Maria Ulan - Differential Geometry, Differential Equations, and Mathematical Physics

This series will serve as a resource for the publication of results and developments presented at summer or winter schools, workshops, tutorials, and seminars. Written in an informal and accessible style, they present important and emerging topics in scientific research for PhD students and researchers. Filling a gap between traditional lecture notes, proceedings, and standard textbooks, the titles included in TSWMS present material from the forefront of research.

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The Summer School Wisla 19: Differential Geometry, Differential Equations, and Mathematical Physics was organized by the Baltic Institute of Mathematics and took place on August 1929, 2019, in the beautiful mountain region of Wisa, Poland.

The subsequent chapters continue the tradition of the previous three by introducing the reader to different topics at the intersection of differential geometry, differential equations, and mathematical physics. They are written in a pedagogical style while simultaneously bringing to attention recent advances made by their authors. The book is aimed at students and researchers who are looking for a concise introduction to the topics covered here. Since all six chapters are written from a geometric perspective, the reader is expected to have some basic knowledge of differential geometry. Below is a summary of each chapter:

Poisson and Symplectic Structures, Hamiltonian Action, Momentum and Reduction is written in a concise form and gives a brief review of well-known material. It covers Poisson and symplectic structures, group actions and orbits, moment maps, and Poisson and Hamiltonian actions. In the end, phase space reduction and PoissonLie structures are discussed.

Notes on Tractor Calculi presents an elementary introduction to tractors based on classical examples, together with glimpses toward modern invariant differential calculus related to a vast class of Cartan geometries, the so-called parabolic geometries. The exposition aims at a quick understanding of basic principles, omitting many proofs or at least their details. Some knowledge in representation theory is assumed.

Symmetries and Integrals is an invitation to the world of symmetries and differential equations. It starts with an introduction to distributions and moves quickly to the challenge of finding explicitly, in quadratures, integral manifolds of completely integrable distributions. While these lecture notes rely heavily on the insight that a geometric understanding of distributions and symmetries gives, they simultaneously focus on the practical aspects of finding, and writing down, exact solutions of differential equations.

Finite Dimensional Dynamics of Evolutionary Equations with Maple uses geometric methods to find exact solutions to partial differential equations appearing in physics, mathematical biology, and mathematical finance. The authors emphasize the computational aspect and provide detailed Maple code. This makes the chapter an excellent introduction to the subject of finite dimensional dynamics by allowing the reader to start using the methods instantaneously.

Critical Phenomena in Darcy and Euler Flows of Real Gases is a survey article on recent results obtained by its authors concerning gas flows through porous media and flows of inviscid gases. Thermodynamics is formulated in terms of contact and symplectic geometry, and the link to measurement theory is emphasized. The methods provided by the geometric formulation of thermodynamics are applied to the analysis of various models of real gases, and special attention is paid to phase transitions. The thermodynamics of the gases under consideration emerges along the gas flow. Explicit methods for finding solutions to the Dirichlet filtration problem and Euler flows are provided. In particular, the locations for different phases of the medium are found.

Differential Invariants for Flows of Fluids and Gases summarizes several of the authors earlier results while simultaneously improving them by taking into account additional structure on the thermodynamic states. Similar to the previous chapter, it relies on the symplectic and contact-geometric formulation of thermodynamics. After adding thermodynamic equations of state to the NavierStokes and Euler equations, the authors compute point symmetries of the equations. The symmetry Lie algebra depends, in general, on the equations of state, and an analysis of possible symmetry Lie algebras is made before the corresponding differential invariants are computed.