Introduction to Symplectic Geometry

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

   PDF

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.

• Symplectic Geometry
• Cotangent Bundles
• G-spaces
• Poisson Manifolds
• supermanifolds

“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)

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

  Jean-Louis Koszul

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

  Yi Ming Zou

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

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