This book treats Jacques Tit's beautiful theory of buildings, making that theory accessible to readers with minimal background. It covers all three approaches to buildings, so that the reader can choose to concentrate on one particular approach. Beginners can use parts of the new book as a friendly introduction to buildings, but the book also contains valuable material for the active researcher.

This book is suitable as a textbook, with many exercises, and it may also be used for self-study.

In the library at Trinity College, Cambridge in 1976, George Andrews of Pennsylvania State University discovered a sheaf of pages in the handwriting of Srinivasa Ramanujan. Soon designated as "Ramanujan 's Lost Notebook," it contains considerable material on mock theta functions and undoubtedly dates from the last year of Ramanujan 's life. In this book, the notebook is presented with additional material and expert commentary.

This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature 1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

Multiplicative invariant theory, as a research area in its own right within the wider spectrum of invariant theory, is of relatively recent vintage. The present text offers a coherent account of the basic results achieved thus far..

Multiplicative invariant theory is intimately tied to integral representations of finite groups. Therefore, the field has a predominantly discrete, algebraic flavor. Geometry, specifically the theory of algebraic groups, enters through Weyl groups and their root lattices as well as via character lattices of algebraic tori.

Throughout the text, numerous explicit examples of multiplicative invariant algebras and fields are presented, including the complete list of all multiplicative invariant algebras for lattices of rank 2.

The book is intended for graduate and postgraduate students as well as researchers in integral representation theory, commutative algebra and, mostly, invariant theory.

From the reviews "This book gives a thorough introduction to several theories that are fundamental to research on modular forms. Most of the material, despite its importance, had previously been unavailable in textbook form. Complete and readable proofs are given... In conclusion, this book is a welcome addition to the literature for the growing number of students and mathematicians in other fields who want to understand the recent developments in the theory of modular forms."
#"Mathematical Reviews"#
"This book will certainly be indispensable to all those wishing to get an up-to-date initiation to the theory of modular forms."
#"Publicationes Mathematicae"#

This edition has been called startlingly up-to-date, and in this corrected second printing you can be sure that it 's even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.

Devoted to the theory of Lie algebras and algebraic groups, this book includes a large amount of commutative algebra and algebraic geometry so as to make it as self-contained as possible. The aim of the book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included, and some recent results are discussed in the final chapters.

This classic geometry text explores the theory of 3-dimensional convex polyhedra in a unique fashion, with exceptional detail. Vital and clearly written, the book includes the basics of convex polyhedra and collects the most general existence theorems for convex polyhedra that are proved by a new and unified method. This edition includes a comprehensive bibliography by V.A. Zalgaller, and related papers as supplements to the original text.

This collection of surveys present an overview of recent developments in Complex Geometry. Topics range from curve and surface theory through special varieties in higher dimensions, moduli theory, K hler geometry, and group actions to Hodge theory and characteristic p-geometry.

Written by established experts this book will be a must for mathematicians working in Complex Geometry

The first survey of its kind, written by internationally known, outstanding experts who developed substantial parts of the field. The book contains an introduction written by Remmert, describing the history of the subject, and is very useful to graduate students and researchers in complex analysis, algebraic geometry and differential geometry.

The content of this monograph is situated in the intersection of important branches of mathematics like the theory of one complex variable, algebraic geometry, low dimensional topology and, from the point of view of the techniques used, com- natorial group theory. The main tool comes from the Uniformization Theorem for Riemannsurfaces, whichrelatesthetopologyofRiemannsurfacesandholomorphic or antiholomorphic actions on them to the algebra of classical cocompact Fuchsian groups or, more generally, non-euclidean crystallographic groups. Foundations of this relationship were established by A. M. Macbeath in the early sixties and dev- oped later by, among others, D. Singerman. Another important result in Riemann surface theory is the connection between Riemannsurfacesandtheir symmetrieswith complexalgebraiccurvesandtheirreal forms. Namely, there is a well known functorial bijective correspondence between compact Riemann surfaces and smooth, irreducible complex projective curves. The fact that a Riemann surface has a symmetry means, under this equivalence, that the corresponding complex algebraic curve has a real form, that is, it is the complex- cation of a real algebraic curve. Moreover, symmetries which are non-conjugate in the full group of automorphisms of the Riemann surface, correspond to real forms which are birationally non-isomorphic over the reals. Furthermore, the set of points xedbyasymmetryishomeomorphictoaprojectivesmoothmodeloftherealform

The goal of the book is to present, in a complete and comprehensive way, a few areas of mathematics interlacing around the Poncelet porism: dynamics of integrable billiards, algebraic geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics. Most important results and ideas connected to Poncelet theorem are presented: classical as well as modern ones, together with historical overview with analysis of classical ideas, their natural generalizations, and natural generalizations. Special attention is payed to realization of the Griffiths and Harris programme about Poncelet-type problems and addition theorems. This programme, formulated three decades ago, is aimed to understanding of higher-dimensional analogues of Poncelet problems and realization of synthetic approach of higher genus addition theorems. This problem is realized in a sequence of papers of the authors, published in last 20 years. Some of those results represent the key contribution to the development of the theory, while this book gives the complete image.

