The title of astrophysicist Livio's latest wide-ranging science survey is a teaser since God rarely makes an appearance; along with the French astronomer Laplace, Livio has no need of that hypothesis. Rather, Livio (The Golden Ratio) is concerned with the contentious question: is mathematics a human invention? Or is it the intricate design of the universe that we are slowly discovering? Scientists in past centuries have argued for the latter, Platonist position. In the last 50 years scientists, calling into question the whole idea of scientific discovery, maintain that we have invented mathematics.
These lecture notes are devoted to an area of current research interest that bridges functional analysis and function theory. The unifying theme is the notion of subharmonicity with respect to a uniform algebra. The topics covered include the rudiments of Choquet theory, various classes of representing measures, the duality between abstract sub-harmonic functions and Jensen measures, applications to problems of approximation of plurisubharmonic functions of several complex variables, and Cole's theory of estimates for conjugate functions.
Stopping Time Techniques for Analysts and Probabilists
This book considers convergence of adapted sequences of real and Banach space-valued integrable functions, emphasizing the use of stopping time techniques. Not only are highly specialized results given, but also elementary applications of these results. The book starts by discussing the convergence theory of martingales and sub-( or super-) martingales with values in a Banach space with or without the Radon-Nikodym property.
The articles in two volumes arose from papers given at the 1991 International Symposium on Geometric Group Theory, and they represent some of the latest thinking in this area. Many of the world's leading figures in this field attended the conference, and their contributions cover a wide diversity of topics. This second volume contains solely a ground breaking paper by Gromov, which provides a fascinating look at finitely generated groups. For anyone whose interest lies in the interplay between groups and geometry, these books will be an essential addition to their library.
These volumes contain survey papers given at the 1991 international symposium on geometric group theory, and they represent some of the latest thinking in this area. Many of the world's leading figures in this field attended the conference, and their contributions cover a wide diversity of topics. Volume I contains reviews of such subjects as isoperimetric and isodiametric functions, geometric invariants of a groups, Brick's quasi-simple filtrations for groups and 3-manifolds, string rewriting, and algebraic proof of the torus theorem, the classification of groups acting freely on R-trees, and much more.
