Topology and Physics of Multiple Structures
Investigation of Topological and Physical Aspects of Objects with Multiple Structures
Tech Area / Field
- PHY-RAW/Radiofrequency Waves/Physics
- PHY-SSP/Solid State Physics/Physics
3 Approved without Funding
Georgian Academy of Sciences / A. Razmadze Mathematical Institute, Georgia, Tbilisi
- McGill University, Canada, QC, Montreal\nKobe University, Japan, Kobe
Project summaryStarting from 70ies of the XXth century, physicists become increasingly interested in algebraic topology and other areas of topology. It becomes apparent that many natural phenomena possess topological properties, which manifests itself especially clearly in quantum physics where, as soon as one encounters fields of complicated mathematical structure with nontrivial singularities, one immediately meets topological questions and, frequently, in an essential way. Similar situations one can encounter also while studying liquid crystals and low temperatore phases of various materials. Thus the above situations are at the same time those in which physical considerations influence greatly topological thinking. Naturally there exist no less important influence in the opposite direction too. In particular, algebraic topology and homotopy theory, topology and geometry of fibrations, theory of singular varieties, symplectic and contact topology and geometry, as well as dimension theory, spaces with two different topologies encountered in Lorenz geometry and many others, are all those topological instruments which are fundamentally used in almost all areas of contemporary physics.
Therefore, the theoretical, as well as in the applied fields of mathematics and physics set with various different structures, particularly: metric, topological, algebraic, bitopological, order, uniformity, etc. often appear. It is known too, that in case of existence two or more structures (of the same name or different) they are not independent, but definite relation i. e. “distribution” exists between them, sometimes of implicative character, i. e. one structure generates the other. Namely the mentioned relations (together with their characteristics) form a strong instrument that can give much more information about the research object, then the consideration each structure separately.
The variety of above mentioned structures and real perspectives of their confluence allowed us to present general project, the aim of which is not only to obtain new results in different directions, but also approach to them from the general point of view, what can be clearly expressed by the following scheme:
As to our theme, the introduction will be carried out as follows:
1. The world-wide energy crisis becoming increasingly deep and acute at present is caused by two reasons. On the one hand, the oil and gas reserves will be sufficient for the mankind for about 50 years. On the other hand, due to the greenhouse effect, the use of traditional types of fuel causes undesirable and very dangerous changes of climatic conditions. Therefore, the problem of utilization of nontraditional sources of ecologically pure energy including low-potential water is becoming more urgent and important.
The reserve of low-potential and ecologically pure energies in the world is great. Among these energies are the energy of geothermal waters, the energy of industrial waste water, the energy of water heated by solar radiation, the energy of warm ocean currents, etc. So far, this practically inexhaustible source of energy has not been used due to the fact that at present, in the world there is no economically sound and efficient converter in which the low-potential energy is used.
Lately, at the Andronikashvili Institute of Physics (Tbilisi, Georgia) the converters have been successfully developed which are capable to operate efficiently using the low-potential energy. The operation of these converters is based on ferromagnetics – paramagnet, ferroelectric – dielectric, austenite – martensitic structural phase transitions, and the working body is the material having the Curie point in the range of low-potential water temperatures.
In above-listed cases, the heat energy of low-potential water is converted into magnetic field energy, electric field energy, and into elastic energy of working body, respectively. It should be noted that the Curie point of the used materials covers a rather wide temperature interval.
2. Topological structure τ given on a set X will be used for defining of a new continuous singular cohomology theory.
A connection with classical singular cohomology will be established in the form of isomorphism, if a given space (X, τ) is CW-complex or paracompact manifold, and the coefficients' group is the real number topological group.
New continuous homology will be constructed. For compact space (X, τ), an isomorphism of continuous homology and the Milnor homology will be proved, when the group of coefficients is R. If dim≤Xn, then the isomorphism will be proved, when the group of coefficients is S1 - one dimensional sphere. Steenrod's duality theorem will be proved for constructed continuous singular cohomologies and continuous homologies.
3. One natural problem in uniform shape theory is to find necessary and sufficient conditions under which a space of some given class has a compactification (remainder) with given topological property.
This problem is interesting and has nontrivial solutions for the following topological properties: n-dimensional (co)homology group, (co)homotopy groups of compactification (remainder) is a given group; shape of compactification (remainder) is shape of given space; cohomological and shape dimensions of compactification (remainder) is given number.
