Series: Sources and Studies in the History of Mathematics and Physical Sciences
2010, 247 p. 101 illus., Hardcover
ISBN: 978-1-84996-004-5
Due: January 2010
Although not so well known today, Book 4 of Pappusf Collection is one of the most important and influential mathematical texts from antiquity, both because of its content and because of its impact on early modern mathematics after 1600. As a kind of textbook in anthology format, the mathematical vignettes form a portrait of mathematics during the Hellenistic "Golden Age", illustrating central problems ? for example, it discusses all three of the famous ancient problems in geometry: squaring the circle; doubling the cube; and trisecting an angle ? varying solution strategies, and the different mathematical styles within ancient geometry.
Historians of mathematics will find it useful for scholarly work on ancient geometry and its reception in the early modern era and it will also serve as a source book for exemplary arguments in ancient geometry. Pappus himself probably intended Collection 4 to be an introductory survey of the classical geometrical tradition ? from the point of view of mathematical methods and strategies ? for readers that had a basic training in elementary geometry (Elements I ? VI). Likewise, this edition can be used as a textbook in advanced undergraduate and graduate courses on the history of ancient geometry.
General Introduction.- Part I - Text and translation;- Greek text and annotated translation.- Part II - Commentary;- 2 Plane Geometry.- 3 Plane Geometry, Archaic Style.- 4 Plane Geometry, Archimedean.- 5 Motion curves and symptoma-mathematics.- 6 Meta-theoretical passage.- 7 Angle trisection.- 8 General angle division.- 9 Quadratix, rectification property.- 10 Analysis for an Archimedean neusis.- Appendices
Series: Monografie Matematyczne , Vol. 70
2010, Approx. 300 p., Hardcover
ISBN: 978-3-0346-0435-2
Due: June 2010
This is a monograph in semi-infinite homological algebra, concentrated mostly on the semi-infinite theory of associative algebraic structures, but including also some material on the semi-infinite homology and cohomology of Lie algebras and topological groups. The main objects of study are the double-sided derived functors SemiExt and SemiTor, and the phenomenon of comodule-contramodule correspondence, connecting them with the more conventional, one-sided Ext and CtrTor. Contramodules, introduced originally by Eilenberg and Moore in 1960's but almost forgotten for four decades, play a very prominent role in this book, with many versions of them introduced and discussed. Among other topics considered in the monograph there are the semi-infinite model category structures and the relative nohomogeneous Koszul duality.
Preface.- Introduction.- 0 Preliminaries and Summary.- 1 Semialgebras and Semitensor Product.- 2 Derived Functor SemiTor.- 3 Semicontramodules and Semihomomorphisms.- 4 Derived Functor SemiExt.- 5 Comodule-Contramodule Correspondence.- 6 Semimodule-Semicontramodule Correspondence.- 7 Functoriality in the Coring.- 8 Functoriality in the Semialgebra.- 9 Closed Model Category Structures.- 10 A Construction of Semialgebras.- 11 Relative Nonhomogeneous Koszul Duality.- Appendix A Contramodules over Coalgebras over Fields.- Appendix B Comparison with Arkhipov's Ext^{\infty/2+*} and Sevostyanov's Tor_{\infty/2+*}.- Appendix C Semialgebras Associated to Harish-Chandra Pairs.- Appendix D Tate Harish-Chandra Pairs and Tate Lie Algebras.- Appendix E Groups with Open Profinite Subgroups.- Appendix F Algebraic Groupoids with Closed Subgroupoids.- Bibliography.- Index.
Series: Oberwolfach Seminars , Vol. 43
2010, Approx. 200 p., Softcover
ISBN: 978-3-0346-0289-1
Due: April 2010
Introductory text to an advanced topic of active research
This book focuses on recent advances in the classification of complex projective varieties. It is divided into two parts. The first part gives a detailed account of recent results in the minimal model program. In particular, it contains a complete proof of the theorems on the existence of flips, on the existence of minimal models for varieties of log general type and of the finite generation of the canonical ring. The second part is an introduction to the theory of moduli spaces. It includes topics such as representing and moduli functors, Hilbert schemes, the boundedness, local closedness and separatedness of moduli spaces and the boundedness for varieties of general type.
