Pol Vanhaecke – författare

In the early 70's and 80's the field of integrable systems was in its prime youth: results and ideas were mushrooming all over the world. It was during the roaring 70's and 80's that a first version of the book was born, based on our research and on lectures which each of us had given. We owe many ideas to our colleagues Teruhisa Matsusaka and David Mumford, and to our inspiring graduate students (Constantin Bechlivanidis, Luc Haine, Ahmed Lesfari, Andrew McDaniel, Luis Piovan and Pol Vanhaecke). As it stood, our first version lacked rigor and precision, was rough, dis- connected and incomplete...In the early 90's new problems appeared on the horizon and the project came to a complete standstill, ultimately con- fined to a floppy. A few years ago, under the impulse of Pol Vanhaecke, the project was revived and gained real momentum due to his insight, vision and determination. The leap from the old to the new version is gigantic. The book is designed as a teaching textbook and is aimed at a wide read- ership of mathematicians and physicists, graduate students and professionals.

2. Divisors and line bundles ...99. 2.1. Divisors ...99. 2.2. Line bundles ...100. 2.3. Sections of line bundles ...101. 2.4. The Riemann-Roch Theorem ...103. 2.5. Line bundles and embeddings in projective space ...105. 2.6. Hyperelliptic curves ...106. 3. Abelian varieties ...108. 3.1. Complex tori and Abelian varieties ...108. 3.2. Line bundles on Abelian varieties ...109. 3.3. Abelian surfaces ...111. 4. Jacobi varieties ...114. 4.1. The algebraic Jacobian ...114. 4.2. The analytic/transcendental Jacobian ...114. 4.3. Abel's Theorem and Jacobi inversion ...119. 4.4. Jacobi and Kummer surfaces ...121. 5. Abelian surfaces of type (1,4) ...123. 5.1. The generic case ...123. 5.2. The non-generic case ...124. V. Algebraic completely integrable Hamiltonian systems ...127. 1. Introduction ...127. 2. A.c.i. systems ...129. 3. Painlev~ analysis for a.c.i, systems ...135. 4. The linearization of two-dkmensional a.e.i, systems ...138. 5. Lax equations ...140. VI. The Mumford systems ...143. 1. Introduction ...143. 2. Genesis ...145. 2.1. The algebra of pseudo-differential operators ...145. 2.2. The matrix associated to two commuting operators ...146. 2.3. The inverse construction ...150.2.4. The KP vector fields ...152. ix 3. Multi-Hamiltonian structure and symmetries ...155. 3.1. The loop algebra 9(q ...155. 3.2. Reducing the R-brackets and the vector field ~ ...157. 4. The odd and the even Mumford systems ...161. 4.1. The (odd) Mumford system ...161. 4.2. The even Mumford system ...163.

In the early 70's and 80's the field of integrable systems was in its prime youth: results and ideas were mushrooming all over the world. It was during the roaring 70's and 80's that a first version of the book was born, based on our research and on lectures which each of us had given. We owe many ideas to our colleagues Teruhisa Matsusaka and David Mumford, and to our inspiring graduate students (Constantin Bechlivanidis, Luc Haine, Ahmed Lesfari, Andrew McDaniel, Luis Piovan and Pol Vanhaecke). As it stood, our first version lacked rigor and precision, was rough, dis- connected and incomplete...In the early 90's new problems appeared on the horizon and the project came to a complete standstill, ultimately con- fined to a floppy. A few years ago, under the impulse of Pol Vanhaecke, the project was revived and gained real momentum due to his insight, vision and determination. The leap from the old to the new version is gigantic. The book is designed as a teaching textbook and is aimed at a wide read- ership of mathematicians and physicists, graduate students and professionals.

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.​

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.​

Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.​