Symplectic Geometry and Fourier AnalysisSymplectic Geometry and Fourier Analysis



Suitable for graduate students in mathematics, this monograph covers differential and symplectic geometry, homogeneous symplectic manifolds, Fourier analysis, metaplectic representation, quantization, Kirillov theory.

Author: Nolan R. Wallach

Publisher: Courier Dover Publications

ISBN: 9780486829623

Category:

Page: 272

View: 807

Suitable for graduate students in mathematics, this monograph covers differential and symplectic geometry, homogeneous symplectic manifolds, Fourier analysis, metaplectic representation, quantization, Kirillov theory. Includes Appendix on Quantum Mechanics by Robert Hermann. 1977 edition.

Geometric AsymptoticsGeometric Asymptotics



The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence.

Author: Victor Guillemin

Publisher: American Mathematical Soc.

ISBN: 9780821816332

Category:

Page: 480

View: 426

Symplectic geometry and the theory of Fourier integral operators are modern manifestations of themes that have occupied a central position in mathematical thought for the past three hundred years - the relations between the wave and the corpuscular theories of light. The purpose of this book is to develop these themes, and present some of the recent advances, using the language of differential geometry as a unifying influence.

Symplectic Geometry and Analytical MechanicsSymplectic Geometry and Analytical Mechanics



The KS transform in hypercomplex form and the quantization of the negative
energy orbit manifold of the Kepler problem. Preprint, Dipartimento di Matematica
, Politecnico di Milano, Italia, 1985. Wallach, N. 1. Symplectic geometry and
Fourier ...

Author: P. Libermann

Publisher: Springer Science & Business Media

ISBN: 9789400938076

Category:

Page: 526

View: 150

Approach your problems from the right end It isn't that they can't see the solution. and begin with the answers. Then one day, It is that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' Brown 'The point of a Pin'. in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thouglit to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sci ences has changed drastically in recent years: measure theory is used (non-trivially) in re gional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homo topy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Symplectic Methods in Harmonic Analysis and in Mathematical PhysicsSymplectic Methods in Harmonic Analysis and in Mathematical Physics



The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency ...

Author: Maurice A. de Gosson

Publisher: Springer Science & Business Media

ISBN: 9783764399924

Category:

Page: 338

View: 448

The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency literature. The topics that are addressed include (but are not limited to) the theory of the Wigner transform, the uncertainty principle (from the point of view of symplectic topology), Weyl calculus and its symplectic covariance, Shubin’s global theory of pseudo-differential operators, and Feichtinger’s theory of modulation spaces. Several applications to time-frequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a non-standard pseudo-differential calculus on phase space where the main role is played by “Bopp operators” (also called “Landau operators” in the literature) is introduced and studied. This calculus is closely related to both the Landau problem and to the deformation quantization theory of Flato and Sternheimer, of which it gives a simple pseudo-differential formulation where Feichtinger’s modulation spaces are key actors. This book is primarily directed towards students or researchers in harmonic analysis (in the broad sense) and towards mathematical physicists working in quantum mechanics. It can also be read with profit by researchers in time-frequency analysis, providing a valuable complement to the existing literature on the topic. A certain familiarity with Fourier analysis (in the broad sense) and introductory functional analysis (e.g. the elementary theory of distributions) is assumed. Otherwise, the book is largely self-contained and includes an extensive list of references.

Symplectic Geometry and Quantum MechanicsSymplectic Geometry and Quantum Mechanics



Let us begin by introducing a notion of Fourier transform for functions (or
distributions) defined on the symplectic phase space (R2nz,σ). Definition 6.6. Let
f ∈S(R2nz). The “symplectic Fourier transform” of f is the function fσ = Fσf defined
by ...

Author: Maurice A. de Gosson

Publisher: Springer Science & Business Media

ISBN: 9783764375751

Category:

Page: 368

View: 985

This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.

Geometric and Harmonic Analysis on Homogeneous SpacesGeometric and Harmonic Analysis on Homogeneous Spaces



It is well-known that Lagrangian submanifolds play a central role in symplectic
geometry. This can easily be traced back to the search for so-called “generating
functions” of (local) symplectomorphisms in the framework of the Hamilton–
Jacobi ...

Author: Ali Baklouti

Publisher: Springer Nature

ISBN: 9783030265625

Category:

Page: 217

View: 918

This book presents a number of important contributions focusing on harmonic analysis and representation theory of Lie groups. All were originally presented at the 5th Tunisian–Japanese conference “Geometric and Harmonic Analysis on Homogeneous Spaces and Applications”, which was held at Mahdia in Tunisia from 17 to 21 December 2017 and was dedicated to the memory of the brilliant Tunisian mathematician Majdi Ben Halima. The peer-reviewed contributions selected for publication have been modified and are, without exception, of a standard equivalent to that in leading mathematical periodicals. Highlighting the close links between group representation theory and harmonic analysis on homogeneous spaces and numerous mathematical areas, such as number theory, algebraic geometry, differential geometry, operator algebra, partial differential equations and mathematical physics, the book is intended for researchers and students working in the area of commutative and non-commutative harmonic analysis as well as group representations.

