CAREER: Multiple Dirichlet series and metaplectic groups

“这项奖励是根据2009年美国复苏和再投资法案（公法111-5）资助的。”近年来，在Weyl群多重狄利克雷级数的研究领域掀起了一股热潮。这些多重狄利克雷级数与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一直处于发展这些联系的一般理论和识别的前沿。他将在数论、自同构形式、元群的表示理论、仿射和无限根系统的理论等新的应用领域进行进一步的研究。这些活动将有助于将Weyl群多重狄利克雷级数这一相对较新的领域置于现代数学中更为中心的位置。因此，中国将承担的研究项目将吸引许多不同领域的专家。在多重狄利克雷级数的新应用中，特别引人注目的是：* Casselman-Shalika公式的形而上学版本（在没有多重性的情况下）*爱森斯坦级数的正交周期公式，它编码了经典形式的深刻和经典性质；例如，发现的最简单的新情况相当于高斯的三平方定理*函数域上的字符和与仿射根的字符之间的联系。结合申请人的研究活动，他将继续指导本科生和高中生。纽约城市学院是一所公认的少数族裔服务机构，拥有多元化的本科学生群体，包括西班牙裔（37%）、黑人（26%）、亚裔（20%）和白人（12%）学生。建议者将这些学生引入研究水平数学。此外，提议者还与哈莱姆儿童协会合作，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，通过让当地学生体验数学和科学领域的前沿研究，为社区服务，处于独特的地位。这项CAREERgrant将有助于扩大这种合作关系的范围。

"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.