A Brief Introduction to Symplectic and Contact Manifolds

Author: Augustin Banyaga
Publisher: World Scientific
ISBN: 9789814696722
Release Date: 2016-08-08

The book introduces the basic notions in Symplectic and Contact Geometry at the level of the second year graduate student. It also contains many exercises, some of which are solved only in the last chapter. We begin with the linear theory, then give the definition of symplectic manifolds and some basic examples, review advanced calculus, discuss Hamiltonian systems, tour rapidly group and the basics of contact geometry, and solve problems in chapter 8. The material just described can be used as a one semester course on Symplectic and Contact Geometry. The book contains also more advanced material, suitable to advanced graduate students and researchers. Contents: Symplectic Vector SpacesSymplectic ManifoldsHamiltonian Systems and Poisson AlgebraGroup ActionsContact ManifoldsSolutions of Selected ExercisesEpilogue: The C0-Symplectic and Contact Topology Readership: Graduate students, researchers and more advanced mathematicians. Symplectic;Contact GeometryKey Features: It is briefThe easy part has been tested and been used for a short courseThe advanced material develops things related to one of the author's research furtherThere is no book, going from the very elementary part to the very advanced level, like this one

Introduction to Symplectic Topology

Author: Dusa McDuff
Publisher: Oxford University Press
ISBN: 9780198794899
Release Date: 2017-03-23
Genre: Mathematics

Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. A significantly revised second edition was published in 1998 introducing new sections and updates on the fast-developing area. This new third edition includes updates and new material to bring the book right up-to-date.

An Introduction to Extremal K hler Metrics

Author: Gábor Székelyhidi
Publisher: American Mathematical Soc.
ISBN: 9781470410476
Release Date: 2014-06-19
Genre: Mathematics

A basic problem in differential geometry is to find canonical metrics on manifolds. The best known example of this is the classical uniformization theorem for Riemann surfaces. Extremal metrics were introduced by Calabi as an attempt at finding a higher-dimensional generalization of this result, in the setting of Kähler geometry. This book gives an introduction to the study of extremal Kähler metrics and in particular to the conjectural picture relating the existence of extremal metrics on projective manifolds to the stability of the underlying manifold in the sense of algebraic geometry. The book addresses some of the basic ideas on both the analytic and the algebraic sides of this picture. An overview is given of much of the necessary background material, such as basic Kähler geometry, moment maps, and geometric invariant theory. Beyond the basic definitions and properties of extremal metrics, several highlights of the theory are discussed at a level accessible to graduate students: Yau's theorem on the existence of Kähler-Einstein metrics, the Bergman kernel expansion due to Tian, Donaldson's lower bound for the Calabi energy, and Arezzo-Pacard's existence theorem for constant scalar curvature Kähler metrics on blow-ups.

The Geometry of Physics

Author: Theodore Frankel
Publisher: Cambridge University Press
ISBN: 9781139505611
Release Date: 2011-11-03
Genre: Mathematics

This book provides a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism (in flat and curved space), thermodynamics, the Dirac operator and spinors, and gauge fields, including Yang–Mills, the Aharonov–Bohm effect, Berry phase and instanton winding numbers, quarks and quark model for mesons. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. The book is ideal for graduate and advanced undergraduate students of physics, engineering or mathematics as a course text or for self study. This third edition includes an overview of Cartan's exterior differential forms, which previews many of the geometric concepts developed in the text.

Cohomological Aspects in Complex Non K hler Geometry

Author: Daniele Angella
Publisher: Springer
ISBN: 9783319024417
Release Date: 2013-11-22
Genre: Mathematics

In these notes, we provide a summary of recent results on the cohomological properties of compact complex manifolds not endowed with a Kähler structure. On the one hand, the large number of developed analytic techniques makes it possible to prove strong cohomological properties for compact Kähler manifolds. On the other, in order to further investigate any of these properties, it is natural to look for manifolds that do not have any Kähler structure. We focus in particular on studying Bott-Chern and Aeppli cohomologies of compact complex manifolds. Several results concerning the computations of Dolbeault and Bott-Chern cohomologies on nilmanifolds are summarized, allowing readers to study explicit examples. Manifolds endowed with almost-complex structures, or with other special structures (such as, for example, symplectic, generalized-complex, etc.), are also considered.

Etale Cohomology Theory

Author: Lei Fu
Publisher: World Scientific
ISBN: 9789814675109
Release Date: 2015-02-27
Genre: Mathematics

Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ℓ-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra. Contents:Descent TheoryEtale Morphisms and Smooth MorphismsEtale Fundamental GroupsGroup Cohomology and Galois CohomologyEtale CohomologyDerived Categories and Derived FunctorsBase Change TheoremsDualityFiniteness Theoremsℓ-Adic Cohomology Readership: Graduate students and researchers in pure mathematics. Key Features:This is a revised version of an earlier edition, in which some errors and misprints are corrected, and some paragraphs are rewritten for better exposition. While the most complete treatment on etale cohomology is in SGA 1, 4, 4 1/2 and 5, which is about 3000 pages long, the existing textbooks on etale cohomology theory are, however, incomplete. This book, at about 600 pages, gives a relatively complete treatment of etale cohomology theoryTo achieve an understanding of this book, the reader is only assumed to be familiar with basic algebraic geometry (up to the level of the first three chapters in Algebraic Geometry by R Hartshorne, Springer-Verlag, 1977) and advanced commutative algebra (up to the level of Commutative Algebra by H Matsumura, Benjamin, New York, 1970)Keywords:Etale Topology;Sheaf;Cohomology;Base Change Theorem;Duality;L-Adic CohomologyReview: Reviews of the First Edition: “A very nice feature of Fu's work is the inclusion of some relevant topics, that are covered only briefly (or not at all) in other references. As is the case throughout the book, proofs are given for almost all results in these chapters. One should also remark that precise hypotheses are explicitly stated in most cases for each result. This shall prove very handy when using the book as a reference.” Mathematical Reviews “The book will certainly be very useful to anybody wishing to understand the key tools and results in étale cohomology theory, together with their proofs… Any reader with adequate background and an interest in seriously studying étale cohomology will find the thoroughness of this book really useful.” MathSciNet