In particular we are here aiming at the characterization of strong (co)shapes and description of (co)homology groups of compactifications and remainders of topological spaces; to solve this problem, the uniform strong shape and coshape theories and uniform strong homological and cohomological theories will be constructed and investigated.
4. The notion of local homeomorphism plays central role in sheaf theory. On the other hand, local homeomorphisms only represented “half of the story”: there are many classes of fibrewise discrete maps which reflect entirely different aspects of set variation and cannot be captured within sheaf theory. In the fifties, Fox investigated certain generalization of branched coverings which he called complete spreads and which had clear features of the “other side” of local homeomorphisms – they are to local homeomorphisms what closed sets are to open sets.
Much more recently Jonathon Funk discovered that this dichotomy goes much deeper: complete spreads are to local homeomorphisms what cosheaves are to sheaves. This brought up intriguing perspective of relating more classes of fibrewise discrete maps to new kinds of nonclassical higher order systems generalizing both intuitionistic type theories with their sheaf models, and hypothetical systems modelled on cosheaves.
One general problem in this direction is description of fibrewise dimension. In this approach, dimensions of fibres of a map are considered globally as a single dimension of a single non-classical space defined over the base.
5. As it is known, the classical cardinal dimension functions in the class of all separable metrizable spaces have following properties: invariantness (P1), normability (P2), monotonicity (P3), σ-additivity (P4), compactificability (P5), the logarithmic property (P6), subadditivity (P7), decomposability (P8), Gδ-enlargement (P9).
These properties (known as the main theorems of dimension theory) play the same role in dimension theory as Eilenberg-Steenrod axioms do in homology theory of general spaces. Following the aims of the project, maximally wide classes of topological spaces will be established, where the classical dimension functions coincide and have all of the properties (P1 – P9) (this question is in close connection with one problem of A. Arhangel’skii and L. Tumarkin). It is expected that (under some natural restrictions) each such class is a subclass of the class of all separable metrizable spaces. Besides, for the classical and some dimension-like functions different classes of spaces (normal, π-compact, locally compact, etc.) will be pointed out, where these functions have various collections of properties (P1 – P9) mentioned above; the classes of spaces, where the classical dimension functions are the unique integervalued invariants, having all (P1 – P9) properties, will also be established. This question is in close connection with the K.Mengers’s old problem on axiomatic characterization of dimension functions in the class of all subspaces of n-dimensional Euclidean space.
Since according to several authors’ results in the class of all completely regular spaces none of classical dimension functions have all nine properties (P1 – P9) mentioned above, the research concerning existence of dimension-like functions defined in the class of all completely regular spaces, or on different subclasses of it, which have all (P1 – P9) properties will be the question of interest. The same question may be considered for any (nonempty) subsystem of the system (P1 – P9) (note the quantity of all such subsystems is equal to 511).
6. Bitopological spaces have a lot of applications in such fundamental fields of mathematics, as analysis, algebraic and differential topologies, mathematical logic, potential theory, computer sciences, theories of linear topological and ordered topological spaces, etc. Representation of bitopological spaces by so called Boolean algebras is of extreme importance. Investigations for development of the theory will be carried out, where localization of bitopological properties and relative variants will be studied. In cases of inclusion, Weston’s C-relativity, Todd’s S- relativity, N- relativity and tangency of topologies, relations between topological, bitopological and almost dimensional functions will be established. Investigations of applied type will be continued: in point set topology - to establish new conditions of maximal connectedness for a strong topology of bitopological space, to study D-spaces and their bitopological modifications, to characterize bitopologically tangency of topologies; in analysis – to establish properties of Baire type and relations between them; in potential theory – to establish new properties of fine topology by help of weak topology and bitopological properties and to discover new relations with ordered topological spaces. Especial consideration should be given for establishment of those classes of spaces, strong topology of which gives possibility of existence of complete bounded computational model, and also to relative separation axioms and relative connectedness, in particular, many relative versions of p-T0, p-T1, p-T2, (i; j)- and p-regularities, (i; j)- and p-complete regularities and p-normality are discussed. Moreover, relative properties of (i; j)- and p-compactness types, including relative versions of (i; j)- and p-paracompactness, (i; j)- and p-Lindelöfness, (i; j)- and p-pseudocompactness will be introduced and investigated together with relative bitopological inductive and covering dimension functions, and, on the other hand, to relative versions of Baire spaces for both the topological and the bitopological case.
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