Differential equations : geometry, symmetries and integrability : the Abel Symposium 2008, proceedings of the Fifth Abel Symposium, Tromsø, Norway, June 17-22, 2008

The Abel Symposium 2008 focused on the modern theory of differential equations and their applications in geometry, mechanics, and mathematical physics. Following the tradition of Monge, Abel and Lie, the scientific program emphasized the role of algebro-geometric methods, which nowadays permeate all mathematical models in natural and engineering sciences. The ideas of invariance and symmetry are of fundamental importance in the geometric approach to differential equations, with a serious impact coming from the area of integrable systems and field theories. This volume consists of original contributions and broad overview lectures of the participants of the Symposium. The papers in this volume present the modern approach to this classical subject. Front Matter....Pages 1-12 Some Canonical Structures of Cartan Planes in Jet Spaces and Applications....Pages 1-20 Transformations of Darboux Integrable Systems....Pages 21-48 Differential Geometric Heuristics for Riemannian Optimal Mass Transportation....Pages 49-73 On Rank Problems for Planar Webs and Projective Structures....Pages 75-106 Niceness Theorems....Pages 107-150 The Polynomial Algebra and Quantizations of Electromagnetic Fields....Pages 151-158 A Bridge Between Lie Symmetries and Galois Groups....Pages 159-172 Focal Systems for Pfaffian Systems with Characteristics....Pages 173-185 Hamiltonian Structures for General PDEs....Pages 187-198 Point Classification of Second Order ODEs: Tresse Classification Revisited and Beyond....Pages 199-221 Classification of Monge–Ampeère Equations....Pages 223-256 On Nonabelian Theories and Abelian Differentials....Pages 257-274 Geometric Aspects of the Quantization of a Rigid Body....Pages 275-285 Shooting for the Eight....Pages 287-310 Compatible Poisson brackets, quadratic Poisson algebras and classical r -matrices....Pages 311-333 Contact Geometry of Second Order I....Pages 335-386 The Abel Symposium 2008 focused on the modern theory of differential equations and their applications in geometry, mechanics, and mathematical physics. This title consists of original contributions and overview lectures of the participants of the Symposium. It includes papers that present the modern approach to this classical subject. Boris Kruglikov, Valentin Lychagin, Eldar Straume, Editors. Includes Bibliographical References.