Many problems in mathematics and its multifarious applications lead to strikingly similar - universal - questions pertaining to random structures. The geometry of random structures is often fractal. Such structures occur particularly in statistical and quantum field theory, with magnetisation and quantum gravity as examples.
The Centre of Excellence in Randomness and Structures investigates such phenomena. The Centre’s specific goal is to understand the analytical and geometric characteristics of random structures. As this research requires expertise in a number of mathematical fields, the Centre of Excellence will bring together a new generation of leading mathematicians to solve these problems.
Random structures also make an unexpected appearance in number theory, including the structure of the sequence of prime numbers. As the noted mathematician Paul Erdős stated: "God may not play dice with the universe, but something strange is going on with the prime numbers." Among other things, the Centre of Excellence explores the random nature of multiplicative functions and the Riemann zeta function.
The Centre will also develop new analytical and geometric methods aimed at understanding, for example, how macroscopic laws of nature follow from microscopic ones. Prospective applications of fundamental importance are in atmospheric (e.g. aerosol particle condensation) and lithosphere (e.g. water percolation in rocks) physics.
The Centre of Excellence also conducts research aimed directly at producing applications by developing high-dimensional statistics as well as randomised algorithms and their geometric understanding for the purposes of computational applications and machine learning.
The Centre of Excellence in Randomness and Structures is headed by Professor Eero Saksman. In addition to the University of Helsinki, the research groups comprising the Centre of Excellence are active at Aalto University, the University of Jyväskylä and the University of Turku.
In recent years, the probabilistic point of view has become increasingly important in a wide range of important problems in pure and applied quantitative sciences. A notable example of this is random geometry, which studies random geometrical objects that arise in classical and quantum statistical physics. Our teams have been at the forefront of mathematical developments in random geometry, which have made Finland a major international center in this new vibrant field. In FiRST, we will continue our groundbreaking work in analytic number theory on the probabilistic nature of arithmetic functions, and will bring therein the latest advances in geometric and harmonic analysis. Finally, homogenization and general multiscale stochastic analysis are ubiquitous tools in mathematical statistical physics, and their development is one of our primary interests.
An essential part of the CoE research is applications of cutting-edge mathematics in physical sciences as well as to statistics. Particular goals of the CoE include mathematical modelling of rock structures in view of applications to geothermal energy and improvement of the predictive accuracy of coagulation-fragmentation models in our pioneering work on "atmospheric mathematics", as well as development of stochastic algorithms applied in machine learning.