Mathematical Sciences: Projections on Banach Spaces

巴拿赫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巴拿赫空间中。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类型的巴拿赫空间。对巴拿赫空间做两件有趣的事情，一是对它进行切片（技术上讲，是将它投影到一个互补的子空间上），二是用有限维的子空间对它进行近似，在有限维的子空间中，人们可以说可以画画和做几何。该项目的首席研究员将研究一个著名的开放问题，即某个标准巴拿赫空间的切片是什么样子，并将研究巴拿赫空间的有限维子空间的几何形状。

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.