Class meets every Friday 10:10-11:50 AM in KAP 245

Organizer: Sergey Lototsky.
Office: KAP 248D.
Phone: (213) 740-2389.

Office Hours: MW 10:30am-noon.
Walk-ins and appointments at other time are welcome.

The objective this semester: To discuss current research projects and future plans of participating students.

Student participants this semester: Diogo Bessam, Ibrahim Ekren, Umit Islak, Christian Keller, Haining Ren.

What we did

  • August 30: organizational meeting, setting plans for the semester. A short introduction to concentration inequalities by Umit.
  • September 6: Diogo’s presentation on Gaussian measures in Banach spaces.
  • September 13: Haining’s presentation on unimodal permutations.
  • September 20: Ibrahim’s presentation on path-dependent partial differential equations (current project) and math finance group at ETH Zurich (future plans).
  • September 27: Diogo’s presentation on the probability research group at the Warwick University. Umit’s presentation on size-biased coupling and concentration of measures.
  • October 4: Christian’s presentation on his research, with emphasis on various topologies in the space of CADLAG processes.
  • October 11: Umit’s presentations on the probability research groups at Oxford and Stanford, and on connections between size-biasing and concentration of measures.
  • October 18: Practice oral exam presentations by Christian and Ibrahim.
  • October 25: Another presentation by Haining on unimodal permutations.
  • November 1: With only Diogo, Ibrahim, and Unmit present, we duscussed general ideas and potential research problems related to various explored and unexplored connections between large deviations, chaos expansion, optimal control, and concentration of measures. There was also a short presentation by Umit about the probability group at the University of Washnigton.
  • November 8: An improptu lecture by Umit about negative association, followed by a presentation by Haining on asymptotic combinatorics.
  • November 15: Ibrahim lead a discussion of a paper by S. Chatterjee about probabilistic representation of the solution of the wave equation.
  • November 22: With only Diogo and Haining present, we got stuck on a very basic question: can it happen that a sum of iid random variables X1+…+Xn has a pdf even though the summands do not? More precisely, we looked at the case when Xk is a Cantor random variable (with cdf given by the Cantor ladder). The characteristic function of Xk is known explicitly, and then we will have the answer in the positive if we can show that some power of the characteristic function is integrable.
  • December 6: This time, Digo, Ibrahim, and Christian were present. We made some progress in our understanding of characteristic functions of Cantor-type distributions, but the underlying question (is the sum of two iid Cantor-midle-third random variables absolutely continuous, and if it is not, how many iid copies should we take?) remains open.