Riassunto
One service mathematics has rendered the 'Eot moi, ... , si j'avait JU comment en revenir. human race. h has put common sense back je n'y serais point aUe:' Jules Verne where it belongs, 011 the topmost shelf nen to the dusty canister labeUed 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. H es viside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other pans and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
Contenuti
1. Simplest Classical Variational Problems.- §1 Equations of Extremals for Functionals.- §2 Geometry of Extremals.- 2.1. The Zero-Dimensional and One-Dimensional Cases.- 2.2. Some Examples of the Simplest Multidimensional Functional. The Volume Functional.- 2.3. The Classical Plateau Problem in Dimension 2.- 2.4. The Second Fundamental Form on the Riemannian Submanifold.- 2.5. Local Minimality.- 2.6. First Examples of Globally Minimal Surfaces.- 2. Multidimensional Variational Problems and Extraordinary (Co)Homology Theory.- §3 The Multidimensional Plateau Problem and Its Solution in the Class of Mapping on Spectra of Manifolds with Fixed Boundary.- 3.1. The Classical Formulations (Finding the Absolute Minimum).- 3.2. The Classical Formulations (Finding a Relative Minimum).- 3.3. Difficulties Arising in the Minimization of the Volume Functional volk for k > 2. Appearance on Nonremovable Strata of Small Dimensions.- 3.4. Formulations of the Plateau Problem in the Language of the Usual Spectral Homology.- 3.5. The Classical Multidimensional Plateau Problem (the Absolute Minimum) and the Language of Bordism Theory.- 3.6. Spectral Bordism Theory as an Extraordinary Homology Theory.- 3.7. The Formulation of the Solution to the Plateau Problem (Existence of the Absolute Minimum in Spectral Bordism Classes).- §4 Extraordinary (Co)Homology Theories Determined for “Surfaces with Singularities”.- 4.1. The Characteristic Properties of (Co)Homology Theories.- 4.2. Extraordinary (Co)Homology Theories for Finite Cell Complexes.- 4.3. The Construction of Extraordinary (Co)Homology Theories for “Surfaces with Singularities” (on Compact Sets).- 4.4. Verifying the Characteristic Properties of the Constructed Theories.- 4.5. Additional Properties of Extraordinary Spectral Theories.- 4.6. Reduced (Co)Homology Groups on “Surfaces with Singularities”.- §5 The Coboundary and Boundary of a Pair of Spaces (X, A).- 5.1. The Coboundary of a Pair (X,A).- 5.2. The Boundary of a Pair (X,A).- §6 Determination of Classes of Admissible Variations of Surfaces in Terms of (Co)Boundary of the Pair(X, A).- 6.1. Variational Classes h(A,L,L?) and h(A,$$\tilde L $$).- 6.2. The Stability of Variational Classes.- §7 Solution of the Plateau Problem (Finding Globally Minimal Surfaces (Absolute Minimum) in the Variational Classes h(A,L,L?) and h(A,$$\tilde L $$ )).- 7.1. The Formulation of the Problem.- 7.2. The Basic Existence Theorem for Globally Minimal Surfaces. Solution of the Plateau Problem.- 7.3. A Rough Outline of the Existence Theorem.- §8 Solution of the Problem of Finding Globally Minimal Surfaces in Each Homotopy Class of Multivarifolds.- 3. Explicit Calculation of Least Volumes (Absolute Minimum) of Topologically Nontrivial Minimal Surfaces.- §9 Exhaustion Functions and Minimal Surfaces.- 9.1. Certain Classical Problems.- 9.2. Bordisms and Exhaustion Functions.- 9.3. GM-Surfaces.- 9.4. Formulation of the Problem of a Lower Estimate of the Minimal Surface Volume Function.- §10 Definition and Simplest Properties of the Deformation Coefficient of a Vector Field.- §11 Formulation of the Basic Theorem for the Lower Estimate of the Minimal Surface Volume Function.- 11.1. Functions of the Interaction of a Globally Minimal Surface with a Wavefront.- 11.2. Formulation of the Basic Volume Estimation Theorem.- §12 Proof of the Basic Volume Estimation Theorem.- §13 Certain Geometric Consequences.- 13.1. On the Least Volume of Globally Minimal Surfaces Passing through the Centre of a Ball in Euclidean Space.- 13.2. On the Least Volume of Globally Minimal Surfaces Passing through a Fixed Point in a Manifold.