Gaultier Lambert: What are Schramm Loewner Evolutions?

In 2000, Oded Schramm introduced a family of curves which grow at random in a simply connected domain of the complex plane whose law is conformally invariant and satisfies the so-called Markov property. He was interested in finding the scaling limits of models which arise in two dimensional statistical physics such as Uniform Spanning Trees, LERW or critical percolation. Since then, the theory of Schramm Loewner Evolutions has been a very active area of research with many applications, the most celebrated being the proof of a conjecture of Mandelbrot that the boundary of Brownian motion has dimension 4/3. 

In this talk, I will present the concept of Loewner growth that Schramm used to defined SLEs, I will explain the definition and basic properties of SLEs in the upper-half plane. Then I will speak about conformal invariance and give a proof of Cardy's formula for SLE_6 in triangles. Finally, if time permits, I will show some pictures and discuss the scaling limit of the percolation exploration path to SLE_6.

I will assume no prerequisite, except that the audience is familiar with the concept of Brownian motion and basic complex analysis such as the Riemann Mapping Theorem.