Nonlinear PDEs, Their Geometry, and Applications: Proceedings of the Wisła 18 Summer School (Tutorials, Schools, and Workshops in the Mathematical Sciences)

Foreword
Preface
Acknowledgements
Contents
Contributors
Acronyms
Part I Lectures
1 Contact Geometry, Measurement,  and Thermodynamics
	1.1 Preface
	1.2 A Crash Course in Probability Theory
		1.2.1 Measure Spaces and Measurable Maps
		1.2.2 Operations Over Measures, Measure Spaces,  and Measurable Maps
		1.2.3 The Lebesgue Integral
		1.2.4 The Radon–Nikodym Theorem
		1.2.5 The Fubini Theorem
		1.2.6 Random Vectors
		1.2.7 Conditional Expectation
		1.2.8 Dependency, Coherence Conditions, and Tensor Product of Random Vectors
	1.3 Measurement of Random Vectors
		1.3.1 Entropy and the Shannon Formula
		1.3.2 Gain of Information
		1.3.3 Principle of Minimal Information Gain
		1.3.4 The Gaussian Distribution
		1.3.5 Central Moments
		1.3.6 Change of Information Gain
		1.3.7 Constraints and Constitutive Relations
		1.3.8 Application to Classical Mechanics and Classical Field Theory
	1.4 Thermodynamics
		1.4.1 Laws of Thermodynamics
		1.4.2 Thermodynamics and Measurement
		1.4.3 Gases
		1.4.4 Thermodynamic Processes and Contact Transformations
	References
2 Lectures on Geometry of Monge–Ampère Equations with Maple
	2.1 Introduction
	2.2 Lecture 1. Introduction to Contact Geometry
		2.2.1 Bundle of 1-Jets
		2.2.2 Contact Transformations
	2.3 Lecture 2. Geometrical Approach to Monge–Ampère Equations
		2.3.1 Non-linear Second-Order Differential Operators
		2.3.2 Multivalued Solutions of Monge–Ampère Equations
		2.3.3 Effective Forms
	2.4 Lecture 3. Contact Transformations of Monge–Ampère Equations
	2.5 Lecture 4. Geometrical Structures
		2.5.1 Pfaffians
		2.5.2 Fields of Endomorphisms
		2.5.3 Characteristic Distributions
		2.5.4 Symplectic Monge–Ampère Equations
		2.5.5 Splitting of Tangent Spaces
	2.6 Lecture 5. Tensor Invariants of Monge–Ampère Equations
		2.6.1 Decomposition of de Rham Complex
		2.6.2 Tensor Invariants
		2.6.3 The Laplace Forms
		2.6.4 Contact Linearization of the Monge–Ampère Equations
	References
3 Geometry of Monge–Ampère Structures
	3.1 About These Lectures
	3.2 Lecture One: What Is It All About?
		3.2.1 Basic Geometric Structures
		3.2.2 Kähler, Special and Other Related Structures
		3.2.3 Holomorphic Symplectic Structures
		3.2.4 Lagrangian, Special Lagrangian and Complex Lagrangian Submanifolds
		3.2.5 Hyperkähler Manifolds
		3.2.6 Generalised Complex Structure
		3.2.7 Notes and Further Reading
	3.3 Lecture Two: Recursion (Nijenuijs) Operators  and Some Related Algebraic Constructions
		3.3.1 Recursion Operators and Its Properties
		3.3.2 Triples of Symplectic Forms
		3.3.3 Notes and Further Reading
	3.4 Lecture Three: Symplectic Monge–Ampère Operators and Equations
		3.4.1 Monge–Ampère Equations
		3.4.2 Geometry of Differential Forms
		3.4.3 Notes and Further Reading
	3.5 Lecture Four: Monge–Ampère Structures
		3.5.1 General Properties
		3.5.2 (4m+2)-Dimensional MA Geometry
		3.5.3 Explicite Examples of Generalised Almost Calabi–Yau on T*mathbbR3 (After B.Banos)
		3.5.4 Notes and Further Reading
	3.6 Lecture Five
		3.6.1 Bi-Lagrangian, Special Lagrangian, Special Kähler and Monge–Ampère Equations
		3.6.2 2d and 3d Rotating Stratified Flows—Dritschel–Viudez Diagnostic MAEs
		3.6.3 Generalised Complex Geometry and Monge–Ampère Structures
		3.6.4 Notes and Further Reading
