|DATE||August 04 (Wed), 2021|
|INSTITUTE||University of Cambridge|
|TITLE||Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace|
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 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 SLE$_4$ uniformizing map and the SLE$_8$ trace, as well as a byproduct of our analysis, concerning the conformal removability of SLE$_4$. The talk is based on joint work with Konstantinos Kavvadias and Jason Miller.