Renormalization for Circle Maps with Singularities and for Directed Polymers

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Renormalization theory was first developed in physics in the context of quantum field theory and statistical mechanics. At the end of 70's in the pioneering work of M. Feigenbaum the renormalization ideas paved their way into the theory of dynamical systems. By now renormalization is one of the most powerful methods in modern theoretical physics and mathematics. The main idea is to look at a system at large scales in a simplified way which takes into account only effective interaction of large blocks and ignores many fine details of a system at small scales. Using the idea of renormalization one can often establish universal scaling laws which describe asymptotic behaviour of many seemingly different systems. The large scale can correspond to either large spatial distances and volumes in statistical mechanics, or to large time in the context of dynamical systems. Remarkably, in many examples such universal behaviour can be studied rigorously. Roughly speaking, it can be described in terms of fixed points of renormalization. Simple fixed points correspond to standard scaling exponents as in the case of diffusion in probability theory. At the same time critical exponents for systems near phase transitions, or for maps with critical points correspond to highly non-trivial renormalization fixed points. Such critical systems are of great interest for renormalization theory. The proposed research project aims to study critical fixed points in two different settings. One of them deals with critical circle maps. The other one is related to the theory of directed polymers. These are very active research areas, and we hope to contribute significantly to the understanding of universal scaling properties in both cases.