Embedding Problems In Symplectic Geometry

Author : Felix Schlenk
Publisher : Walter de Gruyter
Page : 261 pages
File Size : 41,7 Mb
Release : 2008-08-22
Category : Mathematics
ISBN : 9783110199697

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Embedding Problems in Symplectic Geometry by Felix Schlenk Pdf

Symplectic geometry is the geometry underlying Hamiltonian dynamics, and symplectic mappings arise as time-1-maps of Hamiltonian flows. The spectacular rigidity phenomena for symplectic mappings discovered in the last two decades show that certain things cannot be done by a symplectic mapping. For instance, Gromov's famous "non-squeezing'' theorem states that one cannot map a ball into a thinner cylinder by a symplectic embedding. The aim of this book is to show that certain other things can be done by symplectic mappings. This is achieved by various elementary and explicit symplectic embedding constructions, such as "folding", "wrapping'', and "lifting''. These constructions are carried out in detail and are used to solve some specific symplectic embedding problems. The exposition is self-contained and addressed to students and researchers interested in geometry or dynamics.

Author : Michele Audin,Jacques Lafontaine
Publisher : Birkhäuser
Page : 333 pages
File Size : 49,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034885089

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Holomorphic Curves in Symplectic Geometry by Michele Audin,Jacques Lafontaine Pdf

This book is devoted to pseudo-holomorphic curve methods in symplectic geometry. It contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of Gromov's compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to Lagrangian embeddings problems. The chapters are based on a series of lectures given previously by 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 attempt to make this book accessible also to graduate students, the authors provide the necessary examples and techniques needed to understand the applications of the theory. The exposition is essentially self-contained and includes numerous exercises.

Author : Ana Cannas da Silva
Publisher : Springer
Page : 220 pages
File Size : 48,9 Mb
Release : 2004-10-27
Category : Mathematics
ISBN : 9783540453307

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Lectures on Symplectic Geometry by Ana Cannas da Silva Pdf

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Author : Daniele Angella,Costantino Medori,Adriano Tomassini
Publisher : Springer
Page : 262 pages
File Size : 42,8 Mb
Release : 2017-10-12
Category : Mathematics
ISBN : 9783319629148

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Complex and Symplectic Geometry by Daniele Angella,Costantino Medori,Adriano Tomassini Pdf

This book arises from the INdAM Meeting "Complex and Symplectic Geometry", which was held in Cortona in June 2016. Several leading specialists, including young researchers, in the field of complex and symplectic geometry, present the state of the art of their research on topics such as the cohomology of complex manifolds; analytic techniques in Kähler and non-Kähler geometry; almost-complex and symplectic structures; special structures on complex manifolds; and deformations of complex objects. The work is intended for researchers in these areas.

Author : Helmut Hofer,Alberto Abbondandolo,Urs Frauenfelder,Felix Schlenk
Publisher : Springer Nature
Page : 1157 pages
File Size : 54,9 Mb
Release : 2022-12-05
Category : Mathematics
ISBN : 9783031191114

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Symplectic Geometry by Helmut Hofer,Alberto Abbondandolo,Urs Frauenfelder,Felix Schlenk Pdf

Over the course of his distinguished career, Claude Viterbo has made a number of groundbreaking contributions in the development of symplectic geometry/topology and Hamiltonian dynamics. The chapters in this volume – compiled on the occasion of his 60th birthday – are written by distinguished mathematicians and pay tribute to his many significant and lasting achievements.

Author : Boris Hasselblatt
Publisher : Cambridge University Press
Page : 324 pages
File Size : 46,9 Mb
Release : 2007-09-24
Category : Mathematics
ISBN : 9780521875417

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Dynamics, Ergodic Theory and Geometry by Boris Hasselblatt Pdf

Based on the subjects from the Clay Mathematics Institute/Mathematical Sciences Research Institute Workshop titled 'Recent Progress in Dynamics' in September and October 2004, this volume contains surveys and research articles by leading experts in several areas of dynamical systems that have experienced substantial progress. One of the major surveys is on symplectic geometry, which is closely related to classical mechanics and an exciting addition to modern geometry. The survey on local rigidity of group actions gives a broad and up-to-date account of another flourishing subject. Other papers cover hyperbolic, parabolic, and symbolic dynamics as well as ergodic theory. Students and researchers in dynamical systems, geometry, and related areas will find this book fascinating. The book also includes a fifty-page commented problem list that takes the reader beyond the areas covered by the surveys, to inspire and guide further research.

