Deformation invariance of refined Donaldson-Thomas invariants

The modern theory of Hall algebras provides a way to associate an abstract algebraic structure to a given geometry. This method opens up two overlapping areas of research. On the one hand, it is possible to produce geometries for the purpose of presenting preexisting algebraic structures as Hall algebras. Such geometric interpretations of algebraic rules have turned out to lead to fundamental advances, most famously in the theory of quantum groups,  which remain out of the reach of purely algebraic methods. 

On the other hand, it is possible to apply the method to preexisting geometries. In this approach, there is a balance to strike between richness and tractability, a task informed by ideas from theoretical physics as well as classical enumerative geometry. One example is the case of a K3 surface, which are by now classical  objects sitting at the crossroads of almost every branch of mathematics. I will study these K3 Hall surfaces for two reasons. Firstly, a K3 surface is a purely geometric object and it is interesting to see what kind of algebraic structure should arise from them. Secondly, it is possible to deform them — there is an intricate, 19-dimensional space of all K3 surfaces — and their Hall algebras should deform accordingly. This phenomenon is germane to the theory of Hall algebras, as the rule is generally that such constructs should be invariant under deformation, and has not yet been investigated.