Applications of the Geometric Function Theory to Mathematical Physics

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Geometric Function Theory is a classical area of Analysis, initiated by Riemann's proof of his famous mapping theorem. One of the two first Fields medals was awarded in 1936 to L. Ahlfors for his research in the field. In 1985, one of the main conjectures in the Theory, the famous Bieberbach coefficients conjecture, which remained open since 1916, was successfully verified by L. de Branges. One of the main technical tools used in the proof was the so-called Loewner Differential Equation. The latter is a tool invented by C. Loewner in 1923 specifically for the purpose of solving the Bieberbach conjecture. The Loewner Evolution found an unexpected application outside of Geometric Function Theory, in the study of conformally invariant Universality Phenomenon of Statistical Physics. The phenomenon was first observed in the late sixties in the works of L. Kadanoff and K. Wilson (who later received a Nobel prize for his research), and further developed in the eighties by A. Polyakov and his collaborators.Intuitively, the principle states that the long-range behavior of the various lattice models depends only on the "universality class" and is locally invariant under conformal maps. The rigorous mathematical understanding of the phenomenon remained out of reach until a Stochastic version of Loewner Equation, the Schramm Loewner Evolution (SLE), was invented by O. Schramm in 1998. The family was conjectured to be a scaling limit of interfaces of lattice models in different universality classes. G. Lawler, O. Schramm, and W. Werner (who received the Fields medal in 2006 for this work) verified the conjecture for the Loop Erased Random Walk. S. Smirnov, who also received the Fields medal (in 2010), verified the conjecture for the Critical Hexagonal Percolation and Ising model. In the works of O. Schramm and S. Sheffield, the conjecture was established for the level lines of the Gaussian Free Fields. Yet for most of the lattice models, the conjecture is still open. One of the aims of our project is to advance understanding of the conjecture, as well as to establish the relevant rate of convergence. The fine geometric properties of the SLE curves themselves can be formulated in terms of the multifractal spectrum. The values of the spectrum were predicted in the physics literature, in particular, in my previous work. We plan to rigorously establish the values. Another important physical phenomenon is Anderson localization. Informally, it states that, under certain conditions, disorder can suppress the waves. While the physical theory of Anderson localization was one of the main achievements of the physics of condensed matter in the last century, its mathematical counterpart is still under active development. I intend to study two of the mathematical models of the disorder: Jacobi operators with quasiperiodic and random potentials. In the first case, I will mainly concentrate on the Inverse Spectral Problems, the understanding of the properties of a potential from its spectrum. I plan to use of estimates on harmonic measure, one of the classical tools of the Geometric Function Theory. In the random setting, I will work on improving the quantitative estimates on localization. A number of unanswered questions in Geometric Function Theory itself can be stated in terms of the optimal bounds on the multifractal spectrum of planar domains. One of them is the celebrated Brennan's conjecture and its generalizations. I plan to tackle these questions using recently developed methods. I also intend to acquire deeper understanding of the computability properties of Conformal maps and their boundary values.