Author : Urs Frauenfelder,Otto van Koert
Publisher : Springer
Page : 374 pages
File Size : 47,5 Mb
Release : 2018-08-29
Category : Mathematics
ISBN : 9783319722788

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The Restricted Three-Body Problem and Holomorphic Curves by Urs Frauenfelder,Otto van Koert Pdf

The book serves as an introduction to holomorphic curves in symplectic manifolds, focusing on the case of four-dimensional symplectizations and symplectic cobordisms, and their applications to celestial mechanics. The authors study the restricted three-body problem using recent techniques coming from the theory of pseudo-holomorphic curves. The book starts with an introduction to relevant topics in symplectic topology and Hamiltonian dynamics before introducing some well-known systems from celestial mechanics, such as the Kepler problem and the restricted three-body problem. After an overview of different regularizations of these systems, the book continues with a discussion of periodic orbits and global surfaces of section for these and more general systems. The second half of the book is primarily dedicated to developing the theory of holomorphic curves - specifically the theory of fast finite energy planes - to elucidate the proofs of the existence results for global surfaces of section stated earlier. The book closes with a chapter summarizing the results of some numerical experiments related to finding periodic orbits and global surfaces of sections in the restricted three-body problem. This book is also part of the Virtual Series on Symplectic Geometry http://www.springer.com/series/16019

Author : Alexander Cardona,Iván Contreras,Andrés F. Reyes-Lega
Publisher : Cambridge University Press
Page : 395 pages
File Size : 47,5 Mb
Release : 2013-05-09
Category : Mathematics
ISBN : 9781107026834

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Geometric and Topological Methods for Quantum Field Theory by Alexander Cardona,Iván Contreras,Andrés F. Reyes-Lega Pdf

A unique presentation of modern geometric methods in quantum field theory for researchers and graduate students in mathematics and physics.

Author : Dylan Thurston
Publisher : Princeton University Press
Page : 472 pages
File Size : 53,5 Mb
Release : 2020-07-07
Category : Mathematics
ISBN : 9780691185897

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What's Next? by Dylan Thurston Pdf

William Thurston (1946–2012) was one of the great mathematicians of the twentieth century. He was a visionary whose extraordinary ideas revolutionized a broad range of areas of mathematics, from foliations, contact structures, and Teichmüller theory to automorphisms of surfaces, hyperbolic geometry, geometrization of 3-manifolds, geometric group theory, and rational maps. In addition, he discovered connections between disciplines that led to astonishing breakthroughs in mathematical understanding as well as the creation of entirely new fields. His far-reaching questions and conjectures led to enormous progress by other researchers. In What's Next?, many of today's leading mathematicians describe recent advances and future directions inspired by Thurston's transformative ideas. This book brings together papers delivered by his colleagues and former students at "What's Next? The Mathematical Legacy of Bill Thurston," a conference held in June 2014 at Cornell University. It discusses Thurston's fundamental contributions to topology, geometry, and dynamical systems and includes many deep and original contributions to the field. Incisive and wide-ranging, the book explores how he introduced new ways of thinking about and doing mathematics—innovations that have had a profound and lasting impact on the mathematical community as a whole—and also features two papers based on Thurston's unfinished work in dynamics.

Author : Jonny Evans
Publisher : Cambridge University Press
Page : 242 pages
File Size : 44,7 Mb
Release : 2023-07-20
Category : Mathematics
ISBN : 9781009372664

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Lectures on Lagrangian Torus Fibrations by Jonny Evans Pdf

Symington's almost toric fibrations have played a central role in symplectic geometry over the past decade, from Vianna's discovery of exotic Lagrangian tori to recent work on Fibonacci staircases. Four-dimensional spaces are of relevance in Hamiltonian dynamics, algebraic geometry, and mathematical string theory, and these fibrations encode the geometry of a symplectic 4-manifold in a simple 2-dimensional diagram. This text is a guide to interpreting these diagrams, aimed at graduate students and researchers in geometry and topology. First the theory is developed, and then studied in many examples, including fillings of lens spaces, resolutions of cusp singularities, non-toric blow-ups, and Vianna tori. In addition to the many examples, students will appreciate the exercises with full solutions throughout the text. The appendices explore select topics in more depth, including tropical Lagrangians and Markov triples, with a final appendix listing open problems. Prerequisites include familiarity with algebraic topology and differential geometry.

