This book is a collection of survey articles on several topics related to the general notion of integrability. It stems from a workshop on "Mathematical Methods of Regular Dynamics" dedicated to Sophie Kowalevski. Leading experts introduce corresponding areas in depth. The book provides a broad overview of research, from the pioneering work of the nineteenth century to the developments of the 1970s through the present.

The book begins with two historical papers by R. L. Cooke on Kowalevski's life and work. Following are 15 research surveys on integrability issues in differential and algebraic geometry, classical complex analysis, discrete mathematics, spinning tops, Painlevé equations, global analysis on manifolds, special functions, etc. It concludes with Kowalevski's famous paper published in Acta Mathematica in 1889, "Sur le problème de la rotation d'un corps solide autour d'un point fixe".

The book is suitable for graduate students in pure and applied mathematics, the general mathematical audience studying integrability, and research mathematicians interested in differential and algebraic geometry, analysis, and special functions.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

Readership

Graduate students in pure and applied mathematics, the general mathematical audience studying integrability, and research mathematicians interested in differential and algebraic geometry, analysis, and special functions.

Table of Contents

• R. L. Cooke -- The life of S. V. Kovalevskaya
• R. L. Cooke -- Kovalevskaya's mathematical work
• B. Enriquez -- The KZB connection: Parametrizations, flat sections and $q$-deformation
• L. Gavrilov -- Jacobians of singularized spectral curves and completely integrable systems
• L. Haine -- The $q$-hypergeometric equation, Askey-Wilson type solitons and rational curves with singularities
• K. Hikami -- Quantum discrete soliton equations
• E. I. Horozov -- Dual algebras of differential operators
• J.-S. Hu and M. Yan -- A link between two fundamental contributions of Kowalevski
• A. A. Kapaev -- Monodromy deformation approach to the scaling limit of the Painlevé first equation
• V. B. Kuznetsov -- Kowalevski top revisited
• D. Markushevich -- Some algebro-geometric integrable systems versus classical ones
• M. Mazzocco -- Painlevé sixth equation as isomonodromic deformations equation of an irregular system
• A. Nakayashiki and F. A. Smirnov -- Euler characteristics of theta divisors of Jacobians for spectral curves
• E. Previato -- Reduction theory, elliptic solitons and integrable systems
• T. Sasaki and M. Yoshida -- Schwarzian derivatives and uniformization
• A. O. Smirnov -- Elliptic solitons and Heun's equation
• V. V. Sokolov -- Generalized Kowalevski top: New integrable cases on $e$(3) and so(4)

• S. Kowalevski -- Sur le problème de la rotation d'un corps solide autour d'un point fixe