Enumerative geometry and integrable hierarchy

Enumerative geometry typically involves counting solutions to geometric problems, often focusing on the enumeration of curves on a given manifold. An integrable hierarchy is a sequence of exactly solvable mathematical equations that describe complex physical phenomena, such as solitons, arising from nonlinear physics. The deep connection between enumerative geometry and integrable systems dates back to Witten's celebrated conjecture. In this talk, we aim to explore the relationships between these two areas and establish a generalization of Witten's conjecture by considering the underlying decorated Riemann surface. This talk is partially based on joint works with Ce Ji, Chenglang Yang, and Qingsheng Zhang.