**CRM Monograph Series**

Volume: 34;
2014;
203 pp;
Hardcover

MSC: Primary 53;
Secondary 57; 81; 22; 20; 28

**Print ISBN: 978-1-4704-1693-5
Product Code: CRMM/34**

List Price: $96.00

AMS Member Price: $76.80

MAA Member Price: $86.40

**Electronic ISBN: 978-1-4704-1931-8
Product Code: CRMM/34.E**

List Price: $90.00

AMS Member Price: $72.00

MAA Member Price: $81.00

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#### Supplemental Materials

# Function Theory on Symplectic Manifolds

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*Leonid Polterovich; Daniel Rosen*

A co-publication of the AMS and Centre de Recherches Mathématiques

This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards.

I like the spirit of this book. It formulates concepts clearly and explains the relationship between them. The subject matter is important and interesting.

—Dusa McDuff, Barnard College, Columbia University

This is a very important book, coming at the right moment. The book is a remarkable mix of introductory chapters and research topics at the very forefront of actual research. It is full of cross fertilizations of different theories, and will be useful to Ph.D. students and researchers in symplectic geometry as well as to many researchers in other fields (geometric group theory, functional analysis, mathematical quantum mechanics). It is also perfectly suited for a Ph.D.-students seminar.

—Felix Schlenk, Université de Neuchâtel

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students and research mathematicians interested in symplectic geometry/topology, Hamiltonian dynamics, and quantum mechanics.

#### Reviews & Endorsements

To view Yakov Eliashberg's review, click here.

-- Yakov Eliashberg

The intervening chapters, connecting the first chapter's discussion of these 'wonders' to the exciting stuff mentioned above, is concerned with a lot of heavy machinery, but it's all presented in a streamlined fashion: quasi-morphisms, spectral invariants, and symplectic approximation theory make an appearance, for example. [It] is very exciting and deep mathematics: it's really altogether irresistible.

-- MAA Reviews

#### Table of Contents

# Table of Contents

## Function Theory on Symplectic Manifolds

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Three wonders of symplectic geometry 114
- 𝐶⁰-rigidity of the Poisson bracket 2134
- Quasi-morphisms 3548
- Subadditive spectral invariants 4962
- Symplectic quasi-states and quasi-measures 6578
- Applications of partial symplectic quasi-states 89102
- A Poisson bracket invariant of quadruples 103116
- Symplectic approximation theory 119132
- Geometry of covers and quantum noise 127140
- Preliminaries from Morse theory 147160
- An overview of Floer theory 161174
- Constructing subadditive spectral invariants 173186
- Bibliography 185198
- Nomenclature 193206
- Subject index 197210
- Name index 201214
- Back Cover Back Cover1218