Project Leader: Prof. Mariusz Lemańczyk
The project concerns innovative, interdisciplinary fundamental research that connects problems and methods from number theory, dynamical systems, and quantum physics: multiplicative number theory, the randomness of prime numbers, ergodic theory, parabolic and quantum dynamics. It focuses on some of the most pressing problems of modern science, investigated by leading researchers worldwide, including Fields Medalists. The project will be carried out in collaboration with international research centers such as the University of Zurich, the Institute for Advanced Study (Princeton), and the University of Maryland. The research team will be interdisciplinary, composed of faculty from both the Faculty of Mathematics and Computer Science and the Faculty of Physics, Astronomy, and Applied Computer Science.
Problems in number theory are often regarded as the very core of mathematics. Many famous questions concern the still-mysterious set of prime numbers. The properties of this set are expected to be encoded in the statistics of arithmetic functions that preserve the multiplicative structure of natural numbers. The most famous among them (such as the Möbius and Liouville functions) are conjectured to behave similarly to typical samples of independent processes. One way to express this idea is the Riemann Hypothesis; another is Chowla’s famous 1965 conjecture concerning the vanishing correlations of the Liouville function. In 2010, P. Sarnak proposed a new approach, claiming that classical multiplicative functions are uncorrelated with all deterministic observables. This perspective on the randomness of prime numbers gave new momentum to connecting analytic number theory with dynamics, via so-called nonconventional dynamics. In this framework, we seek to understand the dynamical (ergodic) properties of the so-called Furstenberg systems of multiplicative functions. In the past decade, many fundamental questions surrounding Sarnak’s conjecture have been answered, and the conjecture itself has been confirmed for several classes of deterministic systems. However, the full conjecture remains open, leading to new classification problems for multiplicative functions and raising fresh questions concerning nonconventional ergodic theorems and their applications in combinatorics.
Ergodic theory, from its very beginnings, aimed to explain phenomena arising in mathematical physics. In recent years, ergodic-theoretic methods have achieved remarkable success in studying translation dynamics on so-called flat surfaces—both finite and infinite, with periodic (crystalline) structures. Infinite periodic models originate from studies of conductivity in crystals and the geometry of Fermi surfaces. Recently, flat surfaces have also been used to investigate the propagation of light in flat, regular materials periodically “dotted” with impurities (irregularities) that scatter light beams. Even small but periodically distributed irregularities lead to a complete change in the dynamics of light propagation, where advanced methods of abstract ergodic theory play a central role. One of our primary research goals is to gain a deeper understanding of this phenomenon, particularly the distribution of directions in which light rays concentrate within such dotted materials as the light source is rotated, while simultaneously developing new methods in abstract ergodic theory and in number theory related to continued fraction expansions.
Fundamental and applied research on quantum technologies requires precise analysis of the mathematical properties of quantum operations, quantum states in many-body systems, and quantum dynamics, including the detrimental effects of the environment (such as quantum decoherence and/or dissipation). One of the main obstacles in the development of quantum computers, quantum communication, and quantum simulations is preventing the loss of quantum resources (such as quantum coherence and entanglement) in systems that are not perfectly isolated from their surroundings. Overcoming these issues requires a deeper theoretical understanding of quantum phenomena. This, in turn, calls for detailed analysis of quantum channels and their generalizations, known as quantum superchannels and supermaps—key objects in quantum information theory.