演講摘要：

On smooth quasi-projective toric 3-folds, vertices are the contributions from an affine toric chart to the enumerative invariants of Donaldson-Thomas (DT) or Pandharipande-Thomas (PT) moduli spaces. Unlike partition functions, vertices are fundamentally torus-equivariant objects, and they carry a great deal of combinatorial complexity, particularly in equivariant K-theory. In joint work with Nick Kuhn and Felix Thimm, we give two different proofs of the K-theoretic DT/PT vertex correspondence. Both proofs use equivariant wall-crossing in a setup originally due to Toda. A crucial new ingredient is the construction of symmetrized pullbacks of symmetric obstruction theories on Artin stacks, using Kiem-Savvas' étale-local notion of almost-perfect obstruction theory.