Probabilitic Discrepancy Theory and Diophantine Equations

Combinatorics and Number Theory are classical parts of mathematics that are still modern and active fields of research. In particular, during the last decades the development of computer science has opened up new fields of applications (for example in Cryptography) and several new theoretical questions (like complexity problems for algorithms) have come up and have influenced the field of research. In parallel classical methods of combinatorics and number theory are more and more complemented by probabilistic and analytic tools and by tools from dynamical systems in order to quantify various objects of combinatorial and number theoretic nature. This interplay between discrete and continuous mathematics and and between theory and applications has turned out to be very fruitful and has also influenced several active research groups in Austria. The present network joins these forces and unites 10 research projects that are guided by 18 researchers at the TU Wien, at the University of Vienna, at the TU Graz, at the University of Leoben, at the University of Linz and at the University of Salzburg. A major aspect of the network is to employe young researchers as Pre- and Post-Docs. They shall cooperate with several groups within the network and get into contact to international guests that are invited by the network. Conferences, regular research workshops and summer schools will be organized, too. The main scientific subjects covered by the NFN are Enumerative Combinatorics (partitions, hypergeometric series, bijections, tree structures), Digital Problems (numeration systems in cryptography, digital constructions of well- distributed point sequences and their applications, arithmetic dynamical systems, digital functions and fractal geometry), Dynamical systems (dynamical systems related to the distribution of number theoretical sequences, digital expansions, additive number theory and topological problems), Tilings and Fractals (Markov partitions, Rauzy fractal, linear numeration systems, combinatorial aspects of quasicrystals and rhombus Tilings) Discrepancy Theory (special sequences, low discrepancy sequences, metric and topological problems on lacunary sequences, Diophantine approximation), Asymptotic Probability (asymptotic distribution of specific parameters of combinatorial objects such as trees or tilings, functional limit theorems, laws of the iterated logarithm), and Probabilistic analysis of algorithms (for data structures such as trees, priority queues, skip lists etc, limiting distributions).