Intersection cohomology assigns groups which satisfy a generalized form of Poincare duality over the rationals to a stratified singular space. This monograph introduces a method that assigns to certain classes of stratified spaces cell complexes, called intersection spaces, whose ordinary rational homology satisfies generalized Poincare duality. The cornerstone of the method is a process of spatial homology truncation, whose functoriality properties are analyzed in detail. The material on truncation is autonomous and may be of independent interest tohomotopy theorists. The cohomology of intersection spaces is not isomorphic to intersection cohomology and possesses algebraic features such as perversity-internal cup-products and cohomology operations that are not generally available for intersection cohomology. A mirror-symmetric interpretation, as well as applications to string theory concerning massless D-branes arising in type IIB theory during a Calabi-Yau conifold transition, are discussed.

Algorithms in algebraic geometry go hand in hand with software packages that implement them. Together they have established the modern field of computational algebraic geometry which has come to play a major role in both theoretical advances and applications. Over the past fifteen years, several excellent general purpose packages for computations in algebraic geometry have been developed, such as, CoCoA, Singular and Macaulay 2. While these packages evolve continuously, incorporating new mathematical advances, they both motivate and demand the creation of new mathematics and smarter algorithms.

This volume reflects the workshop Software for Algebraic Geometry held in the week from 23 to 27 October 2006, as the second workshop in the thematic year on Applications of Algebraic Geometry at the IMA. The papers in this volume describe the software packages Bertini, PHClab, Gfan, DEMiCs, SYNAPS, TrIm, Gambit, ApaTools, and the application of Risa/Asir to a conjecture on multiple zeta values. They offer the reader a broad view of current trends in computational algebraic geometry through software development and applications."

First Edition sold over 2500 copies in the Americas; New Edition contains three new chapters and two new appendices

This book grew out of the Oberwolfach-SeminarHigherDimensionalAlgebraicGeo- tryorganizedbythetwoauthorsinOctober2008. Theaimoftheseminarwas tointroduce advanced PhD students and young researchers to recent advances and research topics in higher dimensional algebraic geometry. The main emphasis was on the minimal model program and on the theory of moduli spaces. The authors would like to thank the Mathematishes Forshunginstitut Oberwolfach for its hospitality and for making the above mentioned seminar possible, the participants to the seminar for their useful comments, and Alex Kuronya, Max Lieblich, and Karl Schwede for valuable suggestions and conversations. The ?rst named author was partially supported by the National Science Foundation under grant number DMS-0757897 and would like to thank Aleksandra, Stefan, Ana, Sasha, Kristina and Daniela Jovanovic-Haconfor their love and continuos support. The second named author was partially supported by the National Science Foun- tion under grant numbers DMS-0554697 and DMS-0856185, and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics at the University of Wa- ington. He would also like to thank Timea Tihanyi for her enduring love and support throughout and beyond this project and his other co-authors for their patience and und- standing. Contents I Basics 1 1INTRODUCTION 3 1. A. CLASSIFICATION 3 2PRELIMINARIES 17 2. A. NOTATION 17 2. B. DIVISORS 18 2. C. REFLEXIVE SHEAVES 20 2. D. CYCLIC COVERS 21 2. E. R-DIVISORS IN THE RELATIVE SETTING 22 2. F. FAMILIES AND BASE CHANGE 24 2. G. PARAMETER SPACES AND DEFORMATIONS OF FAMILIES 25 3SINGULARITIES 27 3. A."

This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles' resolution of Fermat's Last Theorem.

This book gives an account of theoretical and algorithmic developments on the integral closure of algebraic structures. It gives a comprehensive treatment of Rees algebras and multiplicity theory while pointing to applications in many other problem areas. Its main goal is to provide complexity estimates by tracking numerically invariants of the structures that may occur.

Nilpotent Ue algebras have played an Important role over the last ye!US : either In the domain at Algebra when one considers Its role In the classlftcation problems of Ue algebras, or In the domain of geometry since one knows the place of nilmanlfolds In the Illustration, the description and representation of specific situations. The first fondamental results In the study of nilpotent Ue algebras are obvlsouly, due to Umlauf. In his thesis (leipZig, 1991), he presented the first non trlvlal classifications. The systematic study of real and complex nilpotent Ue algebras was Independently begun by D1xmler and Morozov. Complete classifications In dimension less than or equal to six were given and the problems regarding superior dimensions brought to light, such as problems related to the existence from seven up, of an infinity of non Isomorphic complex nilpotent Ue algebras. One can also find these losts (for complex and real algebras) In the books about differential geometry by Vranceanu. A more formal approach within the frame of algebraiC geometry was developed by Michele Vergne. The variety of Ue algebraiC laws Is an affine algebraic subset In this view the role variety and the nilpotent laws constitute a Zarlski's closed of Irreduclbl~ components appears naturally as well the determination or estimate of their numbers. Theoritical physiCiSts, Interested In the links between diverse mechanics have developed the Idea of contractions of Ue algebras (Segal, Inonu, Wlgner). That Idea was In fact very convenient In the determination of components.

