Mathematical Sciences: Projections on Banach Spaces

Banach空间理论在严格的数学框架内处理无限维现象。它与更具体的应用学科有着历史上的联系，因为，例如，微分方程的解通常被方便地认为存在于适当的Banach空间中。在物理学中，人们经常遇到具有无限多个自由度的系统，这些系统必须在希尔伯特空间中建模，希尔伯特空间是一种特殊的Banach空间。对Banach空间做两件有趣的事情是对它进行切片(从技术上讲，将它投影到一个可补的子空间上)和用有限维子空间来逼近它，在有限维子空间中，人们可以说画画和做几何。这个项目的主要研究人员将致力于一个著名的公开问题，该问题询问某个标准Banach空间的切片是什么样子，并将研究Banach空间的有限维子空间的几何。

Banach space theory treats infinite-dimensional phenomena within the framework of rigorous mathematics. It has historical connections to more concrete applied subjects because, for instance, solutions to differential equations are often conveniently thought of as inhabiting an appropriate Banach space. In physics, one frequently encounters systems with infinitely many degrees of freedom which must be modeled in Hilbert space, a special type of Banach space. Two interesting things to do with a Banach space are to slice it (technically, project it onto a complemented subspace) and to approximate it by finite dimensional subspaces, in which one can so to speak draw pictures and do geometry. The principal investigator for this project will work on a famous open problem that asks what the slices of a certain standard Banach space look like, and will investigate the geometry of finite-dimensional subspaces of Banach spaces.