Wednesday September 12, 2018 4:00 PM Math Tower 5127
 Jack Burkart, Stony Brook University
Intro. Distortion Theorems for Conformal MapsBefore we get to SLE, we will introduce ideas and motivation behind classical Loewner theory. In particular, we will discuss its relationship to the Bieberbach conjecture.
We will also prove classical distortion theorems for conformal maps that we will need in the upcoming weeks, including the Koebe 1/4 theorem and the Koebe distortion theorem.

Wednesday September 19, 2018 4:00 PM Math Tower 5127
 Matthew Dannenberg, Stony Brook University
Caratheodory Convergence, Herglotz Representation, and Hurwitz's TheoremWe will give a characterization of convergence of conformal mappings in terms of convegence of image domains (for some appropriate notion of convergence).
To conclude our prerequisites, we will prove are Hurwitz's theorem that says a sequence of conformal maps converging locally uniformally is conformal or constant, and the Herglotz Representation formula for harmonic functions.

Wednesday September 26, 2018 4:00 PM Math Tower 5127
 Ying Hong Tham, Stony Brook University
Loewner Chains and the Loewner PDEWe will define radial Loewner chains, and describe their various convergence and distortion properties using the tools from the previous weeks.
We will also derive the Loewner PDE, and as an application, prove the Bieberbach conjecture for the case n=3.

Wednesday October 03, 2018 4:00 PM Math Tower 5127
 Tim Alland, Stony Brook University
The Loewner Differential Equation

Wednesday October 10, 2018 4:00 PM Math Tower 5127
 Jack Burkart, Stony Brook University
Slit Domains and Chordal Loewner Theory

Wednesday October 17, 2018 4:00 PM Math Tower 5127
 Silvia Ghinassi, Stony Brook University
Intro to ProbabilityWe will quickly discuss and compare the vocabulary of probability theory with that of measure theory. Then we will introduce independence and conditional expectation.

Wednesday October 24, 2018 4:00 PM Math Tower 5127
 Jack Burkart, Stony Brook University
Optional stopping time and examplesLast week after a quick review of probability we introduced conditional expectations, martingales and stopping times. We will talk about optional stopping times and complete the picture with examples.

Wednesday October 31, 2018 4:00 PM Math Tower 5127
 Matthew Dannenberg, Stony Brook University
Brownian MotionIn this talk we will define Brownian motion carefully, discuss its construction, and discuss many of its useful properties, including the Markov property and its scaling limit properties.

Wednesday November 14, 2018 4:00 PM Math Tower 5127
 Jack Burkart, Stony Brook University
Stochastic Calculus Pt. 2We'll discuss the Stochastic integral as a continuous time martingale.

Wednesday November 28, 2018 4:00 PM Math Tower 5127
 Jessica Maghakian, Stony Brook University
Ito's Formula and Stochastic Differential EquationsWe state and prove Ito's formula and define Stochastic differential equations and state an existence/uniqueness result for them.

Wednesday December 05, 2018 4:00 PM Math Tower 5127
 Silvia Ghinassi, Stony Brook University
SLE: Definition and Basic PropertiesWe will define SLE two ways and discuss some of the basic properties it satisfies.

Wednesday December 12, 2018 4:00 PM Math Tower 5127
 Jae Ho Cho, Stony Brook University
SLE: Transition from simple curves to nonsimple curvesTo finish up the semester, we discuss the proof that SLE$(\kappa)$ transitions from being a simple curve to a nonsimple curve when $\kappa = 4$.

