The K-theoretic DT/PT Vertex Correspondence

Explore a comprehensive lecture on the K-theoretic DT/PT vertex correspondence delivered by Henry Liu from IPMU at the M-Seminar, Kansas State University. Delve into the intricate world of smooth quasi-projective toric 3- and 4-folds, where vertices represent contributions from affine toric charts to enumerative invariants of Donaldson-Thomas (DT) or Pandharipande-Thomas (PT) moduli spaces. Examine the torus-equivariant nature of vertices and their combinatorial complexity in equivariant K-theory. Learn about two distinct proofs of the K-theoretic 3-fold DT/PT vertex correspondence, developed through joint work with Nick Kuhn and Felix Thimm. Understand the application of equivariant wall-crossing in Toda's setup, exploring both a Mochizuki-style master space approach and concepts from Joyce's universal wall-crossing machine. Discover the crucial development of symmetrized pullbacks of symmetric obstruction theories on moduli stacks, utilizing Kiem-Savvas' étale-local notion of almost-perfect obstruction theory. Consider the potential applications of these techniques to related topics such as DT/PT descendent transformations, the DT crepant resolution conjecture, and the 4-fold DT/PT vertex correspondence in this 1 hour and 27 minutes long presentation.