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, Painleve equations, global analysis on manifolds, special functions, etc.It concludes with Kowalevski's famous paper published in ""Acta Mathematica"" in 1889, 'Sur le probleme 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.
The life of S. V. Kovalevskaya by R. L. Cooke Kovalevskaya's mathematical work by R. L. Cooke The KZB connection: Parametrizations, flat sections and $q$-deformation by B. Enriquez Jacobians of singularized spectral curves and completely integrable systems by L. Gavrilov The $q$-hypergeometric equation, Askey-Wilson type solitons and rational curves with singularities by L. Haine Quantum discrete soliton equations by K. Hikami Dual algebras of differential operators by E. I. Horozov A link between two fundamental contributions of Kowalevski by J.-S. Hu and M. Yan Monodromy deformation approach to the scaling limit of the Painleve first equation by A. A. Kapaev Kowalevski top revisited by V. B. Kuznetsov Some algebro-geometric integrable systems versus classical ones by D. Markushevich Painleve sixth equation as isomonodromic deformations equation of an irregular system by M. Mazzocco Euler characteristics of theta divisors of Jacobians for spectral curves by A. Nakayashiki and F. A. Smirnov Reduction theory, elliptic solitons and integrable systems by E. Previato Schwarzian derivatives and uniformization by T. Sasaki and M. Yoshida Elliptic solitons and Heun's equation by A. O. Smirnov Generalized Kowalevski top: New integrable cases on $e$(3) and so(4) by V. V. Sokolov Reprint of the Original Paper: Sur le probleme de la rotation d'un corps solide autour d'un point fixe by S. Kowalevski.