Loewner chains provide a way to encode certain curves in a planar domain

by real-valued "driving" functions. Originally a purely complex analytic

tool to study conformal maps, it has turned out to be very useful in

defining SLE, a "uniformly random" curve in a domain. Not any curve can

be described as a Loewner chain, but the ones that can (we call them

traces) may look very wild and even be space-filling. Intuitively,

traces are characterised by the property that whenever they

self-intersect, they need to "bounce off" instead of "crossing over". In

this talk, I will present three equivalent ways of describing this

property precisely. If time allows, I will present another result about

the dependence of SLE on a parameter \$\kappa\$ that describes its level of

fluctuations.

# Topological characterisations of Loewner traces

Yizheng Yuan, Berlin

PDL C-401