For a Lagrangian submanifold in a CY3, Ekholm and Shende defined a wavefunction living in the HOMFLY-PT skein module of the Lagrangian, which encodes open Gromov-Witten invariants in all genus. In this talk, we study a skein-valued cluster theory that generalizes quantum cluster theory and allows us to compute these wavefunctions in a range of examples. Our results agree with the physical prediction known as the topological vertex. Along the way we introduce a skein dilogarithm and prove a pentagon relation, generalizing previously known forms of the pentagon identity. This talk is based on joint works with Schrader, Zaslow, and Shende.

Constructing dynamical global-in-time solutions to 3D incompressible Euler equation is in general challenging. Recently Guo-Pausader-Widmayer made a breakthrough in this direction by proving asymptotic stability of a uniform-rotating static solution. Their approach heavily depends on the “uniform-rotating” of the background solution. In this talk I will present some recent progress on asymptotic linear stability of a more general columnar flow with non-uniform rotation. This is a joint work with Siqi Ren (Zhejiang U of Technology) and Zhifei Zhang (Peking U).