Mathematical Sciences: Interactions Between Functional Analysis, Harmonic Analysis and Probability

9424396蒙哥马利-史密斯这个项目涉及两个不同方面的工作。第一个涉及对偶是有序的阿贝尔群的调和分析。该项目涉及采用调和分析的传统结果，特别是关于解析性和共轭函数的结果，并将它们推广到新的环境中。这个项目将基于Hahn关于序群的分类定理的综合，并结合现代技术，如鞅理论，Browinean运动和Banach空间的几何。第二个项目涉及半群的谱性质。这项工作对系数为无穷维时微分方程解的渐近行为有一定的应用。泛函分析试图理解研究作为微分方程解出现的函数所需要的那种分析的严格方面。调和分析主要研究函数的傅里叶变换。概率最初是为了理解偶然性事件而发明的理论，现在它是一门与测量理论相关的技术学科。这个项目涉及到这些数学领域的融合。这些结果将对具有在系统论中应用的潜力的广泛的数学家有用。

9424396 Montgomery-Smith This project involves work on two different fronts. The first involves harmonic analysis on abelian groups whose dual is ordered. The project involves taking traditional results from harmonic analysis, specifically those concerning analyticity and the conjugate function, and generalizing them to a new setting. This project will be based on a synthesis of a classification theorem of Hahn on ordered groups together with modern techniques such as martingale theory, Browninan motion and the geometry of Banach spaces. The second project involves spectral properties of semigroups. This work has applications to the asymptotic behavior of differential equations in the case where the coefficients are infinite dimensional. Functional analysis seeks to understand the rigorous side of the kind of analysis that was needed to study functions that arise as solutions of differential equations. Harmonic analysis is concerned with studying the Fourier transform of functions. Probability, theory originally invented to understand events of chance, is now a technical subject related to measure theory. This project involves a blending of theses areas of mathematics. The results will be of use to a broad spectrum of mathematicians with the potential for application in systems theory.