During my stay in Austria I plan to work on the geometry of numbers and to study convex geometry with its
potential applications. I plan to work in collaboration with Professor Peter Gruber (Technische Universität Wien).
The project here proposed has the following main objectives:
1. To continue to study diophantine problems, including decompositions of integer vectors in multidimensional
space and Siegel`s Lemma, using results both from the geometry of numbers and from convex geometry;
2. To work intensively on properties of lattices, on DOTU matrices and on problems of Minkowski and Mordell;
3. To become familiar with algorithmic results about lattice packings of convex polytopes, affine surface area and
valuations, stochastic approximation, approximation of convex bodies, distribution of point sets on Riemannian
manifolds and applications of these to signal processing and numerical integration, stability problems and, possibly,
work in these areas.
In the course of the work further problems may turn up which are worth to be considered.
Convex geometry, which is the main research area of Professor Gruber`s group is close to the geometry of numbers
and should provide help for research in the geometry of numbers. Convex geometry draws tools from different
areas of mathematics, for example from analysis, differential geometry and functional analysis. On the other hand it
has applications both in pure mathematics and in applied fields such as optimization, control theory, calculus of
variations, potential theory, mathematical physics, crystallography and information theory. For these reasons I am
convinced that a research stay in Vienna will give me a better insight into a modern area of research and also into
its applications.