Cluster algebras were first introduced by Fomin and Zelevinsky as an algebraic framework to study total positivity and Lusztig's dual canonical bases in semisimple Lie groups. Later, cluster structures are discovered in many other areas in mathematics - Teichmüller theory, quiver representations, quantum field theory... In the first talk, we will briefly review the background of birational geometry and canonical bases of cluster algebras.
In the second talk, we will discuss the interplay of Weyl groups and cluster structures on families of positive log CY surfaces. On universal families, the action of Weyl groups gives us a simple, conceptual understanding of the Donaldson-Thomas transformation. Restricted to degenerate subfamilies, Weyl groups permute disjoint subfans in the cluster scattering diagrams and enable us to construct a large class of log CY varieties with non-equivalent atlases of cluster torus charts, generalizing the well-known case of the moduli of PGL2-local systems on 1-punctured
genus 1 Riemann surface.