Jean-Louis Koszul - Introduction to Symplectic Geometry

Jointly published with Science Press, Beijing, China

The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Science Press

Jointly published with Science Press, Beijing, China

The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from: Science Press

This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.

The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

En effet, Barbaresco ma dit quil tait curieux de voir le texte qui a t publi daprs un cours sur la gomtrie symplectique que jai donn Tianjin, il y a bien longtemps. Je nai rien pu lui procurer dautre que le petit fascicule rdig en chinois par un assistant de Nankai. Je ne sais pas ce quil vaut. De toutes faons, il ny avait pas beaucoup de choses originales dans ce que jai racont.

Indeed, Barbaresco told me that he was curious to see the text that was published according to a course on symplectic geometry that I gave in Tianjin, long time ago. I could not give anything else but the little notebook written in Chinese by a Nankai assistant. I do not know whether it was worthful. In any case, there were no too many original things in what I was speaking about.

J.-L. Koszul, 02/02/2017.

Above is an excerpt from a message sent to me by Jean-Louis Koszul in February 2017. Earlier, in January 2017, I had written to Koszul about the lectures on Symplectic Geometry he delivered in China. I informed him that the notes of those lectures were missing in my private documentation. I also informed him that Frderic Barbaresco was interested in these notes.

This book is more than an elementary introduction to symplectic structures and their geometry. It highlights the unifying nature of symplectic structures. Often, readings of Koszul works are walks through Algebra, Homological Algebra, Geometry, Differential Geometry, Topology , and Differential Topology . This adage is highlighted by Foreword 3. The lectures in this book were delivered while new developments in Symplectic Geometry were occurring, with the works of A. Weinstein, B. Kostant, V. Guillemin, S. Stenberg, M. Atiyah, R. Bott, and many others. Koszul also emphasizes fruitful exchanges of results and techniques between Symplectic Geometry and other areas. For example, in Chap. is an introduction to quantum symplectic structures. I have often talked with Koszul about his stays in China and in India. He and the editors had planned to include a video with this book. However, on November 28, 2017, Koszul wrote to me :

Depuis que nous sommes installs dans cette maison de retraite, je suis trs mal en point et trs affaibli. Dans ltat o je suis, il nest videmment pas question que je donne un interview, je regrette bien de vous dcevoir.

J.-L. Koszul, 28/11/2017.

Since we have settled in this retirement house, I feel very badly and very weakened. In the situation that I am now, there is obviously no question that I give an interview, I regret to disappoint you.

What is new? As I just mentioned, some of Koszuls work deeply impacts current research in both the Topology of Information and the Geometric Science of Information as well as their applications in Physics. This relates to his work on the geometry of convex cones, on the affine representations of Lie groups and Lie algebras, and on deformations of locally flat manifolds. Koszul knew of the connection through Barbarescos paper Koszul information geometry and Souriau Lie group thermodynamics . He wrote to me :

Pour ce qui est des reprsentations affines, je ne suis pas le premier les avoir manipul. Si jai bon souvenir, elles interviennent dans le travail des russes sur les domaines borns. A part cela, cet article de Barbaresco contient bien de choses que jaimerais comprendre. Je vais essayer de my mettre.

As for the affine representations, I am not the first to have manipulated them. If I remember correctly, they intervene in the work of the Russians on the bounded domains. Apart from that, this Barbaresco article contains many things that I would like to understand. I will try myself to enter into the matter of the article.

J.-L. Koszul .

Thence, I undertook to convince him that the mathematical foundation of the Geometric Science of Information is the algebraic topology of Koszul Geometry, viz the cohomology theory of Koszul Geometry. Subsequently, Koszuls interest in the Geometric Science of Information increased.

Je suis sensible lhonneur que me fait le comit dorganisation du congrs 2013 en minvitant la confrence de Shima et je vous demande de bien vouloir lui transmettre mes remerciements. Jaimerais aussi pouvoir vous dire de transmettre mon acception.

I am sensible to the honor that the organizing committee of the 2013 congress gave me inviting me to the Shima conference and I ask you to forward kindly my thanks. I would also like to tell you to convey my gratitude.

J.-L. Koszul, 20/12/2012 .

On August 29, 2013, H. Shima delivered a keynote conference on Hessian Geometry, whose founder is Jean-Louis Koszul.