	References
4 Introduction to Symbolic Computations in Differential Geometry with Maple
	4.1 Introduction
	4.2 Basic Setup
		4.2.1 Subpackage Tools
	4.3 Calculations with Vectors and Forms
		4.3.1 Computing Symmetries
	4.4 Transformations
		4.4.1 Operations on Transformations
	Reference
Part II Participants Contributions
5 On the Geometry Arising in Some Meteorological Models in Two and Three Dimensions
	5.1 Introduction
	5.2 A Brief Guide in Balanced Meteorological Models
	5.3 Monge–Ampère Geometry
		5.3.1 Monge–Ampère Operators
		5.3.2 Generalized Solutions
		5.3.3 The Problem of Local Equivalence
		5.3.4 Monge–Ampère Structures
	5.4 2d Rotating Stratified Flows—Dritschel–Viudez MAE
		5.4.1 Integrability of the Complex/Product Structure
		5.4.2 Underlying Hypersymplectic Geometry
		5.4.3 2d-Diagnostic Equation of Dritschel–Viudez: Special Choice of Constant Coefficients
		5.4.4 Reduction to Constant Coefficients
		5.4.5 Variation of the Potential and Hyper-Kähler Metrics in 4d
	5.5 Some Examples of 3d-Geostrophic Models
		5.5.1 The Birkett and Thorpe Equation (BT)
		5.5.2 The Hoskins Equation (H)
		5.5.3 The McIntyre-Roulstone Equation (McI)
		5.5.4 The Snyder–Skamarock–Rotunno Equation (SSR)
	5.6 3d Rotating Stratified Flows—Dritschel–Viudez MAE
	References
6 Gas Flow with Phase Transitions: Thermodynamics and the Navier–Stokes Equations
	6.1 Introduction
	6.2 Geometric Representation of Thermodynamic States
	6.3 Van der Waals Gases
		6.3.1 The Equations of State
		6.3.2 Applicable Domains for the Van der Waals Gas
		6.3.3 Phase Transitions
	6.4 Asymptotic Expansions for Solution
		6.4.1 Zeroth-Order Approximation
		6.4.2 First-Order Approximation
	6.5 Phase Transitions Along the Gas Flow
	References
7 Differential Invariants in Thermodynamics
	7.1 Introduction
	7.2 Geometry of Thermodynamics
	7.3 Equivalence of Thermodynamical Systems
	7.4 Differential Invariants
		7.4.1 Differential Invariants Under Aff(V)
		7.4.2 Differential Invariants Under G0 timesAff(V)
	7.5 Application to Gases
		7.5.1 Distinguishing Gases
		7.5.2 Ideal Gas
		7.5.3 Van der Waals Gas
	References
8 Monge–Ampère Grassmannians, Characteristic Classes and All That
	8.1 Grassmannians, Associated with the Lagrangian  and Legendrian Planes
	8.2 Integral or Monge–Ampère Grassmannians
	8.3 Grassmannians for 2- and 3- Effective Forms
		8.3.1 Integral Grassmannians for Monge–Ampère Equations in Dimension 2
		8.3.2 Geometric Structure Associated with 3d- MA Equations
		8.3.3 Integrability and MA Grassmannians in 3d
	8.4 Multidimensional Generalisation of Splitting Construction
		8.4.1 Non-degenerate 2k+1- Forms in Sense of Hitchin
	8.5 Characteristic Classes of Monge–Ampère Equations on a 3-Dimensional Manifolds
		8.5.1 Special Lagrangian Monge–Ampère Characteristic Classes
		8.5.2 Remarks and Speculations About mathbbS
	References
9 Weak Inverse Problem of Calculus  of Variations for Geodesic Mappings  and Relation to Harmonic Maps
	9.1 Geodesic Mappings and Basic Setting
	9.2 Harmonic Mappings
	9.3 Weak Inverse Problem of Calculus of Variations
	9.4 Summary and Conclusions
	References
10 Integrability of Geodesics of Totally Geodesic Metrics
	10.1 Introduction
	10.2 Singularities
	10.3 Geodesics
		10.3.1 Geodesic Equations
		10.3.2 Integrability of Geodesic Equations
	10.4 Einstein–Maxwell Solutions
	10.5 Discussion
	10.6 Conclusions
	References
Index