Project 1
Proving wall-crossing for Calabi–Yau fourfolds
In this vast project, I have chosen to tackle the question of wall-crossing formulae for Calabi-Yau fourfolds, which were originally conjectured by Gross-Joyce-Tanaka. This work had two focal points: providing insight into the wall-crossing framework and being practical in applications. My first work on this subject focuses largely on the first point. Due to an oversight in my proof, it is limited to CY4 quivers and locally CY fourfolds. An approach that was shared with me by Nick Kuhn is capable of resolving the general case. It is expected to appear in a joint work with him, Liu, and Thimm.
One of the interesting outcomes of my research is that one can wall-cross with some nice invariants obtained from multiplicative insertions. This refines the original formulation of Gross-Joyce-Tanaka who only dealt with full virtual fundamental classes. The geometric set-up of such wall-crossing gives naturally rise to a representation theoretic structure I called additive deformations of vertex algebras. Combined with the general wall-crossing theorem, I will use these additive deformations to prove multiple existing conjectures in the literature related to curve and surface counting (see Project 3).





