Mathematical Sciences: Research in Harmonic Analysis

DMS-9501191 PI: Gross Gross将继续研究正域上的算子值贝塞尔函数，高维空间上的超几何函数，拉马努金主定理的多维扩展，并将推广到对称锥的设置，以及与牛顿多项式相关的经典分析和近似理论的构造。此外，Gross将发展一种特殊函数理论，该理论将应用于李群理论中出现的非对称锥，研究与约化李群的退化主级数表示相关的交织算子，并研究与光锥和二阶不定正交群相关的特殊函数。李群理论是为了纪念挪威数学家索夫斯·李而命名的，它一直是20世纪数学的主要主题之一。作为利用系统中固有对称性的数学工具，李群表示理论对数学本身，特别是分析和数论，以及理论物理，特别是量子力学和基本粒子物理产生了深远的影响。

DMS-9501191 PI: Gross Gross will continue his research on operator-valued Bessel functions on domains of positivity, hypergeometric functions on higher-dimensional spaces, multidimensional extensions of Ramanujan's Master Theorem, and will generalize to the setting of symmetric cones, constructs from classical analysis and approximation theory related to Newton polynomials. Also, Gross will develop a theory of special functions that will apply to non-symmetric cones that arise in the theory of Lie groups, investigate intertwining operators associated to degenerate principal series representations of reductive Lie groups, and study special functions associated to light cones and the indefinite orthogonal groups of rank two. The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, the representation theory of Lie groups has had a profound impact upon mathematics itself, particularly in analysis and number theory, and upon theoretical physics, especially quantum mechanics and elementary particle physics.