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About this book

This volume presents lectures given at the Wisła 19 Summer School: Differential Geometry, Differential Equations, and Mathematical Physics, which took place from August 19 - 29th, 2019 in Wisła, Poland, and was organized by the Baltic Institute of Mathematics. The lectures were dedicated to symplectic and Poisson geometry, tractor calculus, and the integration of ordinary differential equations, and are included here as lecture notes comprising the first three chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and mathematical physics. Specific topics covered include:Parabolic geometryGeometric methods for solving PDEs in physics, mathematical biology, and mathematical financeDarcy and Euler flows of real gasesDifferential invariants for fluid and gas flowDifferential Geometry, Differential Equations, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry is assumed.

Table of Contents

Frontmatter

Chapter 1. Poisson and Symplectic Structures, Hamiltonian Action, Momentum and Reduction

Abstract
This manuscript is essentially a collection of lecture notes which were given by the first author at the Summer School Wisła–2019, Poland and written down by the second author. As the title suggests, the material covered here includes the Poisson and symplectic structures (Poisson manifolds, Poisson bi-vectors, and Poisson brackets), group actions and orbits (infinitesimal action, stabilizers, and adjoint representations), moment maps, Poisson and Hamiltonian actions. Finally, the phase space reduction is also discussed. The very last section introduces the Poisson–Lie structures along with some related notions. This text represents a brief review of a well-known material citing standard references for more details. The exposition is concise, but pedagogical. The authors believe that it will be useful as an introductory exposition for students interested in this specific topic.
Vladimir Roubtsov, Denys Dutykh

Chapter 2. Notes on Tractor Calculi

Abstract
These notes present elementary introduction to tractors based on classical examples, together with glimpses towards modern invariant differential calculus related to vast class of Cartan geometries, the so-called parabolic geometries.
Jan Slovák, Radek Suchánek

Chapter 3. Symmetries and Integrals

Abstract
In these lectures, I want to illustrate an application of symmetry ideas to integration of differential equations. Basically, we will consider only differential equations of finite type, i.e. equations with finite-dimensional space Sol of (local) solutions. Ordinary differential equations make up one of the main examples of such equations.
Valentin V. Lychagin

Chapter 4. Finite Dimensional Dynamics of Evolutionary Equations with Maple

Abstract
The theory of finite dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depend on a finite number of parameters among all solutions of evolutionary differential equations.
Alexei G. Kushner, Ruslan I. Matviichuk

Chapter 5. Critical Phenomena in Darcy and Euler Flows of Real Gases

Abstract
This paper is a survey article on some recent results obtained by the authors in the field of gas flows through porous media described by the Darcy law and flows of inviscid gases described by Euler equations. We explicitly formulate thermodynamics in terms of contact and symplectic geometry and show the link between thermodynamics and measurement theory. The methods provided by the geometrical reformulation of thermodynamics are applied to the analysis of various models of real gases, and special attention is paid to phase transitions. We provide explicit methods of finding solutions for the Dirichlet filtration problem and Euler flows and show how thermodynamics of gases under consideration emerges along the gas flow. In particular, we find the locations for different phases of the medium.
Valentin V. Lychagin, Mikhail D. Roop

Chapter 6. Differential Invariants for Flows of Fluids and Gases

Abstract
The paper is an extended overview of the papers. The main extension is a detailed analysis of thermodynamic states, symmetries, and differential invariants. This analysis is based on consideration of Riemannian structure naturally associated with Lagrangian manifolds that represent thermodynamic states. This approach radically changes the description of the thermodynamic part of the symmetry algebra as well as the field of differential invariants.
Anna Duyunova, Valentin V. Lychagin, Sergey Tychkov
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