- 13.3. On the Least Volume of Globally Minimal Surfaces Formed by the Integral Curves of a Field ?.- §14 Nullity of Riemannian, Compact, and Closed Manifolds. Geodesic Nullity and Least Volumes of Globally Minimal Surfaces of Realizing Type.- 14.1. The Definition of the Nullity of a Manifold.- 14.2. The Theorem on the Relation of Nullity with the Least Volumes of Surfaces of Realizing Type.- 14.3. The Proof of the Reifenberg Conjecture Regarding the Existence of a Universal Upper Estimate of the “Complexity” on the Singular Points of Minimal Surfaces of Realizing Type.- §15 Certain Topological Corollaries. Concrete Series of Examples of Globally Minimal Surfaces of Nontrivial Topological Type.- 15.1. Globally Minimal Surfaces Realizing Nontrivial (Co)Cycles in Symmetric Spaces.- 15.2. Compact Symmetric Spaces and Explicit Form of a Geodesic Diffeomorphism.- 15.3. Explicit Computation of the Deformation Coefficient of a Radial Vector Field on a Symmetric Space.- 15.4. An Explicit Formula for the Symmetric Space Geodesic Nullity.- 15.5. Globally Minimal Surfaces of Least Volume (volkX0 = ?k0 in Symmetric Spaces are Symmetric Spaces of Rank 1.- 15.6. Proof of the Classification Theorem for Surfaces of Least Volume in Certain Classical Symmetric Spaces.- 4. Locally Minimal Closed Surfaces Realizing Nontrivial (Co)Cycies and Elements of Symmetric Space Homotopy Groups.- §16 Problem Formulation. Totally Geodesic Submanifolds in Lie Groups.- §17 Necessary Results Concerning the Topological Structure of Compact Lie Groups and Symmetric Spaces.- 17.1. Cohomology Algebras of Compact Lie Groups.- 17.2. Subgroups Totally Nonhomologous to Zero.- 17.3. Pontryagin Cycles in Compact Lie Groups.- 17.4. Necessary Results Concerning Symmetric Spaces.- §18 Lie Groups Containing a Totally Geodesic Submanifold Necessarily Contain Its Isometry Group.- §19 Reduction of the Problem of the Description of (Co)Cycles Realizable by Totally Geodesic Submanifolds to the Problem of the Description of (Co)Homological Properties of Cartan Models.- §20 Classification Theorem Describing Totally Geodesic Submanifolds Realizing Nontrivial (Co)Cycles in Compact Lie Group (Co) Homology.- 20.1. The Statement of the Classification Theorem.- 20.2. The Case of Spaces of Type II.- 20.3. The Case of Spaces of Type I (Co) Homologie ally Trivial Cartan Models. Properties of the Squaring Map of a Symmetric Space.- 20.4. The Case of Spaces of Type I. Spaces SU(k)/SO(k).- 20.5. The Case of Spaces of Type I. Spaces SU(2m)/Sp(m).- 20.6. The Case of Spaces of Type I. Spaces S21-1 = SO(2l)/SO(2l - 1). Explicit Computation of Cocycles Realizable by Totally Geodesic Submanifolds of Type I.- 20.7. The Case of Spaces of Type I. Space E6/F4.- §21 Classification Theorem Describing Cocycles in the Compact Lie Group Cohomology Realizable by Totally Geodesic Spheres.- 21.1. Classification Theorem Formulation.- 21.2. Totally Geodesic Spheres Realizing Bott Periodicity.- 21.3. Realization of Homotopy Group Elements of the Compact Lie Groups by Totally Geodesic Spheres.- 21.4. Necessary Results Concerning the Spinor and Semispinor Representations of an Orthogonal Group.- 21.5. Spinor Representation of the Orthogonal Group SO(8) and the Cayley Number Automorphism Group.- 21.6. Description of Totally Geodesic Spheres Realizing Nontrivial (Co)Cycles in Simple Lie Group Cohomology. The Case of the Group SU(n).- 21.7. The Case of the Groups SO(n) and Sp(2n).- §22 Classification Theorem Describing Elements of Homotopy Groups of Symmetric Spaces of Type I, Realizable by Totally Geodesic Spheres.- 22.1. Classification Theorem Statement.- 22.2. Proof of the Classification Theorem. Relation between the Number of Linearly Independent Fields on Spheres and that of the Elements of Homotopy Groups Realizable by Totally Geodesic Spheres.- 5. Variational Methods for Certain Topological Problems.- §23 Bott Periodicity from the Dirichlet Multidimensional Functional Standpoint.- 23.1. Explicit Description of the Bott Periodicity Isomorphism for the Unitary Group.