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

Rader 该奖项为朗兰兹计划中的一个项目提供资金。相对轨迹公式（由Jacquet发起）是一种新的工具， 学习Langland程序中的自守L-函数和泛函。Rader和S.拉利斯一直在合作证明一些他们认为必要的局部结果，使相对迹公式成为一个定理，而不仅仅是一个例子的集合。他们已经表明，许多结果已知的一般性质和 中心轨道积分推广到某些p-adic对称空间上的球面特征标和双陪集积分的情形。他们计划扩大他们的研究，包括球面子群，如在意义上的M。布里昂虽然这将需要许多年才能完成这样一个项目，它似乎是自然域的相对迹公式，因为（可能）一维表示H发生在不可约光滑无限维表示G有限的多重性。雷德还计划与玛丽-弗朗斯·维涅拉斯合作，调查几何形状的泽列文斯基 某些李代数中幂零轨道的对合这应该有应用程序在建设模块表示Hecke代数后，Kazhdan-Lusztig。 朗兰兹纲领是数论的一部分。数论是研究整数的性质，是数学最古老的分支。从一开始，数论中的问题就为在这门学科的其他不同部分创造新的数学提供了动力。朗格兰纲领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的，但它已经在一些领域有了惊人的应用， 计算机科学和编码理论。

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.