Project 1646

Geometry, education and visualization with applications

Leader: dr Neda Bokan

Abstract

Subject of research

• Manifolds and fiber bundles in the problems of geometry, topology and mathematical physics.
• Aspects of visual mathematics, applications of computer graphics
• In visualisation of results of geometry in the field of education,
• In physics, chemistry, crystallography and engineering sciences.
• Distance learning based on theoretical results in the field of geometry, by way of Internet.
• Integration of empirical knowledge and theoretical knowledge in computer-assisted instruction of geometry.

Description of the work

• Hypersurfaces and submanifolds of smooth manifolds endowed with various structures
• Osserman manifolds
• Representations of groups at the spaces of curvature tensors
• Quantum groups, deformations of enveloping algebras of Lie algebras
• Extension of A. Gray’s investigations on small geodesic spheres
• Transformations and vector fields
• Geometry of differential operators
• Interdependence of topological and geometrical properties of manifolds
• Generalizations of Chen’s curvature invariants
• Homogeneous Riemannian spaces
• Geometrical analysis of a typical quantum system
• Geometry of smooth manifolds endowed with various structures
• Infinitesimal deformations of surfaces and Riemannian spaces
• Development of software for geometry which provides toolbox for graphical representations
• Development of numerical methods for the solutions of visibility, contour problems and other problems in geometry of curves and surfaces
• Development of algorithms for numerical integration and solutions of differential equations for geodesic lines, asymptotic lines, lines of curvature, etc.
• Implementation of the above-mentioned methods and algorithms
• Preparation of the material from various fields of geometry for long distance learning, using the Internet facilities
• Using of computer graphics based on Java applets in long distance learning and electronical publications (textbooks, manuals, etc)
• Development of software for making figures and animations based on formal description of constructions
• Formal description of constructions in hyperbolic plane and in Euclidean plane (Poincare model) described in natural language
• Using software products such as: Mathematica, Flash, Cinderella, etc. for obtaining optimal results in animations

Originality of research

• Synthesis of methods and results from various fields (geometry, informatics, education)
• Analysis of methods and results of learning through Internet
• Participation in projects of DAAD, etc.

Research Goal

• Discovering new geometrical properties of smooth manifolds
• Interaction between local and global invariants of manifolds and bundles constructed on them
• Consideration of various aspects of visual geometry and other fields of mathematics using computer graphics
• Interaction of empirical knowledge and theoretical knowledge in the computer-assisted instruction of geometry
• Preparation of the material for computer-assisted distance learning of geometry, inter alia, using the Internet

State of the Art in Scientific Field

World-wide Situation

• Differential geometry has been intensively developed all over the world by using analytical methods as well as algebraic methods, topology and the like, which was initiated by the necessities of researching certain phenomena in the fields of theoretical mechanics, physics, etc.
• This field is naturally intertwined with other fields of mathematics: Lie groups, quantum groups, global analysis, differential topology, non-commutative geometry, theory of gravitation and theory of relativity, etc.
• Results of differential geometry are used, for instance, in statistics (geometries of small geodesic spheres and tubes), architecture (minimal surfaces) and civil engineering (deformations of surfaces), genetic engineering
• Computer graphics is one of the most important computer fields and it cannot be imagined without the use of differential geometry of curves and surfaces, as well as the application of required numerical methods
• Modern software for creating Internet presentations (Flash, Front Page, etc.) is based on the use of special curves (Bezieržs curves and the like)
• The problem of visualisation and animation is one of the most interesting current problems in the computer science, which renders the interdisciplinary approach to this field indispensable
• The great project of the European Union, established by the Bologne Declaration, devotes special attention to education in the field of geometry by using information technologies

Domestic Situation

• Geometry was intensively developed in our country in the second half of the XXth century
• In that period, starting from the initial phase, we managed to attain a high level of development in geometry, which resulted in a larger number of works published in international journals
• International cooperation was developed, with participation in international projects, financed by:
  - DAAD foundation in Germany,
  - several universities (in Moscow, Rome, Prague, Santiago de Compostella, etc.)
• Several scientific meetings were organised in the field of geometry, out of which some include education problems (and lately problems of visual geometry as well)
• So far, the field of computer geometry has not been developed intensively in our country, neither have its applications in education

Planned Project Results

The cooperation with the company "EKI SOFT" (a section of the Economical Institute) in the production of JAVA applications (applets) based on the computer graphics, which may be used in the Internet presentations

The software production as a support of long distance learning

• testing
• presentation of some courses
• the creation of course participants databases
• evaluation and analysis of participants results
• organisation of geometrical courses using the metod of long distance learning