Mathematical Sciences: Functional Analysis

DMS提案#9622841 PI：南卡罗来纳州玛丽亚·吉拉尔迪博士@哥伦比亚·吉拉尔迪的最终目标是更好地理解泛函分析；她的方法是研究其各个子领域之间的相互作用；目前，她的主要推动力是巴纳赫空间理论的分支。她将继续她在过去几年中取得进展的工作，研究围绕Radon-Nikodym性质的Banach空间理论中的几个(相关的)长期存在的问题。她还将继续研究博克纳-勒贝格空间，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，Banach空间是一个点的集合，除了其他东西外，它还具有测量任意两点之间的“距离”的方法。三维欧几里得(普通)空间就是一个例子。另一个例子是“适当地好的”函数空间(因此，这种类型的Banach空间中的一点是一个函数)，距离适当地被澄清。Banach空间理论的一个实际应用是光盘编码；粗略地说，声音信号是适当的Banach空间中的函数。不幸的是，声音信号包含的信息太多，无法存储在CD上，因此信号由来自Banach空间的接近真实元素的元素近似(这是距离的概念进入的地方)。理解Banach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.