Functional transforms and their kernels on symmetric spaces and Lie groups

简而言之，对称空间指的就是我正在研究的那种“宇宙”。我工作的总体背景是对这个宇宙的谐波分析。调和分析的基本思想，本身是傅里叶分析的推广，是用更基本的函数，即球函数来表示一般函数。历史上，这类研究关注的空间包括实线（一维空间）；平面（二维宇宙）；或者空间（三维宇宙）。一个自然的扩展就是欧几里得n维空间。数学家们想要研究的是更复杂的物体（球体就是一个很自然的例子）。对称空间是足够复杂的对象，作为一种概括，它很有趣，同时又表现得足够好，使它们的研究具有相关性和趣味性（更不用说可访问性了！）。人们不仅可以考虑对称空间上的热扩散等问题，而且利用这些对象来研究概率论也是有意义的。同时，基本函数的行为，即球函数也变得很重要。关于在欧几里得环境中一个众所周知的现象在不同情况下如何表现的自然假设，要么得到证实，要么遭到反驳。这些问题可以通过寻找球面函数的显式公式（如果可能）或使用一般原理来解决。

A symmetric space refers, in plain words, to the kind of "universe" I am studying. The general context of my work is harmonic analysis of this universe. The basic idea of harmonic analysis, itself a generalization of Fourier analysis, is to represent general functions in terms of more basic ones, i.e. spherical functions. Historically, such studies concerned spaces such as the real line (the one-dimensional space); the plane (the two-dimensional universe); or space (the three-dimensional universe). A natural extension was then the Euclidean n-dimensional space. There are more complicated objects mathematicians want to work with (the sphere being a natural example). Symmetric spaces are sufficiently complex objects to be of interest as a generalization, while sufficiently well-behaved to make their study relevant and interesting (not to mention accessible!). Not only can one consider questions such as the heat diffusion on a symmetric space, but it also makes sense to study probability theory using such objects. As well, the behaviour of the basic functions i.e. the spherical functions becomes important. Natural assumptions as to how a well-known phenomenon in the Euclidean setting will behave in a different situation are either confirmed or contradicted. These questions can be addressed by finding explicit formulae for the spherical functions (when possible) or by using general principles.