Mathematical Sciences: Analysis on Nilpotent Lie groups

李群在数学和物理学中自然而然地以对称群的形式出现。一个有趣的例子随手可得，它由一个球体的所有旋转组成，通过连续应用旋转而成一组。幂零李群最简单的非对易例子是由对角线上的上三角三乘三实矩阵组成的Heisenberg群。海森堡群在谐振子的量子力学处理中起着重要的作用。人们对幂零李群的理解已经足够充分，以至于人们可以在经典的欧几里德环境中对它们进行几乎与傅立叶分析相同的详细程度的分析。詹金斯教授的项目有两个主要目标。第一个也是更重要的目标是对幂零情形下的Gelfand对(大体上是一个李群和满足一定交换性条件的自同构子群)进行分类。第二个是找出更多关于球函数相对于这样一对球函数的空间。

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. The simplest noncommutative example of a nilpotent Lie group 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. Nilpotent Lie groups are sufficiently well understood that one can do analysis on them in almost the same level of detail as Fourier analysis in the classical, Euclidean setting. The project of Professor Jenkins has two main objectives. The first and more important goal is to classify Gelfand pairs (roughly, a Lie group together with a compact subgroup of automorphisms satisfying a certain commutativity condition) in the nilpotent case. The second is to find out more about the space of spherical functions relative to such a pair.