# Brief Description of Dr. Dihua Jiang's Research

#### General Introduction

The classical theory of automorphic forms, in particular modular forms,
was initiated by H. Poincar{\'e} and F. Klein. Rooted in the work of Gauss,
Riemann, Jacobi, Eisenstein, it was further developed by Hecke, Siegel, Maass,
Selberg, and then by many others. It became early on a meeting ground for
analysis and number theory, and has developed as an indispensable tool
in analytic number theory, algebraic number theory, Diophantine problems,
arithmetic and algebraic geometry, and recently in infinite dimensional Lie
algebras and mathematical physics. These relations of
automorphic forms with other fields of mathematics can often
be summarized as the modularity problem. The Taniyama-Shimura-Weil
conjecture on elliptic curves is a typical example. About fifty years ago,
it was realized first by Gelfand and Fomin and then in greater generality
by Harish-Chandra that the theory of automorphic forms might be better
understood in terms of harmonic analysis over locally compact topological
groups, especially in terms of the representation theory of complex, real,
and p-adic reductive algebraic groups. Then Langlands gave a systematic
conjectural description of relations between L-functions in number theory or
algebraic geometry and those arising in the theory of automorphic forms.
This is the well-known Langlands Program. It has been actively developed
in the last thirty years. The recent striking advances on the Langlands
Conjectures bring another new exciting moment to this subject.
#### Description of My Work

My Lecture Notes for The Seconed ICCM 2001
(iccm01.ps) (2002)
Selection of My Papers
selected work