The investigators will explore several directions in the geometry of Banach spaces and the geometry of operator spaces. In the linear theory of Banach spaces, the structure of spaces of p-integrable functions, especially the complemented subspaces of these spaces, will be studied. In the non linear theory of Banach spaces, a fundamental problem is to determine when a non linear (uniform, bi-Lipschitz, or coarse) equivalence induces a linear isomorphism. In particular, what (isomorphic) classes of Banach spaces are closed under uniform equivalence? Are the p-integrable functions such a class? When does a Lipschitz factorization of a linear operator induce a linear factorization? Questions in metric geometry that are closely related to the non linear geometry of Banach spaces also will be investigated. In particular, the problem of dimension reduction for Lipschitz embeddings of finite metric spaces in the space of integrable functions will be attacked. Directions in the geometry of operator spaces that will be explored include similarity problems, Grothendieck's theorem, Ramanujan graphs and free groups, and operator valued spaces of analytic functions. It is expected that intuition gained from studying the geometry of Banach spaces will continue to provide guidance in this research in operator spaces and lead to the solutions of old problems about operator algebras.
Banach spaces are vector spaces in which there is a very natural notion of distance. Most mathematical analysts and mathematically oriented engineers model problems in a suitable Banach space, so it is not surprising that the understanding of the geometry of Banach spaces has been useful in many areas of mathematics and engineering. A "pattern" is a finite metric space; that is, a finite collection of points together with a notion of distance between the points that satisfies conditions that agree with our intuition about distance. Recently a number of computer scientists have been investigating patterns and especially how the metric geometry of a Banach space which contains a given pattern can be used to understand better the pattern. One problem that arises in analyzing a pattern that is contained in a (finite dimensional) Banach space is that computations that are polynomial in the number of points in the pattern tend to be exponential in the dimension of the ambient Banach space. This "curse of dimensionality" can sometimes (e.g., through the Johnson-Lindenstrauss "flattening lemma" in the case where the Banach space is a Euclidean space) be corrected through dimension reduction procedure, but to what extent dimension reduction can be effected in other Banach spaces is a major problem in the area which will be investigated. Operator spaces, or "quantized Banach spaces", are Banach spaces in which scalars are replaced by operators on a suitable Hilbert space. Basically operator spaces are to Banach spaces as quantum mechanics is to classical mechanics, and in fact several research directions within operator spaces in this project are meaningful physically. The synergy between operator algebras and the geometry of Banach spaces that lies at the basis of the concept of operator space has led to the solutions by Pisier of similarity problems as well as old problems about C* algebras. Part of this project involves transferring concepts and results in Banach space theory (such as Grothendieck's inequality) into an operator space setting with a view to solving known problems about operators.
