Mathematical Sciences: Harmonic Analysis, Representation Theory and Partial Differential Equations

格林利夫教授将把群表示理论应用于幂零李群及其齐次空间的数学分析问题。特别的重点将放在研究齐次空间上的不变微分算子，用基里洛夫的轨道图描述这些算子的代数，并发展这些空间上的大类微分算子的可解性准则。李群作为对称群在数学和物理学中自然出现。一个很有趣的例子是由一个球体的所有旋转组成的，通过连续地应用旋转组成一个群。在球对称存在的任何情况下，理解这个群体都是重要的。幂零李群最简单的非交换例子（这里最直接感兴趣的一类）是上三角形3乘3实矩阵的海森堡群，对角线上有1。海森堡群作为微分方程的对称群，在谐振子的量子力学处理中起着基础性的作用。微分方程和底层对称群的表示理论之间的密切关系是格林里夫教授项目的核心。

Professor Greenleaf will apply the theory of group representations to problems in mathematical analysis on nilpotent Lie groups and their homogeneous spaces. Particular emphasis will be placed on studying the invariant differential operators on homogeneous spaces, describing algebras of such operators in terms of Kirillov's orbit picture, and developing solvability criteria for large classes of differential operators on these spaces. Lie groups arise naturally in mathematics and physics as groups of symmetries. An interesting example that lies readily at hand consists of all rotations of a sphere, made into a group by applying rotations successively. Understanding this group is important in any situation in which spherical symmetry is present. The simplest noncommutative example of a nilpotent Lie group (the class of most immediate interest here) is the Heisenberg group of upper triangular three-by-three real matrices with ones on the diagonal. The Heisenberg group plays a fundamental role in the quantum mechanical treatment of the harmonic oscillator, as the symmetry group of a certain differential equation. The strong relationship between differential equations and the representation theory of the underlying symmetry group is at the heart of Professor Greenleaf's project.