The cohomology of moduli spaces of 1-dimensional sheaves, together with a special filtration called the perverse filtration, can be used to give an intrinsic definition of (refined) Gopakumar-Vafa invariants. While there are methods to calculate the Betti numbers of these moduli spaces in low degree, understanding the perverse filtration is more challenging. One way to compute it is to fully determine the cohomology ring. In this talk I will explain an approach to describing the cohomology rings of moduli spaces and moduli stacks in terms of generators and relations, which allowed us to determine them for curve classes up to degree 5 (for moduli spaces) and 4 (for moduli stacks). I will explain the P=C conjecture, which is an analogue of the P=W conjecture in a Fano and compact setting. This is joint work with Yakov Kononov, Woonam Lim and Weite Pi.
for more information see: https://personal.math.ubc.ca/~jbryan/Zoominar-UBC-ETH/