Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeperinsights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subjectand closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.

This book deals with the theory of Kac algebras and their dual ity, elaborated independently by M. Enock and J . -M. Schwartz, and by G. . Kac and L. . Vajnermann in the seventies. The sub ject has now reached a state of maturity which fully justifies the publication of this book. Also, in recent times, the topic of "quantum groups" has become very fashionable and attracted the attention of more and more mathematicians and theoret ical physicists. One is still missing a good characterization of quantum groups among Hopf algebras, similar to the character ization of Lie groups among locally...

The Symposium on the Current State and Prospects ofMathematics was held in Barcelona from June 13 to June 18, 1991. Seven invited Fields medalists gavetalks on thedevelopment of their respective research fields. Thecontents of all lectures were collected in the volume, together witha transcription of a round table discussionheld during the Symposium. All papers are expository. Someparts include precise technical statements of recentresults, but the greater part consists of narrative textaddressed to a very broad mathematical public.CONTENTS: R. Thom: Leaving Mathematics for Philosophy.-...