Automorphic forms - computational and p-adic aspects

自守形式是一种特殊的数学函数，具有许多显著的对称性，自19世纪世纪以来，数学家们一直对自守形式感兴趣。这些著名的理论开始继续发展迅速，在世纪，刺激了深刻的见解赫克，艾希勒，韦尔和志村建议深刻的联系与数论和几何。自守形式和其他这些场之间的崇高相互作用，可以用著名的朗兰兹图来概括，这是罗伯特·朗兰兹在20世纪60年代提出的。 从那时起，这些理论指导了许多领域的数学研究。 朗兰兹哲学的哲学基础强烈地影响了我自己的研究。 我的研究目标是推进自守形式的理论，着眼于数论应用。 特别是，我着迷于自守形式和丢番图方程之间的关系-方程只涉及整数和变量，我们只寻求积分解。 虽然丢番图方程的研究可以追溯到古代，但是基于自守形式的现代方法已经彻底改变了丢番图方程的研究。 我的研究深受计算的影响。 像所有的科学研究一样，自守形式的研究因计算机的发展而发生了根本性的变化。 我参与了计算软件的开发和自守形式的实验。 这个软件将是一个有价值的工具，研究人员在许多领域连接到自守形式和朗兰兹结构。

Automorphic forms, special mathematical functions possessing many remarkable symmetries, have fascinated mathematicians since the times of 19th century greats Jacobi, Weierstrass and Dedekind. The theories initiated by these eminent names continued to develop rapidly in the 20th century, spurred on by deep insights of Hecke, Eichler, Weil and Shimura suggesting deep connections with number theory and geometry. The sublime interactions between automorphic forms and these other fields are epitomized by the famous Langlands conjectures, formulated by Robert Langlands in the 1960s. Since then, these conjectures have guided mathematical research in many fields. The philosophy underlying the Langlands conjectures strongly influences my own research. The goal of my research is to advance the theory of automorphic forms with an eye to number theoretic applications. In particular, I am fascinated by the relationship between automorphic forms and Diophantine equations -- equations involving only integers and variables for which we seek only integral solutions. Although the study of Diophantine equations reaches into antiquity, modern methods based on automorphic forms have revolutionized their study. My research is heavily influenced by computation. Like all of scientific research, the study of automorphic forms was fundamentally altered by the development of the computer. I am involved with the development of software for computing and experimenting with automorphic forms. This software will be a valuable tool for researchers in the many fields connected to automorphic forms and the Langlands conjectures.