An Introduction to Symplectic Geometry

Author: Rolf Berndt
Publisher: American Mathematical Soc.
ISBN: 0821820567
Release Date: 2001
Genre: Mathematics

Starts with the basics of the geometry of symplectic vector spaces. Then, symplectic manifolds are defined and explored. In addition to the essential classic results, such as Darboux's theorem, more recent results and ideas are also included here, such as symplectic capacity and pseudoholomorphic curves. These ideas have revolutionized the subject. The main examples of symplectic manifolds are given, including the cotangent bundle, Kahler manifolds, and coadjoint orbits.Further principal ideas are carefully examined, such as Hamiltonian vector fields, the Poisson bracket, and connections with contact manifolds. Berndt describes some of the close connections between symplectic geometry and mathematical physics in the last two chapters of the book. In particular, the moment map is defined and explored, both mathematically and in its relation to physics. He also introduces symplectic reduction, which is an important tool for reducing the number of variables in a physical system and for constructing new symplectic manifolds from old. The final chapter is on quantization, which uses symplectic methods to take classical mechanics to quantum mechanics.This section includes a discussion of the Heisenberg group and the Weil (or metaplectic) representation of the symplectic group. Several appendices provide background material on vector bundles, on cohomology, and on Lie groups and Lie algebras and their representations.

Mixed Hodge Structures

Author: Christiaan Peters
Publisher: Springer Science & Business Media
ISBN: 3540770178
Release Date: 2008-02-27
Genre: Mathematics

This is comprehensive basic monograph on mixed Hodge structures. Building up from basic Hodge theory the book explains Delingne's mixed Hodge theory in a detailed fashion. Then both Hain's and Morgan's approaches to mixed Hodge theory related to homotopy theory are sketched. Next comes the relative theory, and then the all encompassing theory of mixed Hodge modules. The book is interlaced with chapters containing applications. Three large appendices complete the book.

Some Nonlinear Problems in Riemannian Geometry

Author: Thierry Aubin
Publisher: Springer Science & Business Media
ISBN: 9783662130063
Release Date: 2013-03-09
Genre: Mathematics

This book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.

Etale Cohomology PMS 33

Author: James S. Milne
Publisher: Princeton University Press
ISBN: 9781400883981
Release Date: 2016-10-11
Genre: Mathematics

One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

A Survey on Classical Minimal Surface Theory

Author: William Meeks
Publisher: American Mathematical Soc.
ISBN: 9780821869123
Release Date: 2012
Genre: Mathematics

Meeks and Perez present a survey of recent spectacular successes in classical minimal surface theory. The classification of minimal planar domains in three-dimensional Euclidean space provides the focus of the account. The proof of the classification depends on the work of many currently active leading mathematicians, thus making contact with much of the most important results in the field. Through the telling of the story of the classification of minimal planar domains, the general mathematician may catch a glimpse of the intrinsic beauty of this theory and the authors' perspective of what is happening at this historical moment in a very classical subject. This book includes an updated tour through some of the recent advances in the theory, such as Colding-Minicozzi theory, minimal laminations, the ordering theorem for the space of ends, conformal structure of minimal surfaces, minimal annular ends with infinite total curvature, the embedded Calabi-Yau problem, local pictures on the scale of curvature and topology, the local removable singularity theorem, embedded minimal surfaces of finite genus, topological classification of minimal surfaces, uniqueness of Scherk singly periodic minimal surfaces, and outstanding problems and conjectures.

Minimal Submanifolds in Pseudo Riemannian Geometry

Author: Henri Anciaux
Publisher: World Scientific
ISBN: 9789814291248
Release Date: 2011
Genre: Mathematics

Since the foundational work of Lagrange on the differential equation to be satisfied by a minimal surface of the Euclidean space, the theory of minimal submanifolds have undergone considerable developments, involving techniques from related areas, such as the analysis of partial differential equations and complex analysis. On the other hand, the relativity theory has led to the study of pseudo-Riemannian manifolds, which turns out to be the most general framework for the study of minimal submanifolds. However, most of the recent books on the subject still present the theory only in the Riemannian case. For the first time, this textbook provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian geometry, only assuming from the reader some basic knowledge about manifold theory. Several classical results, such as the Weierstrass representation formula for minimal surfaces, and the minimizing properties of complex submanifolds, are presented in full generality without sacrificing the clarity of exposition. Finally, a number of very recent results on the subject, including the classification of equivariant minimal hypersurfaces in pseudo-Riemannian space forms and the characterization of minimal Lagrangian surfaces in some pseudo-Khler manifolds are given.