Automorphic Forms and Lie Algebras

我的研究主要集中在自同构形式和l -函数，以及它们在数论和李代数中的应用。19世纪下半叶最重要的数学思想之一是解析公式经常编码离散信息。例如，一个人可能想要计算一个特定多项式方程的解的个数，但发现解很难找到。另一方面，人们可能想要计算一个模质数方程序列的解的个数，并将答案作为一个序列，即离散信息。我们可以从离散信息中形成所谓的Artin l函数。这个l函数是一个解析函数，可以应用分析，这个l函数编码离散信息。另一个例子是黎曼ζ函数的零提供了计算质数问题的答案。自同构形式是一类具有许多对称性的解析函数。有限维简单李代数作为一种对称度量，在几何和物理中具有重要的基础意义。此后，引入了Kac-Moody代数和Borcherds代数等几种广义李代数。令人惊讶的是，自同构形式出现在这些代数中，因此数论和李代数变得密切相关。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.