CAREER: Analysis of the Geometric Properties of the SLE Curves

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The Schramm-Loewner evolution (SLE) introduced by Oded Schramm in 1999 has been used with great success to prove many outstanding conjectures and predictions from Statistical Physics. The PI has been working on SLE for a number of years, and has made some important contributions to this area. The proposed project focuses on studying the geometric properties of the SLE curves of various kinds. To do this project, the PI will use the stochastic coupling technique he developed earlier. This technique is used to construct more than one SLE curve in single domain in such a way that the curves commute with each other. It was applied to prove the Duplantier's duality conjecture, the reversibility of chordal SLE and whole-plane SLE (for some parameters), and that it is possible to erase loops on a planar Brownian motion to get an SLE curve. In the proposed project, the PI will continue the application of the coupling technique. This includes constructing simple SLE loops on the Riemann sphere, proving the reversibility of the natural parametrization of SLE, and studying the behavior of SLE in multiply connected domains. The proposed research plan will give people better understanding of the behavior of the SLE curves growing in simply and multiply connected domains. This will then increase understanding of a number of two-dimensional Statistical Physics lattice models (e.g. critical percolation, critical Ising model, Gaussian free field, and loop-erased random walk) in different kinds of domains since SLE have been identified as the scaling limits of these models. The project will also have significant impacts on Conformal Field Theory. A significant part of this proposal is its educational component. The PI will develop undergraduate courses on Probability and Complex Analysis at Michigan State University to include the newest development of the PI's research area. He will also teach specialized graduate courses on SLE, and involve graduate students in working under his supervision.