Michéle Audin Knihy

How I have (re-)written this book The book the reader has in hand was supposed to be a new edition of [14]. I have hesitated quite a long time before deciding to do the re-writing work-the first edition has been sold out for a few years. There was absolutely no question of just correcting numerous misprints and a few mathematical errors. When I wrote the first edition, in 1989, the convexity and Duistermaat-Heckman theorems together with the irruption of toric varieties on the scene of symplectic geometry, due to Delzant, around which the book was organized, were still rather recent (less than ten years). I myself was rather happy with a small contribution I had made to the subject. I was giving a post-graduate course on all that and, well, these were lecture notes, just lecture notes. By chance, the book turned out to be rather popular: during the years since then, I had the opportunity to meet quite a few people(1) who kindly pretended to have learnt the subject in this book. However, the older book does not satisfy at all the idea I have now of what a good book should be. So that this „new edition“ is, indeed, another book.

Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. Michèle Audin, professor at the University of Strasbourg, has written a book allowing them to remedy this situation and, starting from linear algebra, extend their knowledge of affine, Euclidean and projective geometry, conic sections and quadrics, curves and surfaces. It includes many nice theorems like the nine-point circle, Feuerbach's theorem, and so on. Everything is presented clearly and rigourously. Each property is proved, examples and exercises illustrate the course content perfectly. Precise hints for most of the exercises are provided at the end of the book. This very comprehensive text is addressed to students at upper undergraduate and Master's level to discover geometry and deepen their knowledge and understanding.

How did Pierre Fatou and Gaston Julia create what we now call Complex Dynamics, in the context of the early twentieth century and especially of the First World War? The book is based partly on new, unpublished sources. Who were Pierre Fatou, Gaston Julia, Paul Montel? New biographical information is given on the little known mathematician that was Pierre Fatou. How did the WW1 injury of Julia influence mathematical life in France? From the reviews of the French version: „Audin’s book is … filled with marvelous biographical information and analysis, dealing not just with the men mentioned in the book’s title but a large number of other players, too … [It] addresses itself to scholars for whom the history of mathematics has a particular resonance and especially to mathematicians active, or even with merely an interest, in complex dynamics. … presents it all to the reader in a very appealing form.“ (Michael Berg, The Mathematical Association of America, October 2009)

Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. The quasi-periodicity of the solutions of an integrable system is a result of the fact that the system is invariant under a (semi-global) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus action. This leads to symplectic toric manifolds (Part B of this book). Physics makes a surprising come-back in Part A: to describe Mirror Symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact toric manifolds (Part C of this book).

"Personne ne se souvient de leurs noms, mais je vais vous dire un ou deux mots de cette passementière qui toute sa courte vie souffrit tellement des dents, de ce marchand de produits chimiques de Saint-Paul que seules de grandes quantités de vin rouge consolaient, de ce menuisier qui sculptait de petits jouets en bois pour l'enfant qu'il attendait, de ce cordonnier qui se souvenait de ce geste touchant, sa femme relevant ses cheveux, elle était morte pendant le siège, de cette tourneuse qui aurait voulu être institutrice, de cette brocheuse qui avait un carnet dans lequel elle notait ce qu'elle faisait ou pensait". Une petite foule de personnages, Marthe, Paul, Maria, Floriss... vivent, aiment, espèrent, travaillent, écrivent, se battent, enfermés dans Paris, pendant les soixante-douze jours qu'a duré la Commune. Comme une rivière bleue est leur histoire, vécue nuit et jour, à travers les fêtes, les concerts, les débats fiévreux, à l'Hôtel de Ville, à la barrière d'Enfer, au Château-d'Eau, sur les fortifications, dans ce Paris de 1871 qui est encore le nôtre. A l'aide de journaux inconnus, de l'état civil et de ses failles, de livres de témoins, le roman de Michèle Audin nous entraîne dans la ville assiégée, derrière quelques-uns des obscurs qui fabriquent cette "révolution qui passe tranquille et belle comme une rivière bleue".

Mit seinen Arbeiten über Faserbündel und Homotopiegruppen gehört Jacques Feldbau zu den Wegbereitern der modernen Topologie. Michèle Audin zeichnet seine Geschichte, sein Leben und Werk nach. Als elsässischer Jude in Clermond-Ferrand verhaftet verstarb Feldbau zwei Wochen vor Ende des zweiten Weltkrieges während der Deportation nach Auschwitz.