Mathematical Research Today and Tomorrow

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.- S.Novikov: Role of Integrable Models in the Development ofMathematics.- S.-T. Yau: The Current State and Prospects ofGeometry and Nonlinear Differential Equations.- A. Connes:Noncommutative Geometry.- S. Smale: Theory of Computation.-V. Jones: Knots in Mathematics and Physics.- G. Faltings:Recent Progress in Diophantine Geometry.

In many fields of scientific research, the highest distinction is the Nobel Prize. In mathematics, there is for historical reasons no Nobel Prize, however, the so-called Fields Medal awarded every 4 years for "Outstanding Discoveries in Mathematics" carries similar prestige and distinction. In this book 7 Fields Medalists have each written down their view of the current development of research in their respective fields. The book will appeal to every mathematician and graduate students of mathematics, and provides a fascinating insight and commentary on present-day mathematics as a growing and moving research discipline.

Leaving mathematics for philosophy.- Rôle of integrable models in the development of mathematics.- The current state and prospects of geometry and nonlinear differential equations.- Noncommutative geometry.- Theory of computation.- Knots in mathematics and physics.- Recent progress in diophantine geometry.- Round-table discussion.