Quantum Length of SLE: Natural Parameterization and Properties

This lecture discusses the quantum length of Schramm-Loewner Evolution (SLE) and its natural parameterization. The instructor begins by explaining the fundamental properties of SLE, particularly focusing on chordal SLE and its laws. The discussion includes the significance of conformal invariance and the domain Markov property, which characterize the family of laws associated with SLE. The instructor introduces the concept of natural parameterization, emphasizing its importance in defining the quantum length of SLE. The lecture also covers the relationship between SLE and random planar maps, illustrating how interfaces in critical models converge to SLE. The instructor highlights the challenges in proving conjectures related to SLE and its coupling with quantum gravity. The lecture concludes with a discussion on the construction of quantum lengths and their equivalence to existing definitions, providing insights into the mathematical framework underlying these concepts.