The book is aimed at advanced graduate students and researchers in algebraic geometry.
I Basics.- 1 Introduction.- 1.A. Classification.- 2 Preliminaries.- 2.A. Notation.- 2.B. Divisors.- 2.C. Reflexive sheaves.- 2.D. Cyclic covers.- 2.E. R-divisors in the relative setting.- 2.F. Vanishing theorems.- 2.G. Families and base change.- 2.H. Parameter spaces and deformations of families.- 3 Singularities.- 3.A. Canonical singularities.- 3.B. Cones.- 3.C. Log canonical singularities.- 3.D. Normal crossings.- 3.E. Pinch points.- 3.F. Semi-log canonical singularities.- 3.G. Pairs.- 3.H. Rational and du Bois singularities.- II Recent advances in the MMP.- 4 Introduction.- 5 The main result.- 5.A. The cone and base point free theorems.- 5.B. Flips and divisorial contractions.- 5.C. The minimal model program for surfaces.- 5.D. The main theorem and sketch of proof.- 5.E. The minimal model program with scaling.- 5.F. PL-flips.- 5.G. Corollaries.- 6 Multiplier ideal sheaves.- 6.A. Asymptotic multiplier ideal sheaves.- 6.B. Extending pluricanonical forms.- 7 Finite generation of the restricted algebra.- 7.A. Rationality of the restricted algebra.- 7.B. Proof of (5.69).- 8 Log terminal models.- 8.A. Special termination.- 8.B. Existence of log terminal models.- 9 Non-vanishing.- 9.A. Nakayama?Zariski decomposition.- 9.B. Non-vanishing.- 10 Finiteness of log terminal models.- III Compact moduli spaces.- 11 Moduli problems.- 11.A. Representing functors.- 11.B. Moduli functors.- 11.C. Coarse moduli spaces.- 12 Hilbert schemes.- 12.A. The Grassmannian functor.- 12.B. The Hilbert functor.- 13 The construction of the moduli space.- 13.A. Boundedness.- 13.B. Constructing the moduli space.- 13.C. Local closedness.- 13.D. Separatedness.- 14 Families and moduli functors.- 14.A. An important example.- 14.B. Q-Gorenstein families.- 14.C. Projective moduli schemes.- 14.D. Moduli of pairs and other generalizations.- 15 Singularities of stable varieties.- 15.A. Singularity criteria.- 15.B. Applications to moduli spaces and vanishing theorems.- 15.C. Deformations of DB singularities.- 16 Subvarieties of moduli spaces.- 16.A. Shafarevichfs conjecture.- 16.B. The Parshin-Arakelov reformulation.- 16.C. Shafarevichfs conjecture for number fields.- 16.D. From Shafarevich to Mordell: Parshinfs trick.- 16.E. Hyperbolicity and boundedness.- 16.F. Higher dimensional fibers.- 16.G. Higher dimensional bases.- 16.H. Uniform and effective bounds.- 16.I. Techniques.- 16.J. Allowing more general fibers.- 16.K. Iterated Kodaira?Spencer maps and strong non-isotriviality.- IV Solutions and hints to some of the exercises.
Series: Grundlehren der mathematischen Wissenschaften , Vol. 339
Set: Minimal Surfaces
2010, Approx. 670 p., Hardcover
ISBN: 978-3-642-11697-1
Due: April 12, 2010
Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume which can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces.
The treatise is a greatly revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296).
The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfacesof zero mean curvature. The final definition of a minimal surface is that of a non-constant harmonic mapping X: O-> R3 which is conformally parametrized on O-> R2 and may have branch points. Thereafter the classical theroy of minimal surfaces is surveyed, comprising many examples, a treatment of BjorlingLs initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto.
