Probabilistic methods in analysis and applications

The concept of uniform distribution modulo one originates from problems in physics and astronomy. Informally speaking, an infinite sequence of real numbers from the unit interval is uniformly distributed modulo one if for every subinterval of [0,1] the relative number of elements of the sequence (up to an index N) in this subinterval converges to its length as N tends to infinity. Here the length of the interval may be interpreted as the expected value of the relative number of elements of a sequence in this interval, and, roughly speaking, an uniformly distributed sequence may be seen as a sequence showing "random" behavior. However, the speed of convergence can vary substantially for different sequences. For this reason the notion of discrepancy was introduced, to allow an evaluation of the "quality" of the distributional behavior of sequences of reals. From a probabilistic point of view, the discrepancy compares the empirical distribution of the first N elements of a sequence to the underlying (uniform) distribution. In 1975 Philipp solved the Erdös-Gal conjecture and showed that the discrepancy of lacunary sequences satisfies a bounded law of the iterated logarithm. This is similar to the Chung-Smirnov LIL for the Kolmogorov-Smirnov statistic, and gives evidence of the famous heuristics that the behavior of lacunary function systems is similar to the behavior of independent, identically distributed random variables. In the last few years there has been substantial progress in this field. Fukuyama developed a new technique to obtain a precise version of Philipp`s result, which suddenly permitted to attack some of the classical, long-standing problems in the field. The notion of discrepancy has a natural generalization to multi-dimensional sequences. For some d-dimensional function f, the Koksma-Hlawka inequality states that the difference between the arithmetic mean of the function values at certain points and its integral can be estimated by the product of the variation of f and the discrepancy of the point set. This is the foundation of the Quasi-Monte Carlo method for numerical integration, which is based on the fact that for a point sequence with small discrepancy the average value of f, evaluated at these sampling points, is a good approximation for the integral of f. In the past, the focus was mainly on finding sequences for which the discrepancy is small for "large" values of N (and for fixed dimension d). Recently a different point of view gained some attention, asking for point sequences with small discrepancy for "small" values of N (as compared to the dimension d). In this case classical deterministic methods do not seem to work, and it is necessary to apply probabilistic methods. In particular, the use of hybrid or weakly dependent random structures (instead of independent ones) seems to be promising with regard to algorithmic aspects. The proposer plans to spend the first year of his Schrödinger scholarship with IAN SLOAN at the University of New South Wales in Sydney, Australia. In this year, he plans to focus his research on problems asking for the existence of low-discrepancy sequences for "small" N (in comparison with d). The second year will be spent with KATUSI FUKUYAMA at Kobe University in Japan, focusing on problems concerning lacunary function systems and metric discrepancy. The third year (return phase) will be spent with GERHARD LARCHER at the University of Linz, Austria. In this the proposer wants to apply the (theoretical) results from the first two years to problems in applied mathematics. In the last year of his Schrödinger scholarship the proposer also wants to write his post- doctoral thesis (Habilitationsschrift), which is necessary for his postdoctoral lecture qualification (Habilitation) in Austria.

My Schrödinger scholarship covered a period of three years, consisting of two years abroad plus one year (the so-called "return phase") in Austria. I spent the first year at the University of New South Wales in Sydney, Australia, the second year at Kobe University in Japan, and the third year at the Johannes Kepler University in Linz.In the first year, in Sydney, I worked in the research group of Ian Sloan, who is one of the leading scientists in the field of numerical mathematics, which is the branch of mathematics concerned with the approximate solution of mathematical problems on a computer. This research group includes several excellent scientists, and it was stimulating and educational to work in such an active and internationally well-connected research group. I started an intensive and ongoing collaboration with Joseph Dick, where we investigated how certain problems from financial and actuarial mathematics can be treated in a more realistic way. The direct method which we developed is new, and has already been adopted by other researchers.I spent the second year at the University of Kobe for an intensive collaboration with Katusi Fukuyama, an expert for certain aspects of Fourier Analysis. In this year I also participated in the International Congress of Mathematicians, the largest conference for mathematics which meets once every four years. In the third year in Linz I worked together with Gerhard Larcher and his group at the Department for Financial Mathematics and Applied Number Theory, working on topics concerning pseudo-randomness of numbers. Simply speaking, the problem there is to quantify whether a given sequence of numbers (such as a sequence of measurements) shows "random" behavior or rather has a specific (non-random) structure.Simultaneously I started an intensive (on-line) collaboration with Kristian Seip from Trondheim and Michel Weber from Strasbourg on the infamous Riemann zeta function, which plays a prominent role in the theory of prime numbers.The three years of my Schrodinger scholarship yielded more than 20 research papers, which were published in international mathematical journals. Furthermore I wrote a habilitation treatise, for which I received the lecture qualification at the TU Graz in October 2014. I gave more than 20 scientific talks in several countries, including South Korea, England, the USA, China and Armenia. Furthermore I received some of the highest Austrian research awards for young scientists, including the FWF START prize, the Hlawka prize of the Austrian Academy of Science, the award of the Austrian Mathematical Society, and the KardinalInnitzer-Förderungspreis.