Ben Davison (IST Austria), Positivity for quantum cluster algebras
Cluster algebras are subalgebras of the field of rational functions in n variables, whose generators are given by recursive application of cluster mutation to the standard monomials. Quantum cluster algebras are a certain very natural quantum deformation of these algebras. Although the definition of cluster mutation is entirely elementary, the question of proving the positivity of the coefficients that appear after repeated cluster mutation has remained open. I'll explain a proof, via a proof of a conjecture of Kontsevich and Efimov regarding the Hodge theory of moduli spaces appearing in cohomological Donaldson-Thomas theory.