Vector-Valued Automorphic Forms and Applications

这项研究建议属于数论领域。这是一个从古希腊时代起就吸引着人类心灵的理论。今天，它仍然是一个以其具有挑战性的问题和复杂的抽象而吸引人们的理论。然而，它在实际世界中也非常有用。例如，数据加密以及互联网通信和商业交易的安全都离不开数论的帮助。*在这个建议中，我研究了向量值自同构形的理论。这些是有趣的物体，在数论、代数、几何和物理的其他分支中都有直接的应用。*在接下来的五年里，我的目标是在我最近与我的学生在这个领域的重要发现的基础上，在一般情况下发展一个完整的矢量值形式理论。**

This research proposal is in the area of number theory. This is a theory that has attracted the human mind since the time of ancient Greece. Today it is still a theory that fascinates people with its challenging problems and sophisticated abstraction. However, it has also been very useful in the practical world. For example, data encryption and the security of internet communication and  business transactions could not be achieved without the help of number theory.***In this proposal, I study the theory of vector-valued automorphic forms. These are intriguing objects that have immediate applications in other branches of number theory, algebra, geometry  as well as physics.***In the next five years, my goal is develop a complete theory of vector-valued forms in the general case, building on important  discoveries I made recently with my students in the area.**