**Title:** Multifractal Analysis of the Schramm-Loewner Evolution

**Speaker:** Professor Gregory Lawler

**Speaker Info:** University of Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

The Schramm-Loewner evolution (SLE) is a conformally invariant process invented by Oded Schramm as a candidate for the scaling limit of planar lattice models in two-dimensional statistical physics. It gives a random fractal curve. To understand the evolution of the curve at time t, one needs to study the derivative near the tip of the path of the conformal map that sends the slit domain to the upper half plane. I will discuss the behavior of this derivative using the reverse-time Loewner flow and give results that can be proved from this analysis, e.g.,--- An alternative proof of Beffara's theorem on the Hausdorff dimension of the paths

--- A parametrization of the curve in a way that matches the fractal structure and should be the limit of the natural parametrization on discrete objects (joint work with Scott Sheffield)

--- The exact Holder continuity of the paths in the capacity parametrization and a multifractal spectrum relating to the local derivatives (joint work with Fredrik Johansson).

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