Dr. Omar Hurtado is awarded a National Science Foundation Mathematical and Physical Sciences Ascending Postdoctoral Research Fellowship (NSF MPS-Ascend) to conduct a program of research and broader impact activities. This fellowship supports the research project entitled "Random Spectral Theory: Anderson Localization and Polynomial Convolutions". The project activities will be conducted at the host institution, University of California, Berkeley, under the mentorship of Dr. Nikhil Srivastava. In the course of this project, he will principally study questions of a probabilistic and spectral nature in two areas: the spectral theory of random Schrodinger operators and the recently developed theory of finite free probability. In the former, he will build on existing work to prove new results demonstrating localization for random operators with very few constraints on the potential, making progress towards demonstrating "universality" of the Anderson localization phenomenon. In the latter, he will attempt to make progress towards proving quantitative versions of qualitative results in the burgeoning theory, in particular proving results analogous to large deviation estimates appearing in the usual "commutative" law of large numbers. In studying finite free probability, he plans to provide opportunities for undergraduate research; various numerical experiments will help inform conjectures as to what asymptotics can be expected, and these are suitable for undergraduate research