Kruglikov / Lychagin / Straume Differential Equations - Geometry, Symmetries and Integrability

The Abel Symposium 2008
1. Auflage 2009
ISBN: 978-3-642-00873-3
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 394 Seiten

Reihe: Abel Symposia

The Abel Symposium 2008 focused on the modern theory of differential equations and their applications in geometry, mechanics, and mathematical physics. Following the tradition of Monge, Abel and Lie, the scientific program emphasized the role of algebro-geometric methods, which nowadays permeate all mathematical models in natural and engineering sciences. The ideas of invariance and symmetry are of fundamental importance in the geometric approach to differential equations, with a serious impact coming from the area of integrable systems and field theories. This volume consists of original contributions and broad overview lectures of the participants of the Symposium. The papers in this volume present the modern approach to this classical subject.

Autoren/Hrsg.

Kruglikov, Boris

Lychagin, Valentin

Straume, Eldar

Inhaltsverzeichnis

1;Preface to the Series;6
2;Preface;7
3;Contents;10
4;Some Canonical Structures of Cartan Planes in Jet Spaces and Applications;14
4.1;1 Jets of Submanifolds;15
4.2;2 The Contact System;18
4.3;3 The Canonical Bracket;20
4.4;4 The Spencer Sequence and Formal Integrability;22
4.5;5 Differential Correspondences;24
4.6;6 Characteristics of Systems of PDE;26
4.7;References;32
5;Transformations of Darboux Integrable Systems;34
5.1;1 Introduction;34
5.2;2 Symmetry Reduction of Exterior Differential Systemsand the Method of Darboux;36
5.3;3 Transformations of Darboux Integrable Systems;39
5.4;4 Linear Equations;41
5.5;5 Internal Equivalences of Some Non-linear PDE;47
5.6;6 Moutard Equations;49
5.7;7 First Order Linear Systems;52
5.8;8 Goursat's Equation;54
5.9;9 The Monge Equations '= [(n)]2;58
5.10;References;60
6;Differential Geometric Heuristics for Riemannian Optimal Mass Transportation;62
6.1;1 Optimal Mass Transportation Diffeomorphims;62
6.2;2 Geometry of the Group of Diffeomorphisms, After Arnol'd;65
6.2.1;2.1 Rearrangement Classes;65
6.2.2;2.2 Tangent Bundle;66
6.2.3;2.3 The Arnol'd Metric;66
6.3;3 The Riemannian Submersion P:DiffProb, after Moser, Ebin--Marsden and Otto;67
6.3.1;3.1 The Submersion;67
6.3.2;3.2 Helmholtz Splitting;68
6.3.3;3.3 Horizontal Lift;69
6.3.4;3.4 The Otto Metric;70
6.3.5;3.5 The L2 Wasserstein Distance;70
6.4;4 Geodesics;71
6.4.1;4.1 A Sufficient Condition for Geodesicity in Diff;71
6.4.2;4.2 Short Horizontal Segments;72
6.4.3;4.3 Horizontal Segments in the Large;78
6.4.3.1;4.3.1 A Reinhart Lemma;78
6.4.3.2;4.3.2 Necessity of Condition G;78
6.5;5 Conclusion: Heuristical Statement;80
6.6;References;85
7;On Rank Problems for Planar Webs and Projective Structures;87
7.1;1 Introduction;87
7.2;2 Planar Webs;88
7.3;3 Basic Constructions;89
7.4;4 Rank;92
7.5;5 Abel's Method;95
7.5.1;5.1 3-Webs;96
7.5.2;5.2 4-Webs of Rank Three;97
7.5.3;5.3 4-Webs of Rank Two;99
7.5.4;5.4 4-Webs of Rank One;100
7.5.5;5.5 4-Webs of Rank Zero;102
7.6;6 Abelian Differential Equations;104
7.7;7 Rank of 4-Webs;106
7.7.1;7.1 The Obstruction;106
7.7.2;7.2 4-Webs of Maximum Rank;108
