Geometric Function Theory and Mathematical Physics

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In recent years, the classical methods of the Geometric Function Theory found numerous applications in the various areas of Mathematical Physics. The proposed research will concentrate on two of these areas. The first one is the Schramm Loewner Evolution (SLE) and its relation to the critical lattice models of Statistical Physics. The proposed research will lead to a better understanding of the fine geometric properties of the SLE curves. One of our aims is to rigorously confirm empirical predictions on the integral mixed spectrum, a major tool that describes the precise behaviour of the harmonic measure and boundary rotation. Other problems in this area are the conformal welding for SLE curves and the Conformal Dimension of the SLE curves, the smallest Hausdorff dimension of their quasiconformal images. SLE was initially introduced as a conjectural scaling limit of certain interfaces for the lattice models of Statistical Physics. In a few situations, such as Percolation, Ising model, Gaussian Free Field, and Loop-Erased Random Walk, this conjecture has been established. I intend to develop a general approach to the rate of convergence results for a wide class of lattice models. This would imply, in particular, that the convergence for the critical Percolation and the Ising model is polynomial in the size of the lattice. I will also look for new observables to establish convergence for other Lattice Models. This study can be viewed as a qualitative theory of critical phenomenon. The second area I will work on is the spectral theory of quasi-periodic Schroedinger and Jacobi operators. I will investigate the Inverse Spectral Problem, one of the central questions of this extremely active field of Mathematical Physics. An important ingredient of our proposed research in this direction is the study of the fine properties of the harmonic measure for planar comb domains. These methods can also be used in the applications of the spectral theory to the KdV equation. Recently, D. Damanik, M. Goldstein, M. Lukic and I were able to verify a particular case of Deift's conjecture on the existence, uniqueness and almost periodicity in time of the solutions of KdV with almost periodic initial conditions. I plan to study the further interplay between quasi-periodicity and integrability. Applications of Geometric Function Theory frequently involve numerical simulations. Understanding the computational complexity of the related problems is thus essential. I will investigate the computability properties of the Green functions and Green maps as well as the Koebe maps and conformal weldings. I also plan to continue working on the fundamental theoretical questions of the Geometric Function Theory, such as Brennan's conjecture and related topics.