# Geometric and Harmonic Analysis

### Overview

Geometric functional analysis is concerned with geometric and linear properties and structure of finite- and infinite-dimensional Banach spaces and their unit balls. An asymptotic point of view is based upon expressing such properties in terms of various quantitative invariants whose limiting behavior is investigated when the dimension or a number of other relevant free parameters tends to infinity. For example, the main probabilistic tools used in the theory are deviation inequalities and the concept of concentration of measure phenomenon, that is in fact an isomorphic form of isoperimetric type inequalities. These and other deep geometric, probabilistic and combinatorial methods developed here are widely used outside the field, in discrete geometry, classical convexity, asymptotic combinatorics and computer science, among others. For example, one area where new applications has been very recently developed is the signal reconstruction and encoding, that is intimately related to the quantitative study of singular numbers of random matrices, including random ±1 matrices and to the study of random sections of convex bodies. More generally, an underlying principle lies in identifying and exploiting “approximate” (or “isomorphic”) symmetries of various problems, thus allowing to detect regularities which could not be tackled by more rigid methods. For example, this approach and related techniques (originating in the local theory of Banach spaces in 1970’s) have been recently successfully used as a starting point in problems on amenability of certain natural Banach algebras which is on the border of abstract harmonic analysis and Banach algebra theory. The groups of asymptotic geometric analysis and discrete geometry in Western Canada (and some researchers in Toronto) have very close ties with the newly established European Network in Phenomena in High Dimensions (PHD) and we presently investigate whether it is possible to formalize these ties in some way. Asymptotic geometric analysis held a number of meetings every year, in PIMS it held two successful summer programs in Vancouver: 3 weeks program in 1999 and 2 months Thematic Summer Programme in 2002.

Abstract harmonic analysis relates to the studies of Banach algebras of spaces of measures or functions associated to (unitary representations of) a locally compact group involving powerful tools from group representations, geometry of Banach space, operator algebras and operator space theory. Two locally compact groups are isomorphic if and only if certain associated Banach algebras (i.e., Fourier algebras or the group algebra) are isometrically isomorphic. Consequently, the study of various Banach algebras and their geometric properties reveals deep structural properties of the underlying group. For example the classical result of B. E. Johnson asserts that the group algebra is amenable if and only if the underlying group is amenable. An analogous result for the Fourier algebra has been proved only very recently by Z. J. Ruan and it involves so-called operator amenability by viewing the Fourier algebra as an operator space. A characterization of amenability in terms of a deep combinatorial property of Følner led to a strong relationship to the recent study of amenable unitary representations of locally compact groups, geometry of Fourier and Fourier-Stieltjes algebra of a group as well as Hahn-Banach type separation and extension properties for closed subgroups of a locally compact group by positive definite functions by experts in Canada and around the world including Kaniuth and Lau.

Modern abstract harmonic analysis has broad impact to related fields such as operator space theory, Banach space theory, operator algebras, Banach algebras, geometric group theory, non-commutative geometry and locally compact quantum groups. In fact operator space theory (sometimes called a quantized functional analysis) has played a very significant role in the development of non-commutative harmonic analysis (including the study of the Fourier algebras, group C algebras and von-Neumann algebras) in recent years.

Recent advances in applied harmonic analysis include the development of a localized Fourier analysis, to provide decompositions of function spaces through simultaneous timefrequency (or space-phase) representations. The short-time Fourier transform is an early instance of this approach, which has been richly extended by the Wigner, wavelet, Gabor, and Stockwell transforms. Deep questions in this area relate to the characterization of the classical Banach spaces (Lp, Sobolev) in terms of the coefficient space for the discrete and continuous transforms and related modulation spaces, as well as the factorization or approximate diagonalization of classical operator families such as the Calderon-Zygmund, pseudodifferential, and Fourier integral operators. Techniques in operator algebras also are applicable to the analysis of this localized version of Fourier analysis: for instance, the work of Larsen, Bratelli and Jorgenson, and others, gives an operator algebraic approach to analyzing wavelet frames, while the work of Balan, Feichtinger, Grochenig, and others demonstrates a deep connection between sampling theory for Gabor transforms and the structure of non commutative tori.

This local Fourier analysis is closely related to the phase space approach to partial differential equations, also known as micro local analysis, which leads to novel applications including in the numerical solutions of partial differential equations, analysis of stochastic PDEs, mathematical wavefield propagation, imaging technology (medical, seismic, and other), and general signal processing. Commercial applications include the development of cellphone technology, and seismic processing tools used in oil and gas exploration. Ongoing research includes developing links between geometric ray theory in wave propagation and the more recent wavefield and path integral methods of time-frequency analysis.

Discrete geometry that was born in ancient times, most probably from the Greek’s studies of regular polyhedra, in recent years become a fast developing area on the border line of mathematics and computer science. It investigates combinatorial and analytic properties of configurations of geometric objects. It offers sophisticated results and techniques of great diversity, and it is a foundation for fields such as computational geometry or combinatorial optimization and also it includes some classical areas such as (analytic) convexity and geometry of numbers. The discrete geometry group at U of C focuses on low dimensional problems in Euclidean as well as in normed spaces including the problem of densest sphere packings, 2 the Boltyanski-Hadwiger illumination problem and its more recent quantitative version, the Kneser-Poulsen conjecture, the Lebesgue covering problem, the Blaschke-Lebesgue problem on the minimum volume of convex bodies of constant width, the Besicovitch covering theorem/ constant and its relatives and the Voronoi conjecture on parallelotopes. Each of these problems has a strong analytic character and most of them are connected to the geometric and linear properties and structure of certain finite dimensional Banach spaces.

Closely connected is additive combinatorics, a rapidly emerging exciting area of analysis. Bringing diverse techniques of harmonic analysis, additive number theory, and discrete geometry together, it aims at one hand at a deeper understanding of longstanding open combinatorial geometry problems such as Erd¨os’ distinct distances problem, and on the other hand, at study of arithmetic progressions, with the recent superb result by Green and Tao that build on Gowers’ proof of Szemeredi’s theorem.