"This proposal will support the applicant as a research fellow at the Mathematical Institute, Oxford University, for two years. The proposal has two objectives.Objective 1 is a joint project with the proposed host scientist, Professor Dominic Joyce. It studies the moduli stacks of coherent sheaves on a Calabi-Yau (C-Y) 3-fold. C-Y 3-folds are a major focus of research in Geometry and String Theory in Mathematical Physics. Donaldson-Thomas (D-T) invariants ""count"" coherent sheaves on C-Y 3-folds, and are interpreted in physics as numbers of ""B-branes"" or ""BPS states"". The host scientist Professor Joyce is well-known for his work on D-T theory. Objective 1 studies foundational properties of moduli stacks of coherent sheaves on C-Y 3-folds, which must be understood to extend D-T theory in several directions: ""categorified"" D-T invariants, extensions to fields other than the complex numbers, and D-T theory for the derived category. We will explore when moduli schemes (and atlases for moduli stacks) can be written as critical loci of holomorphic functions, find an algebraic substitute for this which works over other fields, and apply to Behrend functions and properties of D-T invariants.Objective 2 is a joint project with Oxford faculty member Dr Kobi Kremnizer. It aims to categorify the Verlinde algebra in a novel way using categories of twisted equivariant coherent sheaves on algebraic groups, and to use these to construct a new family of 0-1-2-3 Topological Quantum Field Theories."