Spring 2018 Analysis Student Seminar

This semester, we will spend time defining and developing the basic properties of Brownian motion, and then move on to its connections to harmonic analysis (i.e solving the Dirichlet problem), harmonic measure, and develop some stocahstic calculus at the end time permitting.

References

[BP] "Fractals in Probability and Analysis", by Chris Bishop and Yuval Peres, our primary reference. Link
[GM] "Harmonic Measure", by John Garnett and Donald Marshall.
[La1] "Notes on Probability", by Greg Lawler. A quick intro to probability theory. The first three sections are recommended for those not familiar with probability, and we will draw more from it when needed. Link
[La2] "Stochastic Calculus: An Introduction with Applications", by Greg Lawler, for the last week or two. Link

Schedule

All talks will be given in 5-127 on Wednesdays at 4:00, and we'll try to end around 5:15. The following is the tentative schedule, along with the references we will follow that day.

Date	Speaker	Topic	Reading
Jan. 31	Jack Burkart	Introduction and Definition	[BP] 6.1-6.2
Feb. 7	Jack Burkart	Levy's Construction of Brownian Motion	[BP] 6.2
Feb. 14	Matt Dannenberg	Scaling Relations, Nowhere Differentiability, Holder Continuity	[BP]6.3
Feb. 21	Ben Sokolowsky	Reflection, Conformal Invariance, The Strong Markov Property	[BP] 6.6,7.9
Feb. 28	Silvia Ghinassi	Dimension Results	[BP]6.4, 7.1
Mar. 7	Snow Day	~~Zeros of Brownian Motion~~ Snow	[BP] 6.10
Mar. 14	Spring Break
Mar. 21	Snow Day	~~Law of the Iterated Logarithm~~ Snow	[BP] 7.2
Mar. 28	Snow Day	~~Connections to Harmonic Functions pt. 1~~ Snow	[BP]
Apr. 4	Matt Dannenberg	Law of Iterated Logarithm, Skorokhod's Representation, and Donkster's Invariance Principle	[BP] 7.2-7.3
Apr. 11	Ben Sokolowsky and Jack Burkart	Probabilistic View of Harmonic Functions	[BP] 7.5-7.7
Apr. 18	Mu Zhao	Conditional Probability and Martingales	[La1] Ch. 6
Apr. 25	Jack Burkart	Harmonic Measure and Kakutani's Theorem	[GM], [BP] 7.9
May 1	Matt Dannenberg	Intro to Stochastic Calculus	[La2]