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:
- differential geometry: smooth manifolds and submanifolds endowed with
their geometry and applications in general theory of relativity; geometry
and topology of fiber
bundles; Osserman manifolds; representations of groups on spaces of
homogenous and symmetric spaces; Lie groups and algebras; special
geodesic balls; Chen's curvature invariants; quantum groups and their
applications; knot and
- mathematical physics: quantum and classical models on p-adic, adelic
- probability and statistics: stochastic, stationary and stable processes;
probabilistic and other methods in combinatorics;
- visual mathematics and computing geometry, visualization of geometrical
education; connection between empirical and theoretical knowledge in
teaching of geometry; construction and application of software LinKnot in
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
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:
- international meetings in the fields of
- geometry (International Conference on Differential geometry and its
Contemporary geometry and related topics, 2002,
- visualizations (annual meetings of DAAD project 'Multimedia Technology
Mathematics and Computer
Science Education': 2004, 2005),
- mathematical physics (Summer Schools of Modern Mathematical
2002, 2004, and 2nd International Conference on p-adic Mathematical
- national meetings (very often with international participants) in the
- geometry (Geometrical Seminars, 1980-2004, totally fourteen
The proceedings of many of those meetings were published, some of them by
the world-wide known publishing companies such are World
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:
- Technical University, Berlin,
- Free University, Berlin
- Moscow State University-Chair of Differential Geometry and
- Zusse Institute, Berlin,
- Tor Vergata University, Rome
- Pavle Savić, French-Serbian bilateral cooperation,
- Steklov Mathematical Institute, Moscow,
- Abdus Salam International Centre for Theoretical Physics, Trieste.
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.