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Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory: February 3 – June 5, 2020

Organizers: Kostya Khanin and Misha Lyubich

The goal of the program is to bring together mathematicians and physicists working on various aspects of renormalization in dynamical systems. The idea of Renormalization group emerged in Quantum Field Theory. Later, in the 1960s, it became a major tool in Statistical Mechanics in analysis of phase transitions and critical phenomenon. One can say that the ideas of renormalization group have revolutionized the field. This development culminated in Wilson’s expansion based on his ideas on intrinsic relation between physical parameters in different scales.

In the 1970s the renormalization ideology was transferred to Dynamics in the context of Universality discoveries by Feigenbaum, Coullet and Tresser, and has since become one of the most powerful tools of understanding small scale structure of a large variety of systems. It has become particularly well (and rigorously) developed in the Conformal context, in particular, in the geometric problems related to the celebrated MLC Conjecture on the local connectivity of the Mandelbrot set.

Today, the renormalization ideas have penetrated deeply into many areas of Mathematics and Physics, but an explicit relation between various areas often remains elusive. One of our goals is to look for a unifying approach that would cover various manifestations of the renormalization.

In the first part of the program we will focus on conformal aspects that would include small scale structure of the dynamical and parameter loci of conformal dynamical systems, small scale properties of Brownian motions and harmonic measures, SLE, random conformal welding, and other relevant systems. We shall also discuss other aspects of dynamical renormalization, including related area of renormalization for quasi-periodic Schr ̈odinger operators.

In the second part, we will explore renormalization in various physical situations: QFT, fluid dynamics, and statistical mechanics, including KPZ phenomenon, with the emphasis on the underlying stochastic mechanisms responsible for the statistical
scaling invariance.

There are two workshops associated with this program: Analysis, Dynamics, Geometry and Probability: March 2-6, 2020 and Many Faces of Renormalization: June 1-5, 2020