Applications of Logic, Set Theory and Combinatorics to the Geometry of Banach Spaces

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The study of geometry of infinite dimensional normed linear spaces is the subject of this project. Some such spaces X are elastic, i.e., they stretch to uniformly absorb all spaces that embed into them. One goal of this project is to determine if they must actually be universal, absorbing all spaces of their size. Logic and combinatorics provide some of the tools necessary to attack this problem. Other problems concerning the notion of "partial unconditionality" in normed spaces will also be the subject of this project. These turn out to be related to some fascinating coding problems where one seemingly needs to predict the future in a nondeterministic way to solve the problem. Other problems will be attacked using certain machinery that meshes together the finite and infinite dimensional substructures of a Banach space (a normed linear space without holes).(Non-technical description)This project involves the study of the geometry of normed linear spaces. Ordinary three-dimensional space in which we live is an example of such an object. Distance is measured "as the crow flies". However other geometries and higher dimensions are necessary for applications. A transmitted signal might be a sequence of 0's and 1's. A corrupted signal might differ from the intended one in a number of places (0's become 1's or vice versa). One can measure the degree of correctness of the corrupted signal by counting the number of errors. This corresponds to a "taxi-cab" distance or geometry in n-dimensional space. For example in 2-dimensional space we can measure distance by moving only E-W or N-S, i.e., along roads a taxi-cab could travel. This project will study such alternate geometries using sophisticated tools of combinatorics and logic.Often, the goal is to discover nice substructures among seemingly random larger complicated geometries.