The Diverse World of PDEs: Geometry and Mathematical Physics (Contemporary Mathematics)

This volume contains the proceedings of the Alexandre Vinogradov Memorial Conference on Diffieties, Cohomological Physics, and Other Animals, held from December 13–17, 2021, at the Independent University of Moscow and Moscow State University, Moscow, Russia. The papers are devoted to various interrelations of nonlinear PDEs with geometry and integrable systems. The topics discussed are: gravitational and electromagnetic fields in General Relativity, nonlocal geometry of PDEs, Legendre foliated cocycles on contact manifolds, presymplectic gauge PDEs and Lagrangian BV formalism, jet geometry and high-order phase transitions, bi-Hamiltonian structures of KdV type, bundles of Weyl structures, Lax representations via twisted extensions of Lie algebras, energy functionals and normal forms of knots, and differential invariants of inviscid flows. The companion volume (Contemporary Mathematics, Volume 789) is devoted to Algebraic and Cohomological Aspects of PDEs. Cover Title page Contents The Editors Preface Geometric analysis of metric Legendre foliated cocycles on contact manifolds via SODE structure 1. Introduction 2. Frölicher-Nijenhuis formalism and projective metrizability problem 3. Legendre foliations on contact manifolds 4. Metric Legendre foliated cocycles compatible with SODE structure Conclusions References Special Vinberg cones, invariant admissible cubics and special real manifolds 1. Introduction 2. Nil-algebras and associated Vinberg cones 3. Invariant rational functions of a Vinberg cone 4. Examples of Nil-algebras and associated Vinberg cones 5. Construction of special real manifolds with large isometry groups Acknowledgments References Bundles of Weyl structures and invariant calculus for parabolic geometries 1. Parabolic geometries and the bundle of Weyl structures 2. The invariant calculus 3. The nearly invariant operators References Coverings and pseudosymmetries of differential equations 1. Introduction 2. Coverings of differential equations 3. Infinite jets 4. Mappings of diffieties 5. Pseudosymmetries of differential equations 6. Integrable pseudosymmetries 7. Example of calculating coverings from a control system 8. Commutation of pseudosymmetries 9. Multivector pseudosymmetries 10. Generators of modules of Cartan q-forms for control systems 11. Conclusion References Differential invariants of inviscid flows in pipes Introduction 1. The Euler system 2. Flows in pipes References Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles 1. Introduction 2. Preliminaries 3. Local gauge theories as gauge PDEs 4. Presymplectic gauge PDEs 5. Weak presymplectic gauge PDEs 6. Induced gauge PDEs on submanifolds and boundaries Appendix A. Technical details Acknowledgments References Minimal realizations of the KP hierarchy, its strict version and their reductions 1. Introduction 2. The hierarchies in a wider perspective 3. Reduction to a standard setting 4. The minimal realizations 5. Scaling invariance References Minimal coupling of gravitational and electromagnetic fields in General Relativity Introduction 1. Spacetime 2. Gravitational and electromagnetic fields 3. Observed expressions References Projective-geometric aspects of bi-Hamiltonian structures of KdV type 1. Introduction 2. Preliminaries 3. Conditions of compatibility 4. Conclusion Acknowledgment References Measurement of random operators, jet geometry and high-order phase transitions 1. Introduction 2. Measurement of random operators 3. Jet geometry and probability 4. Random endomorphisms References Metrics admitting projective and c-projective vector fields 1. Introduction 2. Projective structures, their metrizability and degree of mobility 3. Lie’s classical problems 4. Higher dimensional extensions 5. Complex Lie problem References Lax representations via twisted extensions of infinite-dimensional Lie algebras: some new results 1. Introduction 2. Preliminaries and notation 3. Lie algebras of the Kac–Moody type and their extensions 4. Equation associated to a2⋉q_{1,-1} 5. 3D generalized Hunter–Saxton equation 6. Equation associated to a2⋉q_{2,-1} 7. Equation associated to h⊕w 8. Concluding remarks Acknowledgments References Energy functionals and normal forms of knots and plane curves 1. Main definitions 2. Basic properties 3. Two reasonable conjectures 4. Energy functionals for thickened knots and normal forms 5. Physical experiments 6. Circular and almost flat normal forms 7. Normal forms vs. ideal knots Ackowledgements References Back Cover