A complete presentation of a central part of convex geometry, from basics for beginners, to the exposition of current research.

View the abstract.

Starting in the early 1950's, Alberto Calderon, Antoni Zygmund, and their students developed a program in harmonic analysis with far-reaching consequences. The title of these proceedings reflects this broad reach. This book came out of a DePaul University conference honoring Stephen Vagi upon his retirement in 2002. Vagi was a student of Calderon in the 1960's, when Calderon and Zygmund were at their peak. Two authors, Kenig and Gatto, were students of Calderon; one, Muckenhoupt, was a student of Zygmund. Two others studied under Zygmund's student Elias Stein. The remaining authors all have close connections with the Calderon-Zygmund school of analysis. This book should interest specialists in harmonic analysis and those curious to see it applied to partial differential equations and ergodic theory. In the first article, Adam Koranyi summarizes Vagi's work. Four additional articles cover various recent developments in harmonic analysis: Eduardo Gatto studies spaces with doubling and non-doubling measures; Cora Sadosky, product spaces; Benjamin Muckenhoupt, Laguerre expansions; and Roger Jones, singular integrals. Charles Fefferman and Carlos Kenig present applications to partial differential equations and Stephen Wainger gives an application to ergodic theory. The final article records some interesting open questions from a problem session that concluded the conference.

This collection of original papers related to the Israeli GAFA seminar (on Geometric Aspects of Functional Analysis) during the years 2004-2005 reflects the general trends of the theory and are a source of inspiration for research. Most of the papers deal with different aspects of the Asymptotic Geometric Analysis, ranging from classical topics in the geometry of convex bodies to the study of sections or projections of convex bodies.

An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.