This award is made as part of the FY 2014 Mathematical Sciences Postdoctoral Research Fellowships Program. Each of the fellowships supports a research and training project at a host institution in the mathematical sciences, including applications to other disciplines, under the mentorship of a sponsoring scientist. The title of the project for this fellowship to Ilya Gekhtman is "Dynamics on Moduli Space, Patterson-Sullivan Theory, and Random Walks on Groups." The host institution for the fellowship is Yale University, and the sponsoring scientist is Dr. Gregory Margulis.