The main teaching activities of the institute are courses in programming, applied
mathematics, theory of vibrations, machine dynamics, computer science and simulation.
These courses are either for all students or for students in petroleum engineering,
polymer sciences, engineering and mining who want to specialize in applied computer
science or applied mathematics.
- Groups Acting on Graphs
We study automorphism groups of transitive and almost transitive graphs in connection
with the end structure and growth properties of the underlying graph. At the core
of many of these investigations is Gromov's charcterization of groups of polynomial
growth and its generalization to groups acting transitively on graphs.
Topics persued up to now and still being investigated by our group include automorphism
groups of graphs with polynomial growth, groups and graphs with linear growth, s-transitivity,
covering graphs, groups acting on trees and groups of products of graphs.
Furthermore, many of these concepts have been successfully applied to the investigation
of the subgroup structure of free and virtually free groups. Although the primary
goal of these activities are infinite graphs and groups, many important applications
pertain to finite structures, e.g. the construction of graphs with large girth and
contractors or expanders. Of particular interest in this respect are counting methods
for subgroups of given index in free groups and related groups.
In addition we started to investigate transitive directed graphs, in particular
highly arc-transitive digraphs. This is a quite young field of interest which has
close connections to topology. Besides structural properties of those graphs we
are mainly interested in their automorphism groups.
- Products of Graphs
This area is best described by 'The Product Graph Website'
- Algorithms and the Structure of Graphs
The enormous interest in good algorithms for the solution of large systems of linear
equations, both by sequential and parallel methods, has increased the importance
of structural investigations of associated networks. Thus, the research interests
of our group pertaining to products of graphs, isometric embeddings of graphs into
Cartesian products, efficient sequential and parallel algorithms for the decomposition
of graphs into Cartesian products, realizations of metrics by graphs, eigenvalue
methods for the decomposition of graphs and other problems have gained new dimensions.
- Numerical Linear Algebra and Computer Simulation
Numerical simulation of hydrocarbon flow in porous media and turbulent flow in combustion
engines initiated our interest in iterative solvers for large, sparse systems of
linear equations. We investigate multilevel incomplete factorizations of matrices
arising from finite-difference discretizations. Our interest lies in hierarchical
ordering strategies and estimates for resulting condition numbers. These methods
are closely related to algebraic multigrid methods. Promising coarsening strategies
based on minimum spanning trees in the grid are considered.
We also use eigenvalue methods for recursive spectral decomposition of graphs. These
methods are implemented for domain decomposition on distributed-memory parallel
computers. Here we deal with additional constraints like load-balance conditions
or edge sets that must not be cut.
The modeling of flow and transport, and discretication of the resulting partial
differential equations is another area of interest. For example, discretization
strategies in regions where a moving grid glides along stationary grid cells were
developed. The practical application behind this task was to simulate air flow in
rotating fans to optimize the performance of laundry dryers.
Current activities also include the drying of porous refractory bricks and thermal
monitoring of steel slabs.
- Scientific Collaborations
We carry out our investigations in close collaboration with mathematicians all over
the world. The main contacts were established to neighbouring countries as Slovenia
and Hungary, but the list of our co-authors also includes colleagues from Canada,
USA, Iceland, Russia, New Zealand, Germany, Croatia and Czech Republic.