Mathematical Sciences: Analysis on Nilpotent Lie groups

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。幂零李群最简单的非交换例子是上三角形3乘3实矩阵的Heisenberg群，对角线上有1。海森堡群在谐振子的量子力学处理中起着重要作用。幂零李群已经被很好地理解了，人们可以对它们进行分析，其详细程度几乎与经典欧几里得环境中的傅里叶分析相同。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况下的Gelfand对（粗略地说，一个李群和满足某种交换条件的自同构的紧子群）进行分类。第二个是找出更多的球面函数空间相对于这样一个对。

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.