Applications of nonclassical geometry to string theory

String theory is built on a foundation of geometry. This thesis examines several applications of geometry beyond the classical Riemannian geometry of curved surfaces.; The first part considers the use of extended spaces with internal dimensions to each point (“twistors”) to probe systems with a great deal of symmetry but complicated dynamics. These systems are of critical interest in understanding holographic phenomena in string theory and the origins of entropy. We develop a twistor formulation of coset spaces and use this to write simplified actions for particles and strings on anti-de Sitter space, which are easier to quantize than the ordinary (highly nonlinear) actions.; In the second part, we consider two aspects of noncommutative geometry, a generalization of ordinary geometry where points are “fuzzed out” and functions of space become noncommuting operators. We first examine strings with one endpoint on a D-brane in a background magnetic field. (Strings with both endpoints on such a brane are known to behave as though in a noncommutative space) We study their scattering properties and interactions, and show that unlike their double-ended noncommutative counterparts, they have the same ultraviolet divergence properties as commutative strings.; Finally, we examine the bound states of D0-branes in type IIA string theory, generalizing the known membrane bound states formed by strings stretching between pointlike branes. We discover novel states which are extended objects exhibiting a highly noncommutative geometry, and make a preliminary study of their dynamics. Significantly, these states may carry charges which are only conserved modulo a constant, so unlike ordinary brane states, a finite set of stable objects may combine to anihilate.