Book Description

During the last few years, the field of nonlinear problems has undergone great development. This book consisting of the updated Grundlehren volume 252 by the author and of a newly written part, deals with some important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved. Each problem is explained, up-to-date results are given and proofs are presented. Thus, the reader is given access, for each specific problem, to its present status of solution as well as to the most up-to-date methods for approaching it. The main objective of the book is to explain some methods and new techniques, and to apply them. It deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber. ... Read more

Book Description

CONICS AND CUBICS is an accessible introduction to algebraic curves. Its focus on curves of degree at most three keeps results tangible and proofs transparent. Theorems follow naturally from high school algebra and two key ideas, homogeneous coordinates and intersection multiplicities.

By classifying irreducible cubics over the real numbers and proving that their points form abelian groups, the book gives readers easy access to the study of elliptic curves. It includes a simple proof of Bezout's Theorem on the number of intersections of any two curves without common factors.

The book is a text for a one-semester course. The course can serve both as the one undergraduate geometry course taken by mathematics majors in general and as a sequel to college geometry for prospective or current teachers of secondary school mathematics. The only prerequisite is first-year calculus. ... Read more

Reviews (1)

Welcome Addition to the Literature
This book may be the most elementary introduction to algebraic geometry. Still it is roughly senior level and unlike the review above I am not sure that it is suitable for students in secondary education. Nonetheless, it fills a niche that has been largely vacant in the undergraduate literature and I recommend it to serious students and undergraduates alike. ... Read more

Book Description

This monograph develops projective geometries and provides a systematic treatment of morphisms. It is unique in that it does not confine itself to isomorphisms. This work introduces a new fundamental theorem and its applications describing homogeneous coordinates as morphisms of projective geometries by semilinear maps. Other topics treated include three equivalent definitions of projective geometries and their correspondence with certain lattices; quotients of projective geometries and isomorphism theorems; recent results in dimension theory; morphisms and homomorphisms of projective geometries; special morphisms; duality theory; morphisms of affine geometries; polarities; orthogonalities; Hilbertian geometries and propositional systems. The book concludes with a large section of exercises.

Audience: This volume will be of interest to mathematicians and researchers whose work involves projective geometries and their morphisms, semilinear maps and sesquilinear forms, lattices, category theory, and quantum mechanics. This book can also be recommended as a text in axiomatic geometry. ... Read more

Book Description

This book is devoted to a classical topic that has undergonerapid and fruitful development over the past 25 years, namelyBäcklund and Darboux transformations and their applicationsin the theory of integrable systems, also known as solitontheory.

The book consists of two parts. The first is a series ofintroductory pedagogical lectures presented by leading expertsin the field. They are devoted respectively to Bäcklundtransformations of Painlevé equations, to the dressingmethod and Bäcklund and Darboux transformations, and to theclassical geometry of Bäcklund transformations and theirapplications to soliton theory. The second part containsoriginal contributions that represent new developments in thetheory and applications of these transformations.

Both the introductory lectures and the original talks werepresented at an International Workshop that took place inHalifax, Nova Scotia (Canada). This volume covers virtually allrecent developments in the theory and applications ofBäcklund and Darboux transformations. ... Read more

Book Description

Both mathematics and mathematical physics have many active areas of research where the interplay between geometry and quantum field theory has proved extremely fruitful. Duality, gauge field theory, geometric quantization, Seiberg-Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups offer some striking recent examples of modern topics which stand on the borderline between geometry and analysis on the one hand and quantum field theory on the other, where the physicist's and the mathematician's perspective complement each other, leading to new mathematical and physical concepts and results.

This volume introduces the reader to some basic mathematical and physical tools and methods required to follow the recent developments in some active areas of mathematical physics, including duality, gauge field theory, geometric quantization, Seiberg-Witten theory, spectral properties and families of Dirac operators, and the geometry of loop groups. It comprises seven self-contained lectures, which should progressively give the reader a precise idea of some of the techniques used in these areas, as well as a few short communications presented by young participants at the school. ... Read more

Book Description

Although extensively revised, this new edition continues in the fine tradition of its predecessor. Major changes include: a notation that formalizes the distinction between equality and congruence and between line, ray and line segment; a completely rewritten chapter on mathematical logic with inclusion of truth tables and the logical basis for the discovery of non-Euclidean geometries; expanded coverage of analytic geometry with more theorems discussed and proved with coordinate geometry; two distinct chapters on parallel lines and parallelograms; a condensed chapter on numerical trigonometry; more problems; expansion of the section on surface areas and volume; and additional review exercises at the end of each chapter. Concise and logical, it will serve as an excellent review of high school geometry. ... Read more

Reviews (1)

Great text for high school or college
I had used this text in a geometry class I took in college as a math major (class was not required, just needed hours to fill). After using this book and finishing the course, I found it to be much better than the geometry book I used in high school.

