔变体是用几何研究的。这种数论是非常技术性和深奥的，但它在理论计算机科学和编码理论等领域有着惊人的应用。* * *

Gouvea This award funds the investigations of Prof. Fernando Gouvea under the Research in Undergraduate Institutions Program. Prof. Gouvea will study the arithmetic of modular forms and the arithmetic of diagonal hypersurfaces over finite fields. The central topic in the first part of the work is the study of two operators introduced by Atkin. In the second part the investigator will study the simplest cases of arithmetic on abelian varieties. This research falls under the general heading of Number Theory. Number Theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished the driving force to creation of new mathematics in almost all parts of the discipline. One of the philosophies of modern number theory is that many facts about numbers can be studied using calculus and geometry. This approach is very evident in the work of this project since modular forms are studied using calculus and abelian varieties are studied using geometry. This kind of number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory. ***