Semiclassical AnalysisSemiclassical Analysis



This book is an excellent, comprehensive introduction to semiclassical analysis.

Author: Maciej Zworski

Publisher: American Mathematical Soc.

ISBN: 9780821883204

Category:

Page: 431

View: 347

This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. --Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel-Kramers-Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.

Lectures on Symplectic GeometryLectures on Symplectic Geometry



1863: H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms (
2005) Vol. 1864: K. Efstathiou, Metamorphoses of Hamiltonian Systems with
Symmetries (2005) Vol. 1865: D. Applebaum, B.V. R. Bhat, J. Kustermans, J. M. ...

Author: Ana Cannas da Silva

Publisher: Springer Science & Business Media

ISBN: 9783540421955

Category:

Page: 217

View: 107

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.

Fourier Integral OperatorsFourier Integral Operators



Additionally, this book is designed for a one-semester introductory course on Fourier integral operators aimed at a broad audience. This book remains a superb introduction to the theory of Fourier integral operators.

Author: J.J. Duistermaat

Publisher: Springer Science & Business Media

ISBN: 0817681086

Category:

Page: 142

View: 967

This volume is a useful introduction to the subject of Fourier Integral Operators and is based on the author’s classic set of notes. Covering a range of topics from Hörmander’s exposition of the theory, Duistermaat approaches the subject from symplectic geometry and includes application to hyperbolic equations (= equations of wave type) and oscillatory asymptotic solutions which may have caustics. This text is suitable for mathematicians and (theoretical) physicists with an interest in (linear) partial differential equations, especially in wave propagation, rep. WKB-methods.

Harmonic Analysis on Homogeneous SpacesHarmonic Analysis on Homogeneous Spaces



This book is suitable for advanced undergraduate and graduate students in mathematics with a strong background in linear algebra and advanced calculus.

Author: Nolan R. Wallach

Publisher: Courier Dover Publications

ISBN: 9780486836430

Category:

Page: 384

View: 666

This book is suitable for advanced undergraduate and graduate students in mathematics with a strong background in linear algebra and advanced calculus. Early chapters develop representation theory of compact Lie groups with applications to topology, geometry, and analysis, including the Peter-Weyl theorem, the theorem of the highest weight, the character theory, invariant differential operators on homogeneous vector bundles, and Bott's index theorem for such operators. Later chapters study the structure of representation theory and analysis of non-compact semi-simple Lie groups, including the principal series, intertwining operators, asymptotics of matrix coefficients, and an important special case of the Plancherel theorem. Teachers will find this volume useful as either a main text or a supplement to standard one-year courses in Lie groups and Lie algebras. The treatment advances from fairly simple topics to more complex subjects, and exercises appear at the end of each chapter. Eight helpful Appendixes develop aspects of differential geometry, Lie theory, and functional analysis employed in the main text.

Introduction to Symplectic Dirac OperatorsIntroduction to Symplectic Dirac Operators



This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject.

Author: Katharina Habermann

Publisher: Springer

ISBN: 9783540334217

Category:

Page: 125

View: 657

This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research.

Geometric Aspects of Analysis and MechanicsGeometric Aspects of Analysis and Mechanics



M. Poel, Harmonic Analysis on SL(n,R)/GL(n − 1,R), 1986 8. ... Sur les Régions
Interdites du Spectre de l'Opérateur Périodique et Discret de Mathieu, 1988 R.
Sjamaar, Singular Orbit Spaces in Riemannian and Symplectic Geometry, 1990
H.

Author: Erik P. van den Ban

Publisher: Springer Science & Business Media

ISBN: 0817682449

Category:

Page: 372

View: 313

Hans Duistermaat, an influential geometer-analyst, made substantial contributions to the theory of ordinary and partial differential equations, symplectic, differential, and algebraic geometry, minimal surfaces, semisimple Lie groups, mechanics, mathematical physics, and related fields. Written in his honor, the invited and refereed articles in this volume contain important new results as well as surveys in some of these areas, clearly demonstrating the impact of Duistermaat's research and, in addition, exhibiting interrelationships among many of the topics.

Symplectic Techniques in PhysicsSymplectic Techniques in Physics



This work can be used as a textbook for graduate courses, but the depth of coverage and the wealth of information and application means that it will be of continuing interest to, and of lasting significance for mathematicians and ...

