Groups, Invariants, Integrals, and Mathematical Physics: The Wisła 20-21 Winter School and Workshop

Preface
Acknowledgements
Contents
Contributors
Differential Invariants in Algebra
	1 Introduction
	2 Invariants of Binary Forms
		2.1 Algebraic Point of View
		2.2 Differential Point of View
		2.3 Relations Between Algebraic and Differential Invariants
		2.4 Lie Equation
		2.5 Resultants and Discriminants
		2.6 Operations and Structures on Invariants
			2.6.1 Monoid Structure
			2.6.2 Poisson Structure
			2.6.3 Invariant Frame
		2.7 Invariant Coframe
		2.8 Weights
		2.9 Invariants of Binary Forms for n=2,3,4
	3 Quotients
		3.1 Rosenlicht Theorem
		3.2 Algebraicity in Jet Geometry
		3.3 Algebraic Differential Equations
		3.4 Lie-Tresse Theorem
		3.5 Integrability via Quotients
	4 Algebraic Plane Curves
		4.1 Connections and Affine Structures
		4.2 Symmetric Tensors
		4.3 Affine Invariants
		4.4 Invariants of Algebraic Curves
	5 Invariants of Ternary Forms
	References
Lectures on Poisson Algebras
	1 Introduction
	2 Motivation
		2.1 Lagrangian and Hamiltonian Mechanics
		2.2 Hamiltonian Mechanics and Poisson Brackets
	3 Poisson Algebras
		3.1 Subalgebras and Ideals
		3.2 Morphisms and Derivations
	4 Hamiltonian Derivations and Casimirs
		4.1 Exterior Algebra of a Commutative Algebra
	5 Homology and Cohomology
		5.1 Hochschild (Co)Homology
		5.2 Lichnerowicz-Poisson Cohomology
		5.3 Low-Dimensional Poisson Cohomology
			5.3.1 Compatible Poisson Structures
			5.3.2 Interpretation of HP2 (A)
		5.4 Poisson Homology
		5.5 Duality
	6 Polynomial Poisson Algebras
		6.1 Nambu-Jacobi-Poisson Algebras
		6.2 Poisson-Calabi-Yau Algebra
			6.2.1 Low-Dimensional Cohomology of the PCY Algebra
		6.3 Dual Poisson Complex
	7 Graded Poisson Algebras
		7.1 Algebra of Differential Operators
	8 Intermezzo: Tensor, Symmetric and Exterior Algebras
		8.1 Tensor Algebra of a Vector Space
		8.2 Symmetric Algebra of a Vector Space
		8.3 Exterior Algebra of a Vector Space
		8.4 Poisson Structure on a Symmetric Algebra S(g)
	9 Universal Enveloping and PBW Theorem
		9.1 Universal Enveloping Algebra
		9.2 Poincaré-Birkhoff-Witt (PBW) Theorem
		9.3 Universal Enveloping and Differential Operators
	10 Poisson Manifolds
		10.1 Poisson Structure on the Cotangent Bundle
		10.2 Poisson Manifolds
		10.3 Hamiltonian Mapping
		10.4 Poisson Bracket on a Symplectic Manifold
		10.5 Examples of Poisson and Symplectic Manifolds
		10.6 Poisson Manifolds and Lie Theory
		10.7 Symplectic Foliation on g*
			10.7.1 Coadjoint Invariant Functions
	11 Differential Calculus on Poisson Manifolds
		11.1 Coordinate-Free Construction of the Schouten Bracket
	12 Modified Double Poisson Brackets
		12.1 Poisson Brackets for General Associative Algebras
			12.1.1 Ginzburg-Voronov Lemma
			12.1.2 Representation Scheme
			12.1.3 Moduli Space of Representations
		12.2 Double Poisson Brackets
		12.3 Quadratic Double Poisson Brackets
		12.4 Examples and Classification of Low Dimensional Quadratic Double Poisson Brackets
	References
Some Remarks on Multisymplectic and Variational Nature of Monge-Ampère Equations in Dimension Four
	1 Introduction
	2 Preliminary Notions
		2.1 Contact Structure on J1M
		2.2 Symplectic Calculus on the Cartan Distribution
		2.3 Monge-Ampère Operators and Effective Forms
	3 Lagrangians, Variational Problems and the Euler Operator
		3.1 First-Order Lagrangians
		3.2 Euler-Lagrange Equations and the Euler Operator
	4 Effective Forms and the Inverse Variational Problem
		4.1 Plebański, Grant, and Husain Equations
		4.2 Klein-Gordon Equation
	5 Multisymplectic Formulation
		5.1 Plebański, Grant, and Husain Equations
		5.2 Klein-Gordon Equation
	6 Conclusion and Discussion
	References
Generalized Solvable Structures Associated to Symmetry Algebras Isomorphic to gl(2,R) R
	1 Introduction
	2 Preliminaries: Solvable Structures
	3 Generalized Solvable Structures for GL(2,R) R-Invariant Fifth-Order Equations
		3.1 Construction of a Generalized Solvable Structure
	4 Example
	5 Concluding Remarks
	References
Fundamental Groupoids and Homotopy Types of Non-compactSurfaces
	1 Introduction
	2 Striped Surface and Its Graph
		2.1 Seams
		2.2 Foliated Characterization of Striped Surfaces
		2.3 Graph of a Striped Surface
		2.4 Canonical Injection φ:G →Z
	3 Fundamental Groupoids
		3.1 Small Categories
		3.2 Functors
		3.3 Coequalizers
		3.4 Groupoids
		3.5 Fundamental Groupoid
		3.6 Coproducts
		3.7 van Kampen Theorem for Groupoids
		3.8 1-Diagram for Covers by Simply Connected Sets
	4 Proof of Theorem 5.3
	5 Proof of Theorem 5.2
	References
A Geometric Framework to Compare PDEs and Classical FieldTheories
	1 Introduction
		1.1 Previous Attempts to Compare Theories
		1.2 Requirements for the Framework
		1.3 Methods
		1.4 Outline
	2 Notation and Preliminaries
	3 Correspondence and Intersection
		3.1 Motivating Example
		3.2 Formal Definitions
		3.3 Local Description
	4 Consistency Conditions
		4.1 Smoothness Conditions
		4.2 Differential Consistency
	5 Formal Integrability
		5.1 Definitions and Preliminaries
		5.2 Formal Theory
		5.3 Integrability Conditions
		5.4 Explicit Example of the Application of Proposition 16
	6 Shared Structure
		6.1 Definition
		6.2 Solution Transfer
	7 Bäcklund Correspondences
	8 Equivalence Up to Symmetry and Quotient Equations
	9 Application to Electrodynamics and Hydrodynamics
		9.1 Formal Integrability of Maxwell\'s Equations
		9.2 Embedding of Vacuum Electrodynamics in Wave Equations
		9.3 Equivalence Up to Gauge Symmetry
		9.4 Shared Structure of Magneto-Statics and Hydrodynamics
	10 Discussion
		10.1 Conclusion
		10.2 Outlook
	References