Generalized Walsh function systems and digital problems in number theory and in number-theoretical numerics.

This project essentially was concerned with the generation and application of large high-dimensional point sets, with very good uniform distribution properties. Such point sets are used for the approximate evaluation of high- dimensional integrals. The evaluation of such integrals is of great importance in various fields of applications as for example in image-processing (computer tomography), in physics and in financial mathematics (option pricing, risk management). The generation of suitable point sets is (especially in high dimensions) as well from the theoretical point of view as also from the practical point of view an intrinsic problem. One possible method is the method of digital nets. This method was developed about ten years ago by H. Niederreiter in Vienna. In the present project we developed new theoretical results for this method, but especially we also were concerned with the algorithmic implementations (parallel computing) of these point sets. Further we especially investigated the applicability of these methods in financial mathematics. The theoretical investigations also strongly led to the topic of discrete geometry.