- 23.3. Unitary Periodicity and One-Dimensional Functional.- 23.4. The Periodicity Theorem for a Unitary Group is Based on the Dirichlet Functional Two-Dimensional Extremals.- 23.5. The Periodicity Theorem for an Orthogonal Group is Based on the 8-Dimensional Dirichlet Functional Extremals.- §24 Three Geometric Problems of Variational Calculus.- 24.1. Minimal Cones and Singular Points of Minimal Surfaces.- 24.2. The Equivariant Plateau Problem.- 24.3. Representation of Equivariant Singularities as Singular Points of Closed Minimal Surfaces Embedded into Symmetric Spaces.- 24.4. On the Existence of Nonlinear Functions Whose Graphs in Euclidean Space Are Minimal Surfaces.- 24.5. Harmonic Mappings of Spheres in Nontrivial Homotopy Classes.- 24.6. A Rough Outline of Certain Recent Results on the Link of Harmonic Mapping Properties to the Topology of Manifolds.- 24.7. Properties of the Density of Smooth Mappings of Manifolds.- 24.8. The Behaviour of the Dirichlet Functional on the 2-Connected Manifold Diffeomorphism Group. Proof of Theorem 24.6.9.- 24.9. Necessary Topological Condition for the Existence of Nontrivial Globally Minimal Harmonic Mappings.- 24.10. The Minimization of Dirichlet-Type Functionals.- 24.11. Regularity of Harmonic Mappings.- 6. Solution of the Plateau Problem in Classes of Mappings of Spectra of Manifolds with Fixed Boundary. Construction of Globally Minimal Surfaces in Variational Classes h(A,L, L?) and h(A, $$\tilde L $$ )).- §25 The Cohomology Case. Computation of the Coboundary of the Pair (X,A) = ?r(Xr,Ar) in Terms of Those of (Xr,Ar).- §26 The Homology Case. Computation of the Boundary of the Pair (X,A) = ?r(Xr,Ar) in Terms of the Boundaries of (Xr,Ar).- §27 Closedness, Invariance, and Stability of Variational Classes.- 27.1. S-Surgery of Surfaces in a Riemannian Manifold.- 27.2. The Closedness of Variational Classes Relative to the Passage to the Limit.- 27.3. The Invariance of Variational Classes Relative to S-Surgeries of Surfaces.- 27.4. The Stability of Variational Classes.- §28 The General Isoperimetric Inequality.- 28.1. Choice of a Special Coordinate System.- 28.2. Simplicial Points of Surfaces.- 28.3. Isoperimetric Inequality.- §29 The Minimizing Process in Variational Classes and h(A,L,$$\tilde L $$ ).- 29.1. The Minimizing Sequence of Surfaces. Density Functions Related to Surfaces.- 29.2. A Rough Outline of the Minimizing Process.- 29.3. The Constructive Method for the Minimizing Process and the Proof for Its Convergence. First Step.- 29.4. Second and Subsequent Steps in the Minimizing Process.- 29.5. The Theorem on the Coincidence of the Least Stratified Volume with Least ?-Vector in a Variational Class.- §30 Properties of Density Functions. The Minimality of Each Stratum of the Surface Obtained in the Minimization Process.- 30.1. The Value of the Density Function is Always not Less Than Unity on Each Stratum, and Unity only at Regular Points.- 3.2. Each Stratum Is a Smooth Minimal Submanifold, Except Possibly a Set of Singular Points of Measure Zero.- §31 Proof of Global Minimality for Constructed Stratified Surfaces.- 31.1. Proof of the Basic Existence Theorem for a Globally Minimal Surface.- 31.2. The Proof of the Theorem on the Coincidence of the Least Stratified Volume with the Least ?-Vector in a Variational Class.- §32 The Fundamental (Co)Cycles of Globally Minimal Surfaces. Exact Realization and Exact Spanning.- 32.1. Fundamental (Co)Cycle Theorem.- 32.2. Exact Minimal Realization and Exact Minimal Spanning.- 32.3. Minimal Surfaces with Boundaries Homeomorphic to the Sphere.- Appendix I. Minimality Test for Lagrangian Submanifolds in Kähler Manifolds. Submanifolds in Kähler Manifolds. Maslov Index in Minimal Surface Theory.- §1 Definitions.- §3 Certain Corollaries. New Examples of Minimal Surfaces. The Maslov Index for Minimal Lagrangian Submanifolds.- Appendix II. Calibrations, Minimal Surface Indices, Minimal Cones of Large Codimensional and the One-Dimensional Plateau Problem.
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