Quantum cluster transformations and refined DT theory

In this talk I'll describe what a quantum cluster algebra is, and how quantum cluster coefficients arise in the Donaldson-Thomas theory of a quiver with potential.  In particular, we'll be focusing on the link between purity statements on cohomology of vanishing cycles, and positivity in quantum cluster mutation.  In the case of a graded quiver with nondegenerate graded potential, it turns out that we can prove the general positivity conjecture, and even the stronger "Lefschetz property" considered by A. Efimov, by proving the corresponding purity statement on critical cohomology.  This establishes the positivity conjecture in a wide range of examples.