By Mircea Puta
This quantity provides quite a few elements of the geometry of symplectic and Poisson manifolds, and functions in Hamiltonian mechanics and geometric quantization are indicated.
bankruptcy 1 offers a few common evidence approximately symplectic vector house, symplectic manifolds and symplectic aid. bankruptcy 2 offers with the research of Hamiltonian mechanics. bankruptcy three considers a few regular proof referring to Lie teams and algebras which bring about the idea of momentum mappings and the Marsden--Weinstein aid. Chapters four and five contemplate the speculation and the soundness of equilibrium suggestions of Hamilton--Poisson mechanical structures. Chapters 6 and seven are dedicated to the speculation of geometric quantization. This leads, in bankruptcy eight, to themes comparable to foliated cohomology, the idea of the Dolbeault--Kostant advanced, and their functions. A dialogue of the relation among geometric quantization and the Marsden--Weinstein aid is gifted in bankruptcy nine. the ultimate bankruptcy considers extending the speculation of geometric quantization to Poisson manifolds, through the speculation of symplectic groupoids.
each one bankruptcy concludes with difficulties and recommendations, lots of which current major functions and, at times, significant theorems.
For graduate scholars and researchers whose pursuits and paintings contain symplectic geometry and Hamiltonian mechanics.
By Jürgen Jost
Even though Riemann surfaces are a time-honoured box, this booklet is novel in its large standpoint that systematically explores the relationship with different fields of arithmetic. it may function an creation to modern arithmetic as a complete because it develops historical past fabric from algebraic topology, differential geometry, the calculus of diversifications, elliptic PDE, and algebraic geometry. it really is specific between textbooks on Riemann surfaces in together with an creation to Teichmüller conception. The analytic process is also new because it relies at the concept of harmonic maps. For this 2d variation the writer has additional more suitable features of presentation of varied elements of the textual content.
This e-book is an creation to differential manifolds. It provides stable preliminaries for extra complicated themes: Riemannian manifolds, differential topology, Lie thought. It presupposes little history: the reader is barely anticipated to grasp easy differential calculus, and a bit point-set topology. The booklet covers the most subject matters of differential geometry: manifolds, tangent house, vector fields, differential varieties, Lie teams, and some extra subtle issues resembling de Rham cohomology, measure thought and the Gauss-Bonnet theorem for surfaces.
Its ambition is to provide good foundations. specifically, the creation of “abstract” notions equivalent to manifolds or differential varieties is prompted through questions and examples from arithmetic or theoretical physics. greater than one hundred fifty routines, a few of them effortless and classical, a few others extra subtle, can assist the newbie in addition to the extra specialist reader. suggestions are supplied for many of them.
The ebook will be of curiosity to numerous readers: undergraduate and graduate scholars for a primary touch to differential manifolds, mathematicians from different fields and physicists who desire to collect a few feeling approximately this pretty theory.
The unique French textual content advent aux variétés différentielles has been a best-seller in its class in France for plenty of years.
Jacques Lafontaine was once successively assistant Professor at Paris Diderot college and Professor on the college of Montpellier, the place he's almost immediately emeritus. His major study pursuits are Riemannian and pseudo-Riemannian geometry, together with a few points of mathematical relativity. along with his own examine articles, he used to be enthusiastic about numerous textbooks and examine monographs.
This ebook is dedicated to pseudo-holomorphic curve equipment in symplectic geometry. It comprises an creation to symplectic geometry and appropriate concepts of Riemannian geometry, proofs of Gromov's compactness theorem, an research of neighborhood houses of holomorphic curves, together with positivity of intersections, and purposes to Lagrangian embeddings difficulties. The chapters are according to a sequence of lectures given formerly via the authors M. Audin, A. Banyaga, P. Gauduchon, F. Labourie, J. Lafontaine, F. Lalonde, Gang Liu, D. McDuff, M.-P. Muller, P. Pansu, L. Polterovich, J.C. Sikorav. In an try and make this publication obtainable additionally to graduate scholars, the authors give you the precious examples and strategies had to comprehend the functions of the speculation. The exposition is largely self-contained and contains various routines.
