Geometry, supersymmetry and flux backgrounds in string theory

String theory is a putative grand unified theory of quantum gravity where the fundamental objects are loops rather than point particles. In addition to the metric of conventional Einstein gravity it includes a number of additional excitations known as "fluxes". Understanding the role these degrees of freedom play in string theory backgrounds is central both to understanding how low-energy particle physics may emerge from string theory and investigating the remarkable duality between gravitational and non-gravitational theories known as the AdS-CFT correspondence, specifically when there is preserved supersymmetry. Recent work has shown that supersymmetric backgrounds can be understood in terms of a new class of geometrical structures that generalised conventional notions of symplectic or complex geometry. The project will study these new structures and in particular how they capture aspects of the AdS-CFT correspondence. Of particular interest will be constructing new solutions corresponding to wrapped M5-branes and general D3-brane backgrounds. Key points will be to understand the moduli spaces of these structures, their relations to topological theories and BV algebras, and connections to a-maximisation. There are also important connections to differential and algebraic geometry.