Abstract: Donaldson-Thomas theory associates integers (which are virtual counts of sheaves) to a Calabi-Yau threefold X. The simplest example is that of C^3, when the Donaldson-Thomas (DT) invariant of length d sheaves is p(d), the number of plane partitions of d. 

The DT invariants have several refinements, for example a cohomological one, where instead of a DT invariant, one studies a graded vector space with Euler characteristic equal to the DT invariant. 

I will talk about categorical and K-theoretic refinements of DT invariants and of related enumerative invariants, such as Pandharipande-Thomas (PT) and BPS invariants. 

In the first talk, I will focus on the explicit case of C^3. In particular, we show that the K-theoretic DT invariant for d points on C^3 also equals p(d). A central construction here is that of quasi-BPS categories, which is the categorical version of BPS invariants. 

In the second talk, I will talk about a categorical version of the DT/ PT (Pandharipande-Thomas) correspondence for local surfaces. Time permitting, I will also discuss the construction of quasi-BPS categories for quivers with potential and for K3 surfaces.  

The talks are based on joint work with Yukinobu Toda.