Automorphic L-Functions and Langlands Functoriality

最近惊人的结果建立存在的函子对称权力的程度3和4的尖形式在GL（2）作为自守形式在GL（4）和GL（5），以及转移一般尖形式从奇数特殊正交群一般线性群，由调查员和他的合作者，开辟了一个新的前沿自守形式和数论。 他们已经导致了令人惊讶的新的估计对拉马努金-塞尔伯格和佐藤-泰特disturtures，连同大量令人印象深刻的，明确的和新的成果数论，自守形式和几何获得的其他数学家。 研究者探索了GL（2）上形式的更高幂的扩展，以及其他经典群及其单连通相似覆盖上的一般形式的转移。 从GSp（4）到GL（4）的转移的著名的和仍然开放的情况于是成为这个的一个特殊情况。 除了探索新的想法朗兰兹转移超出内窥镜，这种情况已经存在于第三和第四对称的权力，他计划调查任何可能的延伸，他的方法，以无限维群体。 他有一个坚实的办法，以建立稳定的根号码所需的这些转让，通过他的方法，并研究贝塞尔函数和充分的一般性的所有根号码定义从他的方法。 这些根数的子类的稳定性是建立这些转移的最后一个严重问题。 调查员还致力于一些问题的极点的地方交织运营商以及那些自守的L-功能来自他的方法。 自守形式理论是现代数学中一个非常强大、令人兴奋和有前途的部分。 通过一些深入的分析，主要是由于高等研究院的罗伯特·朗兰兹（朗兰兹计划），它试图统一数学不同部分的对象，如数论，分析和几何。 怀尔斯的费马大定理的证明，这是一个结果的平面曲线定义的方程三度有理系数的功能复杂的上半平面，提供了一个很好的例子，这一巨大的计划。 调查员最近的工作与他的合作者导致了新的，惊人的和令人惊讶的对应关系，这种与许多后果数论和几何。 虽然这解决了一些长期存在的重大问题，但还有许多更重要的问题需要回答。 在这个项目中，研究人员使用分析方法，即，连续对象的研究，他已经在他的职业生涯中发展，并已在最近的进展基本，建立新的对应这种对象之间的离散性质与许多应用程序的不同部分的数论和几何。 该项目涉及许多合作和培训研究生和博士后。

Recent striking results establishing the existence of the functorial symmetric powers of degree 3 and 4 for cusp forms on GL(2) as automorphic forms on GL(4) and GL(5), as well as transfer of generic cusp forms from odd special orthogonal groups to general linear groups, by the investigator and his collaborator, has opened a new front in automorphic forms and number theory. They have resulted in surprising new estimates towards Ramanujan--Selberg and Sato--Tate conjectures, together with a large number of impressive, definitive and new results in number theory, automorphic forms and geometry obtained by other mathematicians. The investigator explores extensions of these to higher powers of forms on GL(2) as well as transfers of generic forms on other classical groups and their simply connected similitude coverings. The well-known and still open case of transfer from GSp(4) to GL(4) then becomes a special case of this. Beside exploring new ideas of Langlands on transferring beyond endoscopy, a situation which is already present in the third and fourth symmetric powers, he plans to investigate any possible extension of his method to infinite dimensional groups. He has a solid approach to establishing the stability of root numbers necessary for these transfers, by means of his method, and studies Bessel functions and the full generality of all the root numbers defined from his method. Stability of a subclass of these root numbers is the last serious problem in establishing these transfers. The investigator is also working on a number of problems concerning poles of local intertwining operators as well as those of automorphic L-functions coming from his method. Much of this work is carried out in collaboration with collaborators.The theory of automorphic forms is a very powerful, exciting and promising part of modern mathematics. Through a number of deep conjectures, mainly due to Robert Langlands of the Institute for Advanced Study (Langlands program), it tries to unify objects from different parts of mathematics such as number theory, analysis and geometry. Wiles' proof of Fermat's Last Theorem, which is a consequence of relating plane curves defined by equations of degree three with rational coefficients to functions on complex upper half plane, provides an excellent example of this vast program. The investigator's recent work with his collaborators has led to new, striking and surprising correspondences of this sort with many consequences in number theory and geometry. While this has resolved some very long standing and significant problems, many more important questions need to be answered. In this project, the investigator uses methods of analysis, i.e., the study of continuous objects, that he has developed over his career and have been fundamental in the recent progress, to establish new correspondences of this kind between objects of a discrete nature with many applications to different parts of number theory and geometry. The project involves many collaborations and training for graduate students and postdocs.