Daole Zhang: Hurwitz theory, the ELSV formula, and beyond

Abstract: The ELSV formula, introduced by Ekedahl, Lando, Shapiro, and Vainshtein in 2001, is a foundational result in the study of moduli spaces of curves and Gromov--Witten theory, providing a deep and unexpected connection between combinatorics, algebraic geometry, and theoretical physics. In this thesis, by utilizing the moduli space of relative maