Automorphic forms - computational and p-adic aspects

自同构形式是一种特殊的数学函数，具有许多显著的对称性，自19世纪伟大的雅各比、魏尔斯特拉斯和德德金德时代以来一直吸引着数学家。在黑克、艾希勒、韦尔和下村的深刻见解的推动下，这些著名人物开创的理论在20世纪继续快速发展，表明他们与数论和几何学有着深刻的联系。罗伯特·朗兰兹在20世纪60年代提出的著名的朗兰兹猜想集中体现了自同构形式与这些其他领域之间的崇高相互作用。从那时起，这些猜想指导了许多领域的数学研究。朗兰兹猜想背后的哲学对我自己的研究产生了强烈的影响。 我的研究目标是推进自同构形的理论，着眼于数论应用。特别是，我对自同构形式和丢番图方程之间的关系着迷--只涉及整数和变量的方程，我们只为它们寻求整数解。虽然丢番图方程的研究由来已久，但基于自同构形式的现代方法已经使它们的研究发生了革命性的变化。 我的研究很大程度上受到了计算的影响。像所有的科学研究一样，计算机的发展从根本上改变了对自同构形式的研究。我参与了用于计算和试验自同构形式的软件的开发。该软件将成为与自同构形和朗兰兹猜想相关的许多领域的研究人员的宝贵工具。

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.