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Introduction to Symplectic Geometry

  • Textbook
  • © 2019

Overview

Authors:
  1. Jean-Louis Koszul

    1. Institut Fourier, Université Grenoble Alpes, Gières, Grenoble, France

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  2. Yi Ming Zou

    1. Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, USA

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  • Offers a unique and unified overview of symplectic geometry
  • Highlights the differential properties of symplectic manifolds
  • Great interest for the emerging field of "Geometric Science of Information

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-l
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  2. Some Algebra Basics

    • Jean-Louis Koszul, Yi Ming Zou
    Pages 1-19

  3. Symplectic Manifolds

    • Jean-Louis Koszul, Yi Ming Zou
    Pages 21-55

  4. Cotangent Bundles

    • Jean-Louis Koszul, Yi Ming Zou
    Pages 57-73

  5. Symplectic G-Spaces

    • Jean-Louis Koszul, Yi Ming Zou
    Pages 75-90

  6. Poisson Manifolds

    • Jean-Louis Koszul, Yi Ming Zou
    Pages 91-107

  7. A Graded Case

    • Jean-Louis Koszul, Yi Ming Zou
    Pages 109-116

  8. Back Matter

    Pages 117-121
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Keywords

About this book

This introductory book offers a unique and unified overview of symplectic geometry, highlighting the differential properties of symplectic manifolds. It consists of six chapters:  Some Algebra Basics, Symplectic Manifolds, Cotangent Bundles, Symplectic G-spaces, Poisson Manifolds, and A Graded Case, concluding with a discussion of the differential properties of graded symplectic manifolds of dimensions (0,n). It is a useful reference resource for students and researchers interested in geometry, group theory, analysis and differential equations.This book is also inspiring in the emerging field of Geometric Science of Information, in particular the chapter on Symplectic G-spaces, where Jean-Louis Koszul develops Jean-Marie Souriau's tools related to the non-equivariant case of co-adjoint action on Souriau’s moment map through Souriau’s Cocycle, opening the door to Lie Group Machine Learning with Souriau-Fisher metric.

Reviews

“This book is of great interest for the emerging field of Geometric Science of Information, in which the generalization of the Fisher metric is at the heart of the extension of classical tools from Machine Learning and Artificial Intelligence to deal with more abstract objects living in homogeneous manifolds, groups, and structured matrices.’” (Pablo Suárez-Serrato, zbMATH 1433.53002, 2020)

Authors and Affiliations

  • Institut Fourier, Université Grenoble Alpes, Gières, Grenoble, France

    Jean-Louis Koszul

  • Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, USA

    Yi Ming Zou

About the authors

Jean Louis Koszul, born in 1921, was a French Mathematician. He was a member of 2nd generation of Bourbaki, also a member of French Academy of Sciences.  Jean-Louis Koszul passed away on January 12th 2018, at the age of 97.

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