Author: Victor Guillemin

Publisher: Cambridge University Press

ISBN: 0521389909

Category:

Page: 468

View: 659

Symplectic geometry is very useful for clearly and concisely formulating problems in classical physics and also for understanding the link between classical problems and their quantum counterparts. It is thus a subject of interest to both mathematicians and physicists, though they have approached the subject from different view points. This is the first book that attempts to reconcile these approaches. The authors use the uncluttered, coordinate-free approach to symplectic geometry and classical mechanics that has been developed by mathematicians over the course of the last thirty years, but at the same time apply the apparatus to a great number of concrete problems. In the first chapter, the authors provide an elementary introduction to symplectic geometry and explain the key concepts and results in a way accessible to physicists and mathematicians. The remainder of the book is devoted to the detailed analysis and study of the ideas discussed in Chapter 1. Some of the themes emphasized in the book include the pivotal role of completely integrable systems, the importance of symmetries, analogies between classical dynamics and optics, the importance of symplectic tools in classical variational theory, symplectic features of classical field theories, and the principle of general covariance. This work can be used as a textbook for graduate courses, but the depth of coverage and the wealth of information and application means that it will be of continuing interest to, and of lasting significance for mathematicians and mathematically minded physicists.

Analysis Geometry and Quantum Field TheoryAnalysis Geometry and Quantum Field Theory



The duality in this case is simply an interchange of the radius R ↔ 1/R, or more
precisely the Fourier transform in the circle ... Generalized geometry has the very
nice feature that it subsumes complex and symplectic geometry as particular ...

Author: Clara L. Aldana

Publisher: American Mathematical Soc.

ISBN: 9780821891445

Category:

Page: 258

View: 274

This volume contains the proceedings of the conference ``Analysis, Geometry and Quantum Field Theory'' held at Potsdam University in September 2011, which honored Steve Rosenberg's 60th birthday. The papers in this volume cover a wide range of areas, including Quantum Field Theory, Deformation Quantization, Gerbes, Loop Spaces, Index Theory, Determinants of Elliptic Operators, K-theory, Infinite Rank Bundles and Mathematical Biology.

Microlocal Analysis for Differential OperatorsMicrolocal Analysis for Differential Operators



Our presentation follows essentially the one given by Hörmander ( Hö2 ] ; as for
the symplectic geometry , we have been ... Readers of this book are expected to
be familiar with the theory of distributions , in particular with the Fourier transform
.

Author: Alain Grigis

Publisher: Cambridge University Press

ISBN: 0521449863

Category:

Page: 151

View: 804

This book corresponds to a graduate course given many times by the authors, and should prove to be useful to mathematicians and theoretical physicists.

Representation Theory Complex Analysis and Integral GeometryRepresentation Theory Complex Analysis and Integral Geometry



... a special summer term on “Harmonic analysis, representation theory, and
integral geometry”, hosted by the Max Plank ... and Peter E. Trapa); applications
of symplectic geometry, particularly moment maps, to the study of arithmetic
issues in ...

Author: Bernhard Krötz

Publisher: Springer Science & Business Media

ISBN: 9780817648169

Category:

Page: 275

View: 231

This book is an outgrowth of the special summer session held at the Max Planck Institute for Mathematics and the Hausdorff Research Institute for Mathematics. The contributions in the volume provide a window into a variety of subjects related to reductive groups, including real and complex analysis on homogeneous spaces, arithmetic aspects of moment geometry, geometry of flag varieties, restriction theory of representations, modern aspects of special functions, multiple Dirichlet series, and unfolding identities in the theory of automorphic forms.

Geometric Optics on Phase SpaceGeometric Optics on Phase Space



The appendices link the geometry thus introduced to wave optics through Lie methods. The book addresses researchers and graduate students.

Author: Kurt Bernardo Wolf

Publisher: Springer Science & Business Media

ISBN: 3540220399

Category:

Page: 376

View: 193

Symplectic geometry, well known as the basic structure of Hamiltonian mechanics, is also the foundation of optics. In fact, optical systems (geometric or wave) have an even richer symmetry structure than mechanical ones (classical or quantum). The symmetries underlying the geometric model of light are based on the symplectic group. Geometric Optics on Phase Space develops both geometric optics and group theory from first principles in their Hamiltonian formulation on phase space. This treatise provides the mathematical background and also collects a host of useful methods of practical importance, particularly the fractional Fourier transform currently used for image processing. The reader will appreciate the beautiful similarities between Hamilton's mechanics and this approach to optics. The appendices link the geometry thus introduced to wave optics through Lie methods. The book addresses researchers and graduate students.

The Concise Handbook of AlgebraThe Concise Handbook of Algebra



New results are usually conceived and obtained in contexts where the Lie groups
themselves only serve as tools (e.g., in Riemannian or symplectic geometry) or
are at least are not the prime objects of interest (e.g., in harmonic analysis or ...

Author: Aleksandr Vasilʹevich Mikhalev

Publisher: Springer Science & Business Media

ISBN: 0792370724

Category:

Page: 618

View: 184

Provides a succinct, but thorough treatment of algebra. In a collection that spans about 150 sections, organized in 9 chapters, algebraists are provided with a standard knowledge set for their areas of expertise.