7.7.3;7.3 4-Webs of Maximum Rank and Surfaces of Double Translation;108
7.7.4;7.4 4-Webs of Rank Two;109
7.7.5;7.5 4-Webs of Rank One;111
7.8;8 Planar 5-Webs of Maximum Rank;112
7.9;9 Projective Structures and Planar 4-Webs;113
7.10;References;117
8;Niceness Theorems;119
8.1;1 Introduction and Statement of the Problems;119
8.2;2 Examples;121
8.2.1;2.1 Lots of Compatible Structure Examples;121
8.2.1.1;2.1.1 Groups in the Category of Groups;121
8.2.1.2;2.1.2 Comonoids in the Category of Groups;123
8.2.1.3;2.1.3 Hopf's Theorem on the Cohomology of H-Spaces;123
8.2.1.4;2.1.4 Intermezzo;124
8.2.1.5;2.1.5 Milnor--Moore Theorem (Topological Incarnation);126
8.2.1.6;2.1.6 Theorem ([ch05:bib79 Milnor et al.];127
8.2.1.7;2.1.7 Cartier's Theorem on Nilpotents in Group Schemes;127
8.2.1.8;2.1.8 Nielsen--Schreier Theorem;128
8.2.1.9;2.1.9 Shirshov--Witt Theorem;128
8.2.1.10;2.1.10 Bergman Centralizer Theorem;128
8.2.1.11;2.1.11 H-Spaces;128
8.2.1.12;2.1.12 Bott--Samelson Theorem;128
8.2.2;2.2 Universal Object Examples;128
8.2.2.1;2.2.1 The Universal Enveloping Algebra of a Lie Algebra;129
8.2.2.2;2.2.2 The Group Algebra of a Group;130
8.2.2.3;2.2.3 Free Algebras;130
8.2.2.4;2.2.4 Cofree Coalgebras;130
8.2.2.5;2.2.5 The Classifying Spaces BUn;131
8.2.3;2.3 Niceness Theorems for Hopf Algebras;132
8.2.3.1;2.3.1 The Leray Theorem on Commutative Hopf Algebras;132
8.2.3.2;2.3.2 The Milnor--Moore Theorem on Cocommutative Hopf Algebras;132
8.2.4;2.4 Large vs. Nice;133
8.2.4.1;2.4.1 Big Projective Models are Free;133
8.2.4.2;2.4.2 General Linear Groups in Various Dimensions;133
8.2.4.3;2.4.3 Kuiper's Theorem, [ch05:bib65 Kuiper];133
8.2.5;2.5 Extremal Objects and Niceness;134
8.2.6;2.6 Uniqueness and Rigidity and Niceness;134
8.2.7;2.7 Counterexamples and Paradoxical Objects;135
8.2.7.1;2.7.1 The Alexander Horned Sphere;135
8.2.7.2;2.7.2 The Approximation Property;135
8.2.7.3;2.7.3 The Banach--Tarski Paradox;136
8.2.7.4;2.7.4 Julia and Fatou Sets;137
8.2.7.5;2.7.5 Sorgenfrey Line;137
8.2.7.6;2.7.6 Exotic Spheres;138
8.2.8;2.8 An Excursion into Formal Group Theory;138
8.2.8.1;2.8.1 Lazard Commutativity Theorem;139
8.2.8.2;2.8.2 Universal Formal Groups;139
8.2.8.3;2.8.3 Morphisms;139
8.2.8.4;2.8.4 Logarithms;139
8.2.8.5;2.8.5 p--Typical Formal Groups;140
8.2.8.6;2.8.6 The Universal p--Typical Formal Group, [ch05:bib47 Hazewinkel];140
8.2.8.7;2.8.7 Formal Groups from Cohomology;141
8.2.9;2.9 The Amazing Witt Vectors and Their Gracious Applications15;142
8.2.9.1;2.9.1 Definition of the Functor of the Big Witt Vectors;142
8.2.9.2;2.9.2 Lambda Rings and Sigma Rings;144
8.2.9.3;2.9.3 The Comonad Structure on the Big Witt Vectors;146
8.2.9.4;2.9.4 The Sigma and Lambda Ring Structures on Symm;146
8.2.9.5;2.9.5 Cartier's First Theorem;146
8.2.10;2.10 The Star Example: Symm;147
8.2.11;2.11 Product Formulas;148
8.3;3 Some First Results and Theorems;149
8.3.1;3.1 Freeness Theorems;149
8.3.2;3.2 On the Lazard Universal Formal Group Theorem;150
8.3.3;3.3 Objects and Isomorphisms in Connection with Symm;150
8.3.3.1;3.3.1 The Isomorphism `Ha' Between R(W), the Representing Ringof the Functor of the Big Witt Vectors and U(),the Free Lambda Ring on One Generator;151
8.3.3.2;3.3.2 The Isomorphism `Z' Between R(S) and Symm;151
