Thesis (M.S.)--Rutgers University, 2009. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 17).

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 151-154).

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains vi, 41 p. : ill. (some col.). Vita. Includes abstract. Includes bibliographical references (p. 40-41).

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..

In this thesis we present the connection between vertex operator algebras and modular forms which lies at the heart of Borcherds’ proof of the Monstrous Moonshine conjecture. In order to do so we introduce modular forms, vertex algebras, vertex operator algebras and their partition functions. Each notion is illustrated with examples.

Thesis (Ph. D.)--University of Oregon, 2008. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 40-41). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.