The PI proposes to continue his work on stacks and their applications to arithmetic geometry, algebraic geometry, and noncommutative algebra. One project involves studying the connection between the Hasse principle for geometrically rational varieties over global fields and the period-index problem for Brauer groups of function fields. This is analogous to the connection between Artin's conjecture on the finiteness of the Brauer group for schemes proper over the integers and the Tate-Shafarevich conjecture, and builds on earlier work by the PI on the moduli of twisted sheaves. A second project will continue the joint work carried out by the PI and KovÃ¡cs on higher-dimensional generalizations of the Shafarevich conjecture on non-isotrivial families of curves. Crucial to this project will be a greater understanding of stacks parametrizing morphisms between stacks, whose systematic study was only recently begun. A third project aims to deepen the understanding of stacks and their intrinsic geometry. The PI and his collaborators will study the nature of birational modifications and analytification of stacks, and the categorical information content of stacks. The ultimate goal of the proposed research is to broaden the applications of algebro-geometric and stack-theoretic methods in pure algebra and related fields. Broadly speaking, algebraic geometry is the study of the geometry associated to algebraic objects. One of the most fruitful historical examples is the equation for a circle; the quadratic nature of the equation is closely related to the fact that a line generally intersects a circle in two points. Over the last several millennia, mathematicians have come to understand that the connections between algebra and geometry run far deeper than one might imagine, and this has led to the gradual encroachment of geometric methods into far-flung areas of pure algebra and a profound unification of several seemingly-different subjects. Each new advance in algebraic geometry ultimately finds applications to other areas of mathematics; several abstract areas of the field have turned out to be very computer-friendly, and modern cryptography would be impossible without algebraic geometry. The theory of stacks is relatively young, but it has been the focus of recent attention and is rapidly maturing. The PI's research will be directed toward applying the theory of stacks to a wide range of problems in algebra and geometry, bringing new tools to the geometric analysis of algebraic problems.