Author : A.T. Fomenko
Publisher : CRC Press
Page : 488 pages
File Size : 40,6 Mb
Release : 1995-11-30
Category : Mathematics
ISBN : 2881249019

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Symplectic Geometry by A.T. Fomenko Pdf

Author : Leonid Polterovich,Daniel Rosen,Karina Samvelyan,Jun Zhang
Publisher : American Mathematical Soc.
Page : 128 pages
File Size : 47,7 Mb
Release : 2020-05-11
Category : Education
ISBN : 9781470454951

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Topological Persistence in Geometry and Analysis by Leonid Polterovich,Daniel Rosen,Karina Samvelyan,Jun Zhang Pdf

The theory of persistence modules originated in topological data analysis and became an active area of research in algebraic topology. This book provides a concise and self-contained introduction to persistence modules and focuses on their interactions with pure mathematics, bringing the reader to the cutting edge of current research. In particular, the authors present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, they discuss topological function theory, which provides new insight into oscillation of functions. The book is accessible to readers with a basic background in algebraic and differential topology.

Author : Ivanka Stamova
Publisher : Walter de Gruyter
Page : 241 pages
File Size : 41,9 Mb
Release : 2009-10-16
Category : Mathematics
ISBN : 9783110221824

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Stability Analysis of Impulsive Functional Differential Equations by Ivanka Stamova Pdf

This book is devoted to impulsive functional differential equations which are a natural generalization of impulsive ordinary differential equations (without delay) and of functional differential equations (without impulses). At the present time the qualitative theory of such equations is under rapid development. After a presentation of the fundamental theory of existence, uniqueness and continuability of solutions, a systematic development of stability theory for that class of problems is given which makes the book unique. It addresses to a wide audience such as mathematicians, applied researches and practitioners.

Author : De Gosson Maurice A
Publisher : World Scientific
Page : 308 pages
File Size : 40,5 Mb
Release : 2017-11-10
Category : Science
ISBN : 9781786344168

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Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes by De Gosson Maurice A Pdf

The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical. Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field. Contents: Hamiltonian MechanicsHamilton–Jacobi TheoryMatter Waves, Schrödinger's Equation, and Bohm's Theory The MetatronUncertainties and Quantum BlobsQuantum States and the Density MatrixVarying Planck's ConstantAppendices: The Symplectic GroupThe Metaplectic RepresentationBorn–Jordan QuantizationTwisted Product and Convolution Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics. Keywords: Classical Mechanics;Hamiltonian Mechanics;Quantum Mechanics;Planck's Constant;Mathematical Physics;Algebraic GeometryReview:0

Author : Adolfo Ballester-Bolinches,Ramon Esteban-Romero,Mohamed Asaad
Publisher : Walter de Gruyter
Page : 347 pages
File Size : 46,5 Mb
Release : 2010-10-19
Category : Mathematics
ISBN : 9783110220612

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Products of Finite Groups by Adolfo Ballester-Bolinches,Ramon Esteban-Romero,Mohamed Asaad Pdf

The study of finite groups factorised as a product of two or more subgroups has become a subject of great interest during the last years with applications not only in group theory, but also in other areas like cryptography and coding theory. It has experienced a big impulse with the introduction of some permutability conditions. The aim of this book is to gather, order, and examine part of this material, including the latest advances made, give some new approach to some topics, and present some new subjects of research in the theory of finite factorised groups. Some of the topics covered by this book include groups whose subnormal subgroups are normal, permutable, or Sylow-permutable, products of nilpotent groups, and an exhaustive structural study of totally and mutually permutable products of finite groups and their relation with classes of groups. This monograph is mainly addressed to graduate students and senior researchers interested in the study of products and permutability of finite groups. A background in finite group theory and a basic knowledge of representation theory and classes of groups is recommended to follow it.