This book provides a quick access to computational tools for algebraic geometry, the mathematical discipline which handles solution sets of polynomial equations. Originating from a number of intense one week schools taught by the authors, the text is designed so as to provide a step by step introduction which enables the reader to get started with his own computational experiments right away. The authors present the basic concepts and ideas in a compact way.

The relation between mathematics and physics has a long history, in which the role of number theory and of other more abstract parts of mathematics has recently become more prominent.

More than ten years after a first meeting in 1989 between number theorists and physicists at the Centre de Physique des Houches, a second 2-week event focused on the broader interface of number theory, geometry, and physics.

This book is the result of that exciting meeting, and collects, in 2 volumes, extended versions of the lecture courses, followed by shorter texts on special topics, of eminent mathematicians and physicists.

The present volume has three parts: Conformal Field Theories, Discrete Groups, Renomalization.

The companion volume is subtitled: On Random Matrices, Zeta Functions and Dynamical Systems (Springer, 3-540-23189-7).

Historically, complex analysis and geometrical function theory have been inten sively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last fifty years the theory of holomorphic mappings on complex spaces has been studied by many mathemati cians with many applications to nonlinear analysis, functional analysis, differential equations, classical and quantum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dy namical system: dx / dt + f ( x) = 0, where x is a variable describing the state of the system under study, and f is a vector function of x. The study of such systems when f is a monotone or an accretive (generally nonlinear) operator on the under lying space has been recently the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems (see, for example, [19, 13] and [29]). In a parallel development (and even earlier) the generation theory of one parameter semigroups of holomorphic mappings in en has been the topic of interest in the theory of Markov stochastic processes and, in particular, in the theory of branching processes (see, for example, [63, 127, 48] and [69]).

In various contexts of topology, algebraic geometry, and algebra (e.g. group representations), one meets the following situation. One has two contravariant functors K and A from a certain category to the category of rings, and a natural transformation p: K--+A of contravariant functors. The Chern character being the central exam ple, we call the homomorphisms Px: K(X)--+ A(X) characters. Given f: X--+ Y, we denote the pull-back homomorphisms by and fA: A(Y)--+ A(X). As functors to abelian groups, K and A may also be covariant, with push-forward homomorphisms and fA: A( X)--+ A(Y). Usually these maps do not commute with the character, but there is an element r f E A(X) such that the following diagram is commutative: K(X) A(X) fK j J A K( Y) ------p;-+ A( Y) The map in the top line is p x multiplied by r f. When such commutativity holds, we say that Riemann-Roch holds for f. This type of formulation was first given by Grothendieck, extending the work of Hirzebruch to such a relative, functorial setting. Since then viii INTRODUCTION several other theorems of this Riemann-Roch type have appeared. Un derlying most of these there is a basic structure having to do only with elementary algebra, independent of the geometry. One purpose of this monograph is to describe this algebra independently of any context, so that it can serve axiomatically as the need arises."

This monograph covers one of the divisions of mathematical theory of control which examines moving objects functionating under conflict and uncertainty conditions. To identify this range of problems we use the term "conflict con trolled processes," coined in recent years. As the name itself does not imply the type of dynamics (difference, ordinary differential, difference-differential, integral, or partial differential equations) the differential games falI within its realms. The problems of search and tracking moving objects are also referred to the field of conflict controlled process. The contents of the monograph is confined to studying classical pursuit-evasion problems which are central to the theory of conflict controlled processes. These problems underlie the theory and are of considerable interest to researchers up to now. It should be noted that the methods of "Line of Sight," "Parallel Pursuit," "Proportional N avigation,""Modified Pursuit" and others have been long and well known among engineers engaged in design of rocket and space technology. An abstract theory of dynamic game problems, in its turn, is based on the methods originated by R. Isaacs, L. S. Pontryagin, and N. N. Krasovskii, and on the approaches developed around these methods. At the heart of the book is the Method of Resolving Functions which was realized within the class of quasistrategies for pursuers and then applied to the solution of the problems of "hand-to-hand," group, and succesive pursuit."

A new and complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space, with most of the results being published for the first time. Highlights of the book include a classification of semi-abelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian varieties over number fields, and Eichler integrals in several variables, together with a new approach to Siegel modular forms. A valuable source of reference for researchers and graduate students interested in algebraic geometry, Shimura varieties or diophantine geometry.