The second part of this volume begins with a survey of PlateauLs problem and of some of its modifications. One of the main features is a new completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorem of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components.
Then basic facts of stable minimal suurfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to NitscheLs uniqueness theorem and TomiLs finiteness result.
In addition, a theory of unstable solutions of PlateauLs problems is developed which is based on CourantLs mountain pass lemma. Furthermore, DirichletLs problem for nonparametric H-surfaces is solved, using the solution of PlateauLs problem dor H-surfaces and the pertinent estimates.
Introduction.- Part I. Introduction to the Geometry of Surfaces and to Minimal Surfaces.- 1.Differential Geometry of Surfaces in Three-Dimensional Euclidean Space.- 2.Minimal Surfaces.- 3.Representation Formulas and Examples of Minimal Surfaces.- Part II. Plateaufs Problem.- 4.The Plateau Problem, and its Ramifications.- 5.Stable Minimal- and H-Surfaces.- 6.Unstable Minimal Surfaces.- 7.Graphs with Prescribed Mean Curvature.- 8.Introduction to the Douglas Problem.- Problems.- 9. Appendix 1. On Relative Minimizers of Area and Energy.- Appendix 2. Minimal Surfaces in Heisenberg Groups.- Bibliography.- Index.
Series: Grundlehren der mathematischen Wissenschaften , Vol. 340
Set: Minimal Surfaces
2010, Approx. 565 p., Hardcover
ISBN: 978-3-642-11699-5
Due: April 11, 2010
"Regularity of Minimal Surfaces" begins with survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaceswith fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to deriva a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas.
This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of PlateauLs problem for H-surfaces in a Riemannian manifold.
A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed.
The final chapter on branch points presents a new approach to the Osserman-Gulliver-Alt theorem that area minimizing solutions of PlateauLs problem have no interior branch points.
Introduction.- Part I. Boundary Behaviour of Minimal Surfaces.- 1.Minimal Surfaces with Free Boundaries.- 2.The Boundary Behaviour of Minimal.- 3.Singular Boundary Points of Minimal Surfaces.- Part II. Geometric Properties of Minimal Surfaces.- 4.Enclosure and Existence Theorems for Minimal Surfaces and H-Surfaces. Isoperimetric Inequalities.- 5.The Thread Problem.- 6.Branch Points.- Bibliography.- Index.
Series: Grundlehren der mathematischen Wissenschaften , Vol. 341
Set: Minimal Surfaces
Originally published as part of volume 296 in the series: Grundlehren der mathematischen Wissenschaften
2010, Approx. 550 p., Hardcover
ISBN: 978-3-642-11705-3
Due: April 12, 2010
Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two chapters deal with existence, regularity and uniqueness theorems for minimal surfaces with partially free boundaries. Here one of the main features is the possibility of "edge-crawling" along free parts of the boundary.
The third chapter deals with a priori estimates for minimal surfaces in higher dimensions and for minimizers of singular integrals related to the area functional. In particular, far reaching Bernstein theorems are derived.
The second part of the book contains what one might justly call a "global theory of minimal surfaces" as envisioned by Smale. First, the Douglas problem is treated anew by using Teichmuller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to PlateauLs problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented.
Of particular interest are the so-called "forced Jacobi fields", which play an important role both for the index theorems and for the branch point theory developed in Vol.2 (Nr. 340).
Introduction.- Part I. Free Boundaries and Bernstein Theorems.- 1.Minimal Surfaces with Supporting Half-Planes.- 2.Embedded Minimal Surfaces with Partially Free Boundaries.- 3.Bernstein Theorems and Related Results.- Part II. Global Analysis of Minimal Surfaces.- 4.The General Problem of Plateau: Another Approach.- 5.The Index Theorems for Minimal Surfaces of Zero and Higher Genus.- 6.Euler Characteristic and Morse Theory for Minimal Surfaces.- Bibliography.- Index.