Title: Categorical Donaldson-Thomas theory of C^{3}

Abstract: I will explain some results in categorical Donaldson-Thomas theory of C^{3}. An important construction is that of quasi-BPS categories, which are a categorical approximation of BPS (also called Gopakumar-Vafa) invariants in enumerative geometry. Quasi-BPS categories are categories of matrix factorizations on certain (twisted) noncommutative resolutions of singularities constructed by Špenko-Van den Bergh. We expect quasi-BPS categories to be indecomposable.

We construct semiorthogonal decompositions of the derived category of coherent sheaves on the stack of commuting matrices (alternatively, the Hall algebra of C^{2}) and of a categorification of Donaldson-Thomas invariants of C^{3} using quasi-BPS categories. We compute (versions of) the K-theory of quasi-BPS categories. Along the way, we propose a K-theoretic version for C^{3} of the Bridgeland-King-Reid and Haiman theorem that the Hilbert scheme of d points on C^{2} is derived equivalent to the orbifold C^{2d}/S_{d}.

This is joint work with Yukinobu Toda.