Topological characterisations of Loewner traces

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.