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Introduction to symplectic geometry. (English) Zbl 1433.53002

Singapore: Springer (ISBN 978-981-13-3986-8/hbk; 978-981-13-3987-5/ebook). l, 121 p. (2019).
This book serves as a lasting tribute to a pioneer of various fields, and highlights Koszul’s contributions to the development of symplectic geometry. Several forewords, written in earnest appreciation, allow the readers to observe the importance of Koszul’s ideas, as well as their lasting impact.
These heartfelt and insightful introductions also serve to record the history of Koszul, his influence on those who knew him closely and the respect he inspired from fellow mathematicians. As historical homages, they show the humanity of mathematical research, often a solitary intellectual pursuit.
The numerous contributions of Koszul, in his texts and work, can be best understood through the perspective of Barbaresco in Foreword 2. The original proofs and developments of Koszul’s book can also be seen there. Connections to geometric science of information are highlighted, as Koszul’s antisymmetric bilinear map, \[ c_{\mu}(a,b)=\{ \langle \mu, a \rangle , \langle \mu, b \rangle \} - \langle \mu , \{ a,b \} \rangle, \] generalizes Fisher’s metric from Information Geometry. Here \((a,b)\) are elements of a Lie algebra \(\mathfrak{g}^{\ast}\), and \(\mu\) is the moment map of a Hamiltonian action on \(\mathfrak{g}^{\ast}\).
As Barbaresco puts it:
“Koszul’s book introduces original developments and proofs that are of major interest to various communities. 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.”
The third foreword, by Charles-Michel Marle, neatly recounts the content and main features of the book. Several historical annotations about specific contributions are surely very interesting for specialists. Perhaps the point that most distinguishes this book, an explanation of Poisson geometry and Koszul’s contributions to that field are presented here.
Once the mathematical contents begin, the first three chapters efficiently present the basics needed to proceed to more advanced topics. Algebraic preliminaries about skew forms, symplectic manifolds and their local coordinates, and tangent bundles and their canonical structures are included.
The fourth chapter deals with Hamiltonian actions, moment maps, and equivariance. Chapter five introduces Poisson manifolds, describes the symplectic structure on the associated foliation, and exemplifies this theory with the structures that appear in the dual of a Lie algebra. Finally, the sixth chapter makes connections with supermanifolds. These last four chapters are the most original in terms of content, and distinguish it from other introductions to these topics.
This text could certainly be used for a graduate level course. The final three chapters link to areas of active research, while the elegant presentation of the first three allow the readers a general introduction.

MSC:

53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53D05 Symplectic manifolds (general theory)
53D17 Poisson manifolds; Poisson groupoids and algebroids
58A32 Natural bundles
62B11 Information geometry (statistical aspects)
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