Automorphic forms and L-functions

19世纪下半叶最重要的数学思想之一是，解析公式通常对离散信息进行编码。例如，一个人可能想要计算一个特定方程的解的数量，但发现很难找到解。另一方面，人们可能想要计算一系列方程的解的数量，并将答案作为一个序列。数学家称这类问题为“离散”问题。微积分中出现的函数并不是“离散的”，而是“分析的”，对这些函数来说，微积分运行得很好。令人惊讶的是，正确的解析函数往往能为离散问题提供答案。L函数是由与算术几何对象(例如伽罗瓦群和椭圆曲线)或模形式相关联的局部数据形成的一种生成函数。这些L函数提供了正确类型的解析函数。例如，Riemann Zeta函数的零点为素数计数问题提供了答案。另一个猜想的例子是，椭圆曲线的L函数的零点提供了椭圆曲线是否有无穷多个有理解的答案。在过去的三十年里，这些例子已经被系统化成数论的一个分支，称为朗兰兹程序。朗兰兹计划是一项雄心勃勃的计划，旨在统一数论、调和分析和几何。它开辟了数学的新前沿，并为解决旧问题提供了新的见解和技术。事实上，安德鲁·怀尔斯对费马大定理的求解是朗兰兹计划的成果之一。我的研究将对数论、表象理论产生深远的影响。

One of the most important mathematical ideas of the second half of the 19th century is that an analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, one might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kinds of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kinds of analytic functions often provide the answers to the discrete problems. An L-function is a type of a generating function formed out of local data associated with either an arithmetic-geometric object (such as Galois groups and elliptic curves) or a modular form. These L-functions provide the right kinds of analytic functions. For example, zeros of the Riemann zeta functions provide the answer to the problem of counting prime numbers. Another conjectural example is that the zero of the L-functions of an elliptic curve provide the answer to whether the elliptic curve has an infinitely many rational solutions.In the past thirty years, these examples have been systematized into a branch of number theory called the Langlands program. The Langlands program is an ambitious plan to unify number theory, harmonic analysis and geometry. It opened a new frontier in mathematics, and has given new insights and techniques in solving old problems. In fact, the solution of Fermat's last theorem by Andrew Wiles is one of the achievements of the Langlands program. My research will have far-reaching consequences in number theory, representation theory.