Hilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Rado. The book gives a concise introduction to variational methods and presents an overview of areas of current research in the field.

The fourth edition...

Foliations is one of the major concepts of modern geometry and topology meaning a partition of topological space into a disjoint sum of leaves. This book is devoted to geometry and topology of surface foliations and their links to ergodic theory, dynamical systems, complex analysis, differential and noncommutative geometry. This comprehensive book addresses graduate students and researchers and will serve as a reference book for experts in the field.

The present volume is a translation, revision and updating of our book (pub lished in French) with the title "Geometrie Algebrique Reelle." Since its pub lication in 1987 the theory has made advances in several directions. There have also been new insights into material already in the French edition. Many of these advances and insights have been incorporated in this English version of the book, so that it may be viewed as being substantially different from the original. We wish to thank Michael Buchner for his careful reading of the text and for his linguistic corrections and stylistic...

The aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research.

Smooth Four-Manifolds and Complex Surfaces

Slightly more than 25 years ago, the first text devoted entirely to sieve meth ods made its appearance, rapidly to become a standard source and reference in the subject. The book of H. Halberstam and H.-E. Richert had actually been conceived in the mid-1960's. The initial stimulus had been provided by the paper of W. B. Jurkat and Richert, which determined the sifting limit for the linear sieve, using a combination of the, A2 method of A. Selberg with combinatorial ideas which were in themselves of great importance and in terest. One of the declared objectives in writing their book was to...

During the last 30 years there have been several attempts at extending the notion of entropy to noncommutative dynamical systems. The authors present in the book the two most successful approaches to the extensions of measure entropy and topological entropy to the noncommutative setting and analyze in detail the main models in the theory.

The book addresses mathematicians and physicists, including graduate students, who are interested in quantum dynamical systems and applications of operator algebras and ergodic theory. Although the authors assume a basic knowledge of operator...

Neron models were invented by A. Neron in the early 1960s in order to study the integral structure of abelian varieties over number fields. Since then, arithmeticians and algebraic geometers have applied the theory of Neron models with great success. Quite recently, new developments in arithmetic algebraic geometry have prompted a desire to understand more about Neron models, and even to go back to the basics of their construction. The authors have taken this as their incentive to present a comprehensive treatment of Neron models. This volume of the renowned "Ergebnisse" series provides a...

The text of this book has its origins more than twenty- ve years ago. In the seminar of the Dutch Singularity Theory project in 1982 and 1983, the second-named author gave a series of lectures on Mixed Hodge Structures and Singularities, accompanied by a set of hand-written notes. The publication of these notes was prevented by a revolution in the subject due to Morihiko Saito: the introduction of the theory of Mixed Hodge Modules around 1985. Understanding this theory was at the same time of great importance and very hard, due to the fact that it uni es many di erent theories which are quite...