Automorphic Forms and Lie Algebras

我的研究集中在自守形式和L-函数，以及它们在数论和李代数中的应用。 19世纪后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，一个人可能想计算一个特定多项式方程的解的数量，但发现很难找到解。另一方面，人们可能想计算一个方程序列模素数的解的数量，并将答案作为一个序列，即离散信息。人们可以从离散信息中形成所谓的Artin L函数。这个L函数是一个可以应用分析的解析函数，并且这个L函数对离散信息进行编码。另一个例子是黎曼zeta函数的零点提供了计算素数问题的答案。 自守形式是具有许多对称性的某些解析函数。有限维单李代数作为对称性的度量在几何和物理学中具有重要意义。从那时起，一些广义李代数被引入，如Kac-Moody代数和Borcherds代数。令人惊讶的是，自守形式出现在这些代数中，因此数论和李代数密切相关。我的研究将进一步发展这种关系。

My research centers on automorphic forms and L-functions, and their applications in Number Theory and Lie Algebras. 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 polynomial equation, but discovers 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 modulo prime numbers and have the answers as a sequence, namely, discrete information. One can form the so-called Artin L-function out of the discrete information. This L-function is an analytic function where one can apply analysis, and this L-function encodes the discrete information. Another example is that zeros of the Riemann zeta function provide the answer to the problem of counting prime numbers. Automorphic forms are certain analytic functions which have many symmetries. Finite dimensional simple Lie algebras are of fundamental importance in geometry and physics as a measure of symmetry. Since then, several generalized Lie algebras were introduced such as Kac-Moody algebras and Borcherds algebras. Amazingly, automorphic forms occur in these algebras and thus Number Theory and Lie Algebras became intimately related. My research will develop this relationship further.