Intro; 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 Formula1.3.2 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 Geometry2.2.1 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 Equations2.6.1 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 Reading3.3 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