The principal investigator will study conformal mappings generated by the Loewner differential equation, and random conformal and quasiconformal mappings. The Loewner equation relates a continuously increasing sequence of planar simply connected domains to a real-valued function, the driving term of the equation. The correspondence is by means of conformal maps onto a standard domain (such as a disc). Through this mechanism complicated two-dimensional shapes can be encoded by seemingly simpler objects, namely, real-valued functions of a real variable. The correspondence between a shape and its driving term is complicated and leaves many open questions. The aim of this project is to provide a better understanding of this correspondence. For instance, the principal investigator will study the continuity of the sets under deformations of the driving terms. In light of Oded Schramm's SLE and the spectacular work of Lawler, Schramm, Werner, Smirnov and others, random driving terms (in particular, Brownian motion) are especially interesting and will be a focus of the research.
Conformal mappings are often used to change coordinates from one region to a simpler region, such as a disc. They have applications in many areas within mathematics and to several branches of physics. On small scale, conformal maps look like rotations and dilations. Hence it is plausible that rotation- and dilation-invariant mathematical models of physical phenomena (e.g., Brownian motion, percolation, crystal growth, electrodeposition) are invariant under conformal coordinate changes. Theoretical physicists have long used this heuristic and obtained predictions for many of these models. Oded Schramm's discovery of the stochastic Loewner evolution (SLE, the Loewner equation driven by one-dimensional Brownian motion) and Smirnov's work on percolation have put this philosophy on a firm mathematical basis. The results obtained in recent years have generated a lot of excitement in both the mathematics and the physics communities. They have also created a new bridge between the two disciplines. A goal of this research is to shed new light on the mathematical side of this emerging theory.
Under this grant, the PI investigated properties of the stochastic and the deterministic Loewner Equation, and of random conformal and quasiconformal maps. Since its invention by Oded Schramm in the year 2000, SLE has become a very powerful tool to describe scaling limits of numerous statistical physics lattice models (such as percolation or the Ising model). Despite its growing importance in probability theory and statistical physics, our understanding of even the most basic properties is still very limited. The intellectual merit of the PI's research, and his joined work with his students and collaborators, consists of answers to fundamental questions concerning SLE. For instance, continuity of the traces in general simply connected domains has been shown, and smoothness of traces for smooth driving terms has been established. On the other hand, it has been shown that the traces do not neccessarily change continuous in the determinstic setting, whereas they do vary continuously in the stochastic setting. Another result obtained under this grant is the parabolicity of the uniform infinite planar triangulation, a combinatorial object related to the study of quantum gravity. The broader impact lies in the training and supervision of graduate students and postdoctoral scholars, the co-organization of numerous conferences and summer schools, and the dissemination of ideas and results through research papers and overview articles.