8.3.3.3;3.3.3 The Isomorphism `S' from R(S) to Rrat (GL);152
8.3.3.4;3.3.4 On a Possible Isomorphism `L' Between R(S) and H(BU;Z);153
8.3.3.5;3.3.5 On the Isomorphism `Du' Between H (BU;Z) and H (BU;Z);153
8.3.3.6;3.3.6 On the Isomorphism `SP' Between H (BU;Z) and Symm;154
8.3.3.7;3.3.7 The Isomorphism `F' Between R(W) and U();155
8.3.3.8;3.3.8 On the Isomorphisms `M1' and `M3' Between R(S), K(Pk) and U();156
8.3.3.9;3.3.9 On the Object E(Z) and the Isomorphisms `Ho1' and `Ho2';156
8.3.3.10;3.3.10 The K-Theory of Endomorphisms;157
8.3.3.11;3.3.11 Leftovers;159
8.4;References;159
9;The Polynomial Algebra and Quantizations of Electromagnetic Fields;163
9.1;1 Quantizations and Braidings of Zn-Graded Modules;164
9.2;2 The Algebra of Polynomials C[ x1,…,xn];164
9.2.1;2.1 Quantizations of Derivations of C[ x1,…,xn];166
9.2.2;2.2 Quantizations of Connections and Curvatures of C[ x1,…,xn];167
9.3;3 Electromagnetic Fields;169
9.4;References;170
10;A Bridge Between Lie Symmetries and Galois Groups;171
10.1;1 Introduction;171
10.2;2 Lie Symmetries of Differential Equations;172
10.3;3 Symmetries of Algebraic Equations;173
10.3.1;3.1 First Example;175
10.3.2;3.2 Second Example;177
10.3.3;3.3 Third Example;178
10.4;4 Derivation of the Galois Groups from Symmetries;179
10.4.1;4.1 First Example;179
10.4.2;4.2 Second Example;180
10.4.3;4.3 Third Example;181
10.4.4;4.4 A New Definition of Galois Groups;181
10.5;5 The Galois Representation of Lie Symmetries of Differential Equations;182
10.6;References;184
11;Focal Systems for Pfaffian Systems with Characteristics;185
11.1;1 Introduction;185
11.2;2 Focal Systems: General Results;186
11.3;3 Focal Systems for First-Order Partial Differential Equations in the Plane;188
11.4;4 Focal Systems for Second-Order Hyperbolic Partial Differential Equations in the Plane;192
11.5;References;197
12;Hamiltonian Structures for General PDEs;198
12.1;1 Introduction;198
12.2;2 Notation: Infinite Jets and Differential Equations;200
12.3;3 Cotangent Bundle to an Equation;202
12.4;4 Examples;207
12.5;References;208
13;Point Classification of Second Order ODEs: Tresse Classification Revisited and Beyond;210
13.1;1 Scalar Differential Invariants;211
13.2;2 Tresse Classification Revisited;213
13.2.1;2.1 Relative Differential Invariants of Second Order ODEs;213
13.2.2;2.2 Specifications;215
13.3;3 Classification of Second Order ODEs;217
13.3.1;3.1 Dimensional Count;217
13.3.2;3.2 Absolute Differential Invariants;218
13.3.3;3.3 Equivalence Problem;219
13.4;4 Singular Stratum: Projective Connections;222
13.4.1;4.1 The Original Approach of Tresse;223
13.4.2;4.2 The Second Tresse Approach;224
13.4.3;4.3 Lie Equations;225
13.5;5 Application to Symmetries;228
13.6;References;231
14;Classification of Monge--Ampère Equations;233
14.1;1 Lychagin's Approach to Monge--Ampère Equations;235
14.1.1;1.1 Differential Operators;235
14.1.2;1.2 Effective Differential Forms;236
14.1.3;1.3 Contact Equivalence, Symmetries and Multivalued Solutions;236
14.1.4;1.4 Symplectic Geometry of Equations;238
14.2;2 Geometry of Monge--Ampère Equationson Two-Dimensional Manifolds;239
14.2.1;2.1 Geometric Structures on J1M;239
14.2.2;2.2 Geometric Structures on TM;241
