Project 144032

Geometry, education and visualization with applications

Leader: dr Zoran Rakić


This project is a continuation of consecutive successful geometric projects established by academician Mileva Prvanović more then thirty years ago. In the beginning the classical differential geometry was developed. Later in eighties, research was enlarged by topics in modern differential geometry intertwined with global analysis, topology and algebra. The leader of geometric project in period 1991-2005 was prof. Neda Bokan. The project was obtained current name at the beginning of the previous project period, 2002-2005, motivated by new trends and our cooperation with Technical University of Berlin and Zusse Institute, Berlin. At the beginning of current project period some colleagues from fields of mathematical physics and theory of probability and statistics joined us and enlarged again topics of our research. The subject of the research now covers several topics in differential geometry its applications in the visualization and education, as well as mathematical physics and theory of probability and statistics.

Subject of research

The subject of this research consists of selected topics in differential geometry its applications in the visualization and education, as well as mathematical physics and theory of probability and statistics. The topics of our research are in the field of:

Description and importance of the research

Differential geometry has been intensively developed all over the world by using analytical methods as well as methods of algebra (theory of representations of groups and algebras, etc.), topology, 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, 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.
Research in the field of p-adic, adelic, and non-commutative geometry belong to a new and actual field of contemporary mathematical physics. Their importance is in the investigations non-Archimedean and non-commutative properties of space-time geometry and quantum phenomena on very small distances. It is the continuation of a successful research of MNTR project (2002-2005), 1426 'Quantum models on non-commutative and adelic spaces'.
Investigations in the fields of stochastic processes, Brownian motion and etc., are today of the great importance since they are applicable in combinatorics, physics, insurance and financial mathematics, etc.
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.
The problem of visualization 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 Bologna Declaration, devotes special attention to education in the field of geometry by using information technologies.
In that period, starting from the initial phase, we managed to attain a high level of development in differential geometry which resulted in a larger number of works published in international journals. Several scientific international and national meetings were organized by the members of this project, such are:

The proceedings of many of those meetings were published, some of them by the world-wide known publishing companies such are World Scientific, American Institute of Physics, etc.
Actuality and scientific value of those projects, in the previous period, has been confirmed by institutional and personal contacts with eminent institution and individuals. International cooperation was developed, with participation in international projects, financed by German foundation DAAD. We emphasize only the institutional cooperation with:

Expected Results

Discovering of new geometrical properties of: smooth manifolds (semi-Riemannian, endowed with various connections and additional structures) and their applications; Lie groups and algebras; quantum groups. Finding interactions between local and global invariants of manifolds and bundles constructed on them. Research on quantum and classical methods on p-adic, adelic and non-commutative spaces. Calculating the corresponding n-dimensional Feynman path integrals and corrections of standard phenomenological quantities. Discovering of new asymptotic properties of stochastic processes. Work on the software LinKnot and its application in knot and link theory. Interaction of empirical and theoretical knowledge in the computer-assisted teaching of geometry. Standardization and preparation of the material for computer-assisted distance learning of geometry, using Internet. Publishing of obtained results, organization of international scientific meetings and continuation of successful international cooperations (project financed by German foundation DAAD), and further implementation of obtained results in practice.