CAREER: Multiple Dirichlet series and metaplectic groups

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)资助的。近年来，Weyl群多重Dirichlet级数领域出现了一系列的研究活动。这多个狄利克雷级数和几个不同的数学领域之间出现了令人惊讶和意想不到的联系。PI一直走在发展一般理论和确定这些联系的前沿。他将继续追求进一步的发展，从而产生相互关联的数论、自同构形式、亚普勒群的表示理论和仿射和无限根系理论的新应用。这些活动将有助于将相对较新的领域Weyl群多重Dirichlet级数置于现代数学中更中心的位置。因此，Chinta将承担的研究项目将引起许多不同领域的专家的兴趣。在多重Dirichlet级数的新应用中，特别引人注目的是*Casselman-Shalika公式的亚普勒版(在没有重数的情况下)*Eisenstein级数的正交周期的公式，它编码了经典形式的深层和经典性质，例如，发现的最简单的新情况等价于Gauss的三平方定理*函数域上的特征和和仿射根系统的特征之间的联系*与提出者的研究活动相结合，将是他对本科生和高中生的持续指导。纽约城市学院是一所公认的少数族裔服务机构，本科生群体多样化，由西班牙裔(37名)、黑人(26名)、亚裔(20名)和白人(12名)学生组成。提出者将向这些学生介绍研究性水平的数学。此外，提倡者还与哈莱姆儿童协会进行了合作，以指导来自代表性不足社区的高中生。城市学院位于哈莱姆区，通过让该地区的学生体验数学和科学领域的前沿研究，它在服务社区方面具有独特的地位。这笔CAREER赠款将有助于扩大这一伙伴关系的范围。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)." In recent years there has been a flurry of activity in the field of Weyl group multiple Dirichlet series. Surprising and unexpected connections between these multiple Dirichlet series and several different fields of mathematics have emerged. The PI has been at the forefront of the development of the general theory and identification of these connections. He will pursue the further developments which lead to new applications interrelating number theory, automorphic forms, the representation theory of metaplectic groups and the theory of affine and infinite root systems.These activities will serve to place the relatively new field of Weyl group multiple Dirichlet series into a more central position in modern mathematics.As such, the research projects Chinta will undertake will interest specialists in many different fields. Particularly striking among the new applications of multiple Dirichlet series are * A metaplectic version of the Casselman-Shalika formula (in the absence of multiplicity one) * A formula for orthogonal periods of Eisenstein series which encodes deep and classical properties of classical forms, e.g. the simplest new case discovered is equivalent to Gauss's three squares theorem * Connections between characters sums over function fields and characters of affine root systemsIntegrated with the research activities of the proposer will be his continued mentorship of undergraduate and high school students. City College of New York is a recognized minority serving institution with a diverse undergraduate student body consisting of Hispanic (37\%), Black (26\%), Asian (20\%) and White(12\%) students. The proposer will introduce these students to research level mathematics. In addition, the proposer has entered into a collaboration with the Harlem Children's Society, in order to mentor high school students from underrepresented communities. Situated in Harlem, City College is in a unique position to serve its community by enabling area students to experiencecutting edge research in mathematics and the sciences. This CAREERgrant will help extend the scope of this partnership.