14.3;3 Tensor Invariants of Equations;241
14.4;4 The Laplace Forms;244
14.5;5 Monge--Ampère Equations with Constant Coefficients;247
14.6;6 Contact Linearization of the Monge--Ampère Equations;248
14.6.1;6.1 +=-=0;248
14.6.2;6.2 +=0 and -=0;249
14.6.3;6.3 One of the Laplace Forms Is Zero and the Another One Is Not;250
14.6.4;6.4 Normal Form vxxvyy=k(x,y)v+f(x,y);250
14.7;7 Equivalence of Monge--Ampère Equations to Linear Equations with Constant Coefficients;250
14.8;8 Equivalence Problem for Nondegenerate Equations;252
14.9;9 Symplectic Equations and Operators;254
14.9.1;9.1 Tensor Invariants of Nondegenerate Symplectic Equations;254
14.9.2;9.2 Absolute Parallelism for Symplectic Equations;255
14.9.3;9.3 Absolute Parallelism for Symplectic Operators;258
14.9.4;9.4 The Tricomi Operator;263
14.10;References;264
15;On Nonabelian Theories and Abelian Differentials;267
15.1;1 Introduction;267
15.2;2 Topological Solution of Dispersionless Toda;271
15.3;3 Nekrasov Partition Function;274
15.4;4 The Gromov--Witten Potential of P1;276
15.5;5 Nonabelian Theory and Abelian Integrals;278
15.6;6 Different Functional Formulations;280
15.7;7 Conclusion;283
15.8;References;283
16;Geometric Aspects of the Quantization of a Rigid Body;285
16.1;1 Introduction;285
16.2;2 Geometric Quantization;286
16.3;3 Covariant Quantum Mechanics;288
16.4;4 Rigid Body;291
16.5;References;294
17;Shooting for the Eight: A Topological Existence Proof for a Figure-Eight Orbit of the Three-Body Problem;296
17.1;1 Introduction;296
17.2;2 Equation of Motion and Reduction;298
17.3;3 A Wazewski Set;303
17.4;4 The Shooting Argument;309
17.5;5 Proofs of the Lemmas;313
17.6;References;319
18;Compatible Poisson Brackets, Quadratic Poisson Algebras and Classical r-Matrices;320
18.1;1 Introduction;320
18.2;2 Compatible Brackets and Linearization: General Case;323
18.3;3 Quadratic Algebras and Poisson--Lie Groups;328
18.3.1;3.1 First and Second Sklaynin Brackets;328
18.3.2;3.2 ``Twisted Reflection Equation Algebra'' and Its Linearizations;334
18.4;4 Conclusion and Discussion;341
18.5;References;342
19;Contact Geometry of Second Order I;343
19.1;1 Introduction;343
19.2;2 Geometry of Jet Spaces;345
19.2.1;2.1 Spaces of Contact Elements;345
19.2.2;2.2 Second Order Contact Manifolds;346
19.2.3;2.3 Derived Systems and Cauchy Characteristic Systems;348
19.2.4;2.4 Review of Tanaka Theory;348
19.2.5;2.5 Symbol Algebra of (L(J),E);350
19.3;3 Symbols of Second Order Equations;352
19.3.1;3.1 Symbol Algebras;352
19.3.2;3.2 Case n=2;354
19.3.3;3.3 Involutive Symbols;356
19.3.4;3.4 Typical Symbols;358
19.4;4 PD Manifolds of Second Order;359
19.4.1;4.1 Realization Theorem;359
19.4.2;4.2 First Reduction Theorem;361
19.4.3;4.3 Construction of (R(X); DX1,DX2);364
19.4.4;4.4 Symbol Subspaces;367
19.5;5 Parabolic Geometries Associated with PD-Manifolds of Second Order;368
19.5.1;5.1 Differential Systems Associated with Simple Graded Lie Algebras (Parabolic Geometries);368
19.5.2;5.2 Standard Contact Manifolds;371
19.5.3;5.3 Classical Type Examples;372
19.5.4;5.4 Exceptional Type Examples;374
19.6;6 Examples of First Reduction Theorem;376
19.6.1;6.1 Typical Class of Type f3(r);376
19.6.2;6.2 G2-Geometry;378
19.6.3;6.3 Other Examples;381
19.7;References;393