This book, like many high school texts, breaks geometry down into constructions using only a straight-edge and compass and then introduces definitions and theorems (some proved, some left to prove as exercises). The part that hooked me in was the fill-in-the-blank-step proofs that were partially completed but let the reader finish so that he or she will become familiar with "Statement/Reason" geometric proofs.

Finally, after ten chapters which deal with such topics as quadrilaterals, parallel lines, polygon areas, similar and congruent triangles, and circles, the authors introduce logic by methods of truth tables, Euler circles, and tautologies.

Last but not least, a chapter connecting algebra to 2-Dimensional (Cartesian) geometry offer the bridge between the "abstract" world of algebra and the "concrete" world of geometry. ... Read more

Book Description

This volume contains the proceedings of an AMS special sessionheld at the 1999 Joint Mathematics Meetings in San Antonio. Theparticipants were an international group of researchers studyingsingularities from algebraic and analytic viewpoints. Thecontributed papers contain original results as well as someexpository and historical material. This volume is dedicated toOscar Zariski, on the one hundredth anniversary of his birth.

Topics include the role of valuation theory in algebraic geometrywith recent applications to the structure of morphisms;algorithmic approaches to resolution of equisingular surfacesingularities and locally toric varieties; weak subintegralclosures of ideals and Rees valuations; constructions ofuniversal weakly subintegral extensions of rings; direct-sumdecompositions of finitely generated modules; construction andexamples of resolution graphs of surface singularities;Jacobians of meromorphic curves; investigation of spectralnumbers of curve singularities using Puiseux pairs; Gröbnerbasis calculations of Hochschild homology for hypersurfaces withisolated singularities; and the theory of characteristic classesof singular spaces---a brief history with conjectures and openproblems. ... Read more

Book Description

This self-contained and relatively elementary introduction to functions of several complex variables and complex (especially compact) manifolds is intended to be a synthesis of those topics and a broad introduction to the field. Part I is suitable for advanced undergraduates and beginning postgraduates whilst Part II is written more for the graduate student. The work as a whole will be useful to professional mathematicians or mathematical physicists who wish to acquire a working knowledge of this area of mathematics. Many exercises have been included and indeed they form an integral part of the text. The prerequisites for understanding Part I would be met by any mathematics student with a first degree and together the two parts provide an introduction to the more advanced works in the subject. ... Read more

Book Description

Comprising a selection of expository and research papers, Harmonic Analysis and Integral Geometry grew from presentations offered at the July 1998 Summer University of Safi, Morocco-an annual, advanced research school and congress. This lively and very successful event drew the attendance of many top researchers, who offered both individual lectures and coordinated courses on specific research topics within this fast growing subject.Harmonic Analysis and Integral Geometry presents important recent advances in the fields of Radon transforms, integral geometry, and harmonic analysis on Lie groups and symmetric spaces. Several articles are devoted to the new theory of Radon transforms on trees.With its related presentations addressing recent developments in various aspects of these intriguing areas of study, Harmonic Analysis and Integral Geometry becomes an important addition not only to the Research Notes in Mathematics series, but to the general mathematics literature. ... Read more

Book Description

Groupoids often occur when there is symmetry of a nature notexpressible in terms of groups. Other uses of groupoids caninvolve something of a dynamical nature. Indeed, some of themain examples come from group actions. It should also be notedthat in many situations where groupoids have been used, the mainemphasis has not been on symmetry or dynamics issues. While theimplicit symmetry and dynamics are relevant, the groupoidrecords mostly the structure of the space of leaves and theholonomy. More generally, the use of groupoids is very muchrelated to various notions of orbit equivalance.

This book presents the proceedings from the Joint Summer ResearchConference on "Groupoids in Analysis, Geometry, and Physics"held in Boulder, CO. The book begins with an introduction toways in which groupoids allow a more comprehensive view ofsymmetry than is seen via groups. Topics range from foliations,pseudo-differential operators, $KK$-theory, amenability, Fellbundles, and index theory to quantization of Poisson manifolds.Readers will find examples of important tools for working withgroupoids.

This book is geared to students and researchers. It is intendedto improve their understanding of groupoids and to encouragethem to look further while learning about the tools used. ... Read more

Book Description

ISBN: 0-87567-034-2 ... Read more

Book Description

The subject of this book is Osserman semi-Riemannian manifolds, and in particular, the Osserman conjecture in semi-Riemannian geometry. The treatment is pitched at the intermediate graduate level and requires some intermediate knowledge of differential geometry. The notation is mostly coordinate-free and the terminology is that of modern differential geometry. Known results toward the complete proof of Riemannian Osserman conjecture are given and the Osserman conjecture in Lorentzian geometry is proved completely. Counterexamples to the Osserman conjuncture in generic semi-Riemannian signature are provided and properties of semi-Riemannian Osserman manifolds are investigated. ... Read more

Book Description

Energy of knots is a theory that was introduced to create a "canonical configuration" of a knot — a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a "canonical configuration" of a knot in each knot type. It also considers this problems in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments. ... Read more