Relative Gromov-Witten theory and vertex operators

In this thesis, we report on two projects applying representation theoretic techniques to solve enumerative and geometric problems, which were carried out by the author during his pursuit of Ph.D. at Columbia. We first study the relative Gromov-Witten theory on T*P¹ x P¹ and show that certain equivariant limits give relative invariants on P¹ x P¹. By formulating the quantum multiplications on Hilb(T*P¹) computed by Davesh Maulik and Alexei Oblomkov as vertex operators and computing the product expansion, we demonstrate how to get the insertion operator computed by Yaim Cooper and Rahul Pandharipande in the equivariant limits. Brenti proves a non-recursive formula for the Kazhdan-Lusztig polynomials of Coxeter groups by combinatorial methods. In the case of the Weyl group of a split group over a finite field, a geometric interpretation is given by Sophie Morel via weight truncation of perverse sheaves. With suitable modifications of Morel's proof, we generalize the geometric interpretation to the case of finite and affine partial flag varieties. We demonstrate the result with essentially new examples using sl₃ and sl₄.