Automorphic forms - computational and p-adic aspects

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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.