By Reinhard Laubenbacher
The tales of 5 mathematical trips into new geographical regions, pieced jointly from the writings of the explorers themselves. a few have been guided by way of mere interest and the joys of event, others via simpler explanations. In each one case the result used to be an unlimited growth of the identified mathematical global and the realisation that also larger vistas stay to be explored. The authors inform those tales by means of guiding readers during the very phrases of the mathematicians on the center of those occasions, supplying an insightinto the artwork of imminent mathematical difficulties. The 5 chapters are thoroughly self sustaining, with various degrees of mathematical sophistication, and should allure scholars, teachers, and the intellectually curious reader. through operating via a number of the unique resources and supplementary workouts, which debate and resolve -- or try and remedy -- an exceptional challenge, this ebook is helping readers detect the roots of recent difficulties, principles, and ideas, even complete topics. scholars also will see the hindrances that prior thinkers needed to transparent with the intention to make their respective contributions to 5 vital topics within the evolution of arithmetic.
The current paintings grew out of a examine of the Maslov classification (e. g. (37]), that's a primary invariant in asymptotic research of partial differential equations of quantum physics. one of many many in terpretations of this category used to be given by way of F. Kamber and Ph. Tondeur (43], and it exhibits that the Maslov type is a secondary attribute type of a fancy trivial vector package endowed with a true aid of its constitution team. (In the fundamental paper of V. I. Arnold concerning the Maslov type (2], it's also mentioned with no information that the Maslov classification is attribute within the classification of vector bundles pointed out pre viously. ) consequently, we would have liked to check the total variety of secondary attribute periods taken with this interpretation, and we gave a brief description of the implications in (83]. It became out entire exposition of this idea used to be quite long, and, furthermore, I felt that many strength readers must use loads of scattered references so one can locate the mandatory info from both symplectic geometry or the speculation of the secondary attribute sessions. at the otherhand, either those topics are of a miles higher curiosity in differential geome try to topology, and within the purposes to actual theories.
By Peter W. Michor
This ebook treats the basics of differential geometry: manifolds, flows, Lie teams and their activities, invariant idea, differential types and de Rham cohomology, bundles and connections, Riemann manifolds, isometric activities, and symplectic and Poisson geometry. The format of the cloth stresses naturality and functoriality from the start and is as coordinate-free as attainable. Coordinate formulation are continually derived as additional details. a few appealing strange features of this publication are as follows: preliminary submanifolds and the Frobenius theorem for distributions of nonconstant rank (the Stefan-Sussman idea) are mentioned. Lie teams and their activities are handled early on, together with the slice theorem and invariant conception. De Rham cohomology contains that of compact Lie teams, resulting in the examine of (nonabelian) extensions of Lie algebras and Lie teams. The Frolicher-Nijenhuis bracket for tangent package valued differential varieties is used to specific any form of curvature and moment Bianchi identification, even for fiber bundles (without constitution groups). Riemann geometry starts off with a cautious therapy of connections to geodesic buildings to sprays to connectors and again to connections, going through the second one and 3rd tangent bundles. The Jacobi move at the moment tangent package is a brand new point coming from this standpoint. Symplectic and Poisson geometry emphasizes workforce activities, momentum mappings, and rate reductions. This publication provides the cautious reader operating wisdom in quite a lot of themes of contemporary coordinate-free differential geometry in no longer too many pages. A prerequisite for utilizing this e-book is an effective wisdom of undergraduate research and linear algebra.
The winding quantity is likely one of the most elementary invariants in topology. It measures the variety of instances a relocating element $P$ is going round a hard and fast aspect $Q$, only if $P$ travels on a direction that by no means is going via $Q$ and that the ultimate place of $P$ is equal to its beginning place. this straightforward concept has far-reaching functions. The reader of this ebook will find out how the winding quantity can assist us express that each polynomial equation has a root (the basic theorem of algebra), warrantly a good department of 3 items in house by way of a unmarried planar reduce (the ham sandwich theorem), clarify why each basic closed curve has an inside of and an out of doors (the Jordan curve theorem), relate calculus to curvature and the singularities of vector fields (the Hopf index theorem), let one to subtract infinity from infinity and get a finite resolution (Toeplitz operators), generalize to provide a primary and gorgeous perception into the topology of matrix teams (the Bott periodicity theorem). most of these topics and extra are built beginning basically from arithmetic that's universal in final-year undergraduate classes. This publication is released in cooperation with arithmetic complicated examine Semesters.