# Math 252A: Topics in Complex Analysis, Fall 2016

**Time and place: ** MWF, 12pm-12:50pm, MS 5148.

**Instructor: ** Mario Bonk

**Office: ** MS 6137

**Office hours: ** Mo 1pm-2pm, We 11am-12am, Th 1pm-2pm.

**E-mail: ** mbonk at math.ucla.edu

**Phone: ** (310) 825-4948

**Course material:** This course is an introduction to the Stochastic Loewner Equation (SLE). The course will start with a review of some more advanced topics relevant for a treatment of the Loewner equation such as boundary behavior of conformal maps, Caratheodory kernel convergence, and Vitali's convergence theorem. Existence and uniqueness results for the classical
Loewner-Kufarev equation will be thoroughly discussed and applications will be given.
Then we will move to the probabilistic SLE-setting, where the driving term of the Loewner equation is a (scaled) Brownian motion. We will establish some of the properties of SLE (such as phase transition and Hoelder continuity of the associated conformal maps) and discuss further directions.

**Prerequisites:** Required is a solid foundation in classical complex analysis (including topics such as the Riemann mapping theorem and Koebe's distortion theorem) and
probability (martingales and Brownian motion). A knowledge of stochastic differential equations is not necessary, but the relevant facts will be discussed. Students who want to participate in this course are supposed to familiarize themselves with course notes from a previous course with a similar topic. They are available here:
Notes 1. The material in these notes will be partially
reviewed, but the proofs be mostly skipped.

**Course Grades:** Course grades are based on regular attendance and active participation in the course. Homework may be assigned occasionally.

**Literature:**
G.F. Lawler, Conformally Invariant Processes in the Plane, Amer. Math. Soc., Providence, RI, 2005.

N. Berestycki and J.R. Norris, Notes on the Schramm-Loewner,
available at:
Notes 2.

M. Zinsmeister, Stochastic Loewner Equation, available at:
Notes 3.