Advanced Lectures in Mathematics – serie

This volume presents a selection of work based upon lectures given by distinguished mathematicians at the Yau Mathematical Sciences Center at Tsinghua University, and at the Tsinghua Sanya International Mathematics Forum.Mathematics is very diversified and broad, but there are some basic underlying themes. The editors hope that the diversity of material presented here conveys the richness and unity of mathematics.

Algebraic geometry is, loosely speaking, concerned with the study of zero sets of polynomials (over an algebraically closed field). As one often reads in prefaces of int- ductory books on algebraic geometry, it is not so easy to develop the basics of algebraic geometry without a proper knowledge of commutative algebra. On the other hand, the commutative algebra one needs is quite difficult to understand without the geometric motivation from which it has often developed. Local analytic geometry is concerned with germs of zero sets of analytic functions, that is, the study of such sets in the neighborhood of a point. It is not too big a surprise that the basic theory of local analytic geometry is, in many respects, similar to the basic theory of algebraic geometry. It would, therefore, appear to be a sensible idea to develop the two theories simultaneously. This, in fact, is not what we will do in this book, as the "commutative algebra" one needs in local analytic geometry is somewhat more difficult: one has to cope with convergence questions. The most prominent and important example is the substitution of division with remainder. Its substitution in local analytic geometry is called the Weierstraft Division Theorem. The above remarks motivated us to organize the first four chapters of this book as follows. In Chapter 1 we discuss the algebra we need. Here, we assume the reader attended courses on linear algebra and abstract algebra, including some Galois theory.

Ruled varieties are unions of a family of linear spaces. They are objects of algebraic geometry as well as differential geometry, especially if the ruling is developable.This book is an introduction to both aspects, the algebraic and differential one. Starting from very elementary facts, the necessary techniques are developed, especially concerning Grassmannians and fundamental forms in a version suitable for complex projective algebraic geometry. Finally, this leads to recent results on the classification of developable ruled varieties and facts about tangent and secant varieties.Compared to many other topics of algebraic geometry, this is an area easily accessible to a graduate course.

The book first describes connections between some basic problems and technics of combinatorics and statistical physics. The discrete mathematics and physics terminology are related to each other. Using the established connections, some exciting activities in one field are shown from a perspective of the other field. The purpose of the book is to emphasize these interactions as a strong and successful tool. In fact, this attitude has been a strong trend in both research communities recently. It also naturally leads to many open problems, some of which seem to be basic. Hopefully, this book will help making these exciting problems attractive to advanced students and researchers.

This volume covers in a comprehensive way and at an elementary level essentially all the theorems and techniques which are commonly used and needed in any branches of mathematics, particularly in complex and in real analytic geometry, in commutative algebra, in algebraic geometry and in real algebraic geometry. In particular it presents Rueckert's complex nullstellensatz, Risler's real nullstellensatz, Tougeron's implicit function theorem and Artin's approximation theorem, to name a few. Up to now a student of any of the subjects mentioned above usually had to learn about power series within the framework of the vast theory of the subject. This had the advantage that he sees many theorems about power series as special instances of more general facts, but on the other hand made it highly time consuming to understand all these theorems. The present book opens another path - one gets acquainted with power series in a direct and elementary way, and then disposes of a good box of tools and examples to penetrate any of the subjects mentioned above, as well as some others.

This work illustrates two basic principles in the calculus of variations - the questions of existence of solutions, and closely related, the problem of regularity of minimizers. Chapter one studies variational problems for nonquadratic energy functionals defined on suitable classes of vector-valued functions where nonlinear constraints are also incorporated. Problems of this type arise for mappings between Riemannian manifolds or in nonlinear elasticity. Using direct methods for the existence of generalized minimizers is rather easy to establish, and it is then shown that regularity holds up to a set of small measure. Chapter two contains a short introduction into geometric measure theory, which serves as a basis for developing an existence theory for (generalized) manifolds with prescribed mean curvature form and boundary in arbitrary dimensions and co-dimensions. A major aspect of the book is that it concentrates on techniques, and presents methods which turn out to be useful for applications in regularity theorems, as well as for existence problems.

In recent years, algorithmic graph theory has become increasingly important since it serves as a link between discrete mathematics and theoretical computer science. This textbook introduces students of mathematics and computer science to the interrelated fields of graph theory, algorithms and complexity. No specific previous knowledge is assumed. The central theme of the book is a geometrical problem dating back to Jakob Steiner. This problem, now called the Steiner tree problem, was initially of importance only within the context of land surveying. Recent applications, as diverse as VLSI-layout and the study of phylogenetic trees, have, however, lead to significant interest in the problem. The resulting progress has uncovered fascinating connections to and among graph theory, the study of algorithms and complexity. The single problem thus serves to bind and motivate these areas. The book's topics include: exact algorithms; computational complexity; approximation algorithms; limits of approximability; randomness helps; the Manhattan Steiner problem; heuristics; packing of Steiner trees; and applications.A fundamental feature of the book is that each chapter ends with an "excursion" into some related area. These excursions reinforce the concepts and methods introduced for the Steiner tree problem by putting them in a broader context.

Dieses Buch entstand nach einer einsemestrigen Vorlesung an der Humboldt-Universitat Berlin im Studienjahr 1996/ 97 und ist eine Einfuhrung in die Theorie der Spinoren und Dirac-Operatoren uber Riemannschen Mannigfaltigkeiten. Vom Leser werden nur die grundlegenden Kenntnisse der Algebra und Geometrie im Umfang von zwei bis drei Jahren eines Mathematik- oder Physikstudiums erwartet. Ein Anhang gibt eine Einfuhrung in das aktuelle Gebiet der Seiberg-Witten-Theorie.

Among the intuitively appealing aspects of graph theory is its close connection to drawings and geometry. The development of computer technology has become a source of motivation to reconsider these connections, in particular geometric graphs are emerging as a new subfield of graph theory. Arrangements of points and lines are the objects for many challenging problems and surprising solutions in combinatorial geometry. The book is a collection of beautiful and mostly very recent results from the intersection of geometry, graph theory and combinatorics.

The aims of this book are threefold:We start with a naive description of a Markov chain as a memoryless random walk on a finite set. This is complemented by a rigorous definition in the framework of probability theory, and then we develop the most important results from the theory of homogeneous Markov chains on finite state spaces.Chains are called rapidly mixing if all of the associated walks, regardles of where they started, behave similarly already after comparitively few steps: it is impossible from observing the chain to get information on the starting position or the number of steps done so far. We will thoroughly study methods which have been proposed in the last decades to investigate this phenomenon.A number of examples will be studied to indicate how the methods treated in this book can be applied.

The purpose of coding theory is the design of efficient systems for the transmission of information. The mathematical treatment leads to certain finite structures: the error-correcting codes. Surprisingly problems which are interesting for the design of codes turn out to be closely related to problems studied partly earlier and independently in pure mathematics. In this book, examples of such connections are presented. The relation between lattices studied in number theory and  geometry and error-correcting codes is discussed. The book provides at the same time an introduction to the theory of integral lattices and modular forms and to coding theory.In the 3rd edition, again numerous corrections and improvements have been made and the text has been updated.