Asymptotic structures in Banach spaces

University Of Texas At Austin, Austin TX

Investigators

Abstract

The geometry of a separable infinite dimensional Banach space X can be studied and better understood through a study of its spreading models and asymptotic structure (those finite dimensional bases that can be found inside X, but arbitrarily spread out). Knowledge of these structures does not always pass through to infinite dimensional information about X but sometimes, surprisingly, it does. The author will study a number of open problems of this nature using the tools of infinite combinatorics, analysis and logic. For example if a Banach space has only one spreading model, must it contain a copy of one of the classical Banach spaces? This project concerns the study of the geometry of normed linear spaces. The easiest example of such a space is ordinary three dimensional Euclidean space. However one may, even in two or three dimensions, have other geometries than Euclidean. For example in the "taxicab" space distances between points in the plane are computed by traveling the shortest route along roads that run only horizontally or vertically. In this geometry the set of all points equidistant from a fixed point is diamond shaped rather than a circle. Applications of such alternate geometries are numerous in physics, engineering, signal processing and many other sciences. The state of a system or a signal may be given by a sequence of numbers and one may have to have a way of computing the distance between two states or signals to see how close they are. And in these applications one has to often use finite dimensional spaces of larger dimension than three or even spaces of infinite dimension. The author will be exploring the latter case by studying the "asymptotic" structures of these alternate geometries. This is a method of linking finite and infinite dimensional structure. The techniques to be used are a combination of analysis, infinitary combinatorics and logic. The problems that arise could also impact and motivate development in these latter areas as well.