RUI: Mathematical Sciences: Spherical Characters on P-adic Coset Spaces and the Relative Trace Formula

雷德这个奖项为朗兰兹计划中的一个项目提供资金。由Jacquet提出的相对迹公式是研究自同构的L函数和朗格兰程序中的函数性的新工具。Rader和S.Rallis一直在合作证明一些他们认为必要的局部结果，以使相对迹公式成为一个定理，而不仅仅是一组例子。他们已经证明了许多已知的特征标积分和中心轨道积分的结果推广到某些p-进对称空间上的球特征标积分和重陪集积分的情况。他们计划扩大他们的研究范围，将M.Brion意义上的球形子群包括在内。尽管完成这样一个项目需要很多年，但它似乎是相对迹公式的自然域，因为(很可能)H的一维表示出现在G的有限重数的不可约光滑无限维表示中。Rader还计划与Marie-France Vigneras合作，研究某些李代数中幂零轨道的几何Zlevinskii对合。这在构造Kazhdan-Lusztig之后的Hecke代数的模表示方面应该有应用。朗兰兹计划是数论的一部分。数论是研究全体数的性质的学科，是数学中最古老的分支。从一开始，数论中的问题就为在该学科的其他不同部分创造新的数学提供了动力。朗兰德的计划是一种将数论与微积分联系起来的一般哲学；它体现了研究整数的现代方法。现代数论是非常技术性和深刻性的，但它在理论计算机科学和编码理论等领域有着惊人的应用。

Rader This award provides funding for a project in the Langlands program. The Relative Trace Formula (as initiated by Jacquet) is a new tool for studying automorphic L-functions and functoriality in the Langland's program. Rader and S. Rallis have been collaborating to prove some of the local results they think necessary to make the Relative Trace Formula a theorem instead of just a collection of examples. They have shown that many of the results known for usual character and central orbital integrals generalize to the case of spherical characters and double coset integrals on certain p-adic symmetric spaces. They plan to expand their research to include spherical subgroups such in the sense of M. Brion. Although it will take many years to complete such a project, it seems to be the natural domain for relative trace formulae, because (probably) one dimensional representations of H occur in irreducible smooth infinite dimensional representations of G with finite multiplicity. Rader also plans to complete a collaborative effort with Marie-France Vigneras investigating the geometric Zelevinskii involution for nilpotent orbits in certain Lie algebras. This should have applications in constructing modular representations of Hecke algebras, after Kazhdan-Lusztig. The Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. Modern number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.