This award is made as part of the FY 2017 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 Lien-Yung Kao is " Thermodynamic formalism and its applications to deformation spaces." The host institution for the fellowship is University of Chicago, and the sponsoring scientist is Amie Wilkinson.