# Lukas Schoug: Regularity of the SLE_4 uniformizing map and the SLE_8 trace

**Time: **
Thu 2021-09-16 15.15 - 16.15

**Location: **
3418, Lindstedtsvägen 25, KTH Matematik

**Lecturer: **
Lukas Schoug (Cambridge)

The Schramm–Loewner evolution (SLE) is a one-parameter family of random planar fractal curves, which has been of considerable interest since their introduction by Schramm in 1999, as they arise as scaling limits in several two-dimensional statistical mechanics models at criticality. Choosing the parameter \(\kappa\) to be either 4 or 8 results in special behaviour, as \(\kappa = 4\; (\kappa = 8)\) is the largest (resp. smallest) \(\kappa\) such that \(\mathrm{SLE}_\kappa\) curves are simple (resp. space-filling). As such, regularity results in those cases differ significantly from the cases of other values of \(\kappa\). We will discuss recent results on the modulus of continuity of the \(\mathrm{SLE}_4\) uniformizing map and the \(\mathrm{SLE}_8\) trace, as well as a byproduct of our analysis, concerning the conformal removability of \(\mathrm{SLE}_4\). The talk is based on joint work with Konstantinos Kavvadias and Jason Miller.