Donaldson–Thomas invariants of threefold flops

This thesis is about a class of complex algebraic threefolds known as flops, which are an important part of the Minimal Model Program in birational geometry. Threefold flops are commonly studied via their enumerative invariants, and here we focus on one such type of invariant: refined Donaldson–Thomas invariants. We develop theoretical aspects of refined Donaldson–Thomas theory for threefold flops, which allow us to understand their stability conditions and cyclic A∞-deformation theory. With these new methods, we are able to sidestep common computational barriers in the field and fully determine the Donaldson–Thomas invariants for an infinite family of flops, which includes many new examples. Our results show that a refined version of the strong-rationality conjecture of Pandharipande–Thomas holds in this setting, and also that refined Donaldson–Thomas invariants are not sufficiently fine to determine flops. Where possible we work motivically, computing invariants in the Grothendieck ring of varieties, but we also produce Hodge theoretic realisations of the invariants