Mathematical Sciences: Analysis on Nilpotent Lie Groups

李群理论，以挪威人的荣誉命名 数学家Sophus Lie，一直是 世纪数学。 作为数学工具， 利用系统中固有的对称性，李理论 对数学本身和理论产生了深远的影响 物理学，特别是量子力学和基本粒子 物理学 詹金斯教授提出的研究利用了 幂零李群的表示理论来研究某些 微分算子类 其中包括薛定谔 运算符，这些运算符决定了 量子力学系统中的定律 如果有差别 算子，构造一个适当的李群，其中， 算子在泛包络中无限小地实现 代数 这种构造导致卷积半群， 概率测度，其微分算子为 无穷小生成元 这样，詹金斯教授利用 这些半群的理论来研究谱， 给定算子的相应本征函数展开。

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, Lie theory has had a profound impact upon mathematics itself and theoretical physics, especially quantum mechanics and elementary particle physics. The research proposed by Professor Jenkins utilizes the representation theory of nilpotent Lie groups to study certain classes of differential operators. These include Schrodinger operators, which are the operators that determine the dynamical laws in quantum mechanical systems. Given a differential operator, one constructs an appropriate Lie group in which that operator is realized infinitesimally in the universal enveloping algebra. This construction leads to a convolution semigroup of probability measures having the differential operator as infinitesimal generator. In this way, Professor Jenkins uses the theory of these semigroups to study the spectrum and corresponding eigenfunction expansion of the given operator.