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.
Inhaltsverzeichnis
1. Simplest Classical Variational Problems. - §1 Equations of Extremals for Functionals. - §2 Geometry of Extremals. - 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. - §4 Extraordinary (Co)Homology Theories Determined for Surfaces with Singularities . - §5 The Coboundary and Boundary of a Pair of Spaces (X, A). - §6 Determination of Classes of Admissible Variations of Surfaces in Terms of (Co)Boundary of the Pair(X, A). - §7 Solution of the Plateau Problem (Finding Globally Minimal Surfaces (Absolute Minimum) in the Variational Classes h(A, L, L?) and h(A, $$\tilde L $$
)). - §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. - §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. - §12 Proof of the Basic Volume Estimation Theorem. - §13 Certain Geometric Consequences. - §14 Nullity of Riemannian, Compact, and Closed Manifolds. Geodesic Nullity and Least Volumes of Globally Minimal Surfaces of Realizing Type. - §15 Certain Topological Corollaries. Concrete Series of Examples of Globally Minimal Surfaces of Nontrivial Topological Type. - 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. - §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. - §21 Classification Theorem Describing Cocycles in the Compact Lie Group Cohomology Realizable by Totally Geodesic Spheres. - §22 Classification Theorem Describing Elements of Homotopy Groups of Symmetric Spaces of Type I, Realizable by Totally Geodesic Spheres. - 5. Variational Methods for Certain Topological Problems. - §23 Bott Periodicity from the Dirichlet Multidimensional Functional Standpoint. - §24 Three Geometric Problems of Variational Calculus. - 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). - §28 The General Isoperimetric Inequality. - §29 The Minimizing Process in Variational Classes and h(A, L, $$\tilde L $$
). - §30 Properties of Density Functions. The Minimality of Each Stratum of the Surface Obtained in the Minimization Process. - §31 Proof of Global Minimality for Constructed Stratified Surfaces. - §32 The Fundamental (Co)Cycles of Globally Minimal Surfaces. Exact Realization and Exact Spanning. - 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.
