Mathematical Sciences: Functional Analysis

DMS提案#9622841 PI: Dr. Maria Girardi南卡罗莱纳大学@哥伦比亚Girardi的最终目标是更好地理解功能分析；她的方法是研究其各个子领域之间的相互作用；此时，她的主要研究方向是巴拿赫空间理论的分支。她将继续她的工作，在过去的几年里，她在Banach空间理论中几个（相关的）长期存在的问题上取得了进展，这些问题以Radon-Nikodym属性为中心。她还将继续研究Bochner-Lebesgue空间，并开始一个将泛函分析和同调代数联系在一起的新项目。粗略地说，巴拿赫空间是一个点的集合，其中有一种方法可以测量任意两点之间的“距离”。一个例子是三维欧几里得（普通）空间。另一个例子是“合适的”函数空间（因此这种类型的巴拿赫空间中的一个点是一个函数），距离适当地明确。巴拿赫空间理论的一个实际应用是光盘编码；粗略地说，声音信号是合适巴拿赫空间中的一个函数。不幸的是，声音信号包含太多的信息，无法存储在CD上，因此信号由巴拿赫空间中接近真实元素的元素来近似（这就是距离概念的来源）。理解巴拿赫空间的结构有助于上述和其他实际应用。

DMS Proposal #9622841 PI: Dr. Maria Girardi University of South Carolina @ Columbia Girardi's ultimate goal is to gain a better understanding of Functional Analysis; her approach is to examine the interplay between its various sub-areas; at this time, her main thrust is in the branch of Banach Space Theory. She will continue her work, in which she has made progress over the past years, on several (related) long-standing problems in Banach space theory that are centered around the Radon-Nikodym Property. She will also continue examining the Bochner-Lebesgue space and begin a new project that ties together Functional Analysis and Homological Algebra. Roughly speaking, a Banach space is a collection of points that has, among other things, a way to measure the "distance" between any two points. An example is the 3-D Euclidean (ordinary) space. Another example is the space of "suitably nice" functions (so a point in this type of Banach space is a function) with distance suitably clarified. A practical application of Banach space theory is compact disc coding; a sound signal is, roughly speaking, a function in a suitable Banach space. Unfortunately the sound signal contains too much information to store on the CD so the signal is approximated by an element from the Banach space that is close to the true element (this is where the notion of distance enters). Understanding the structure of Banach spaces aids in the above, and other, practical applications.