The investigations in this project are at the interface of geometry and physics.  In one direction, our knowledge of the geometric nature of the fundamental forces and particles of physics will be expanded, much as Einstein explained gravity as a manifestation of the curvature of space and time.  In another direction, physical ideas and insights will be applied to advance our knowledge about the geometry of certain curved spaces.  For the past thirty years, this general area of mathematical investigation has produced many important applications to other areas of mathematics, as well as sometimes surprising applications to physics, and to science and engineering more generally.  Doctoral students will be trained to assist in mathematical aspects of these investigations, adding highly skilled workers to the STEM workforce.Several problems in algebraic geometry inspired by string theory will be investigated, in addition to problems in string theory. One set of problems involves the study of three important algebro-geometric curve-counting invariants motivated by string theory: Gopakumar-Vafa (or BPS) invariants, Donaldson-Thomas invariants, and Pandharipande-Thomas invariants, as well as their motivic and refined extensions. Some of these theories are known to describe equivalent data, while others are only conjecturally equivalent.  New techniques and examples will be investigated to extend the reach of these theories.  Important goals are to establish some of the conjectures, at least in certain situations, and to put the definition of the refined BPS invariants of string theory on a more firm mathematical foundation by connecting the differing approaches. In another direction, the investigation of the small quantum product of a toric variety will be reopened with the goal of answering a question about its structure that was almost answered in the 1990s. Finally, the geometry-physics dictionary for F-theory and its dual descriptions in terms of M-theory and type II string theory will be investigated with the objectives of making the dictionary more precise and then deducing applications to both algebraic geometry and physics.