Randomness and Determinism in Analysis and Number Theory

This project is concerned with fundamental research in the mathematical fields of number theory, analysis, and probability theory. Among the problems studied in the project are the following: How well can real numbers be approximated by rational numbers (that is, by fractions). How well can that be done for certain specific real numbers, and how well can it be done for typical real numbers? Which properties does the decimal expansion of real numbers have, which are also typically observed for purely random sequences? What special role do simple irrational numbers such as the Golden Mean or the square-root of 2 play in this context? This research project has connections with theoretical computer science (notions of pseudo-randomness, generation of pseudo-random numbers), numerical mathematics (low-discrepancy sampling, Quasi-Monte Carlo method), and theoretical physics (quantum field theory).

This project is concerned with fundamental research in the mathematical fields of number theory, analysis, and probability theory. Among the problems studied in the project are the following: How well can real numbers be approximated by rational numbers (that is, by fractions). How well can that be done for certain specific real numbers, and how well can it be done for "typical" real numbers? Which properties does the decimal expansion of real numbers have, which are also typically observed for purely random sequences? What special role do simple irrational numbers such as the Golden Mean or the square-root of 2 play in this context? This research project has connections with theoretical computer science (notions of pseudo-randomness, generation of pseudo-random numbers), numerical mathematics (low-discrepancy sampling, Quasi-Monte Carlo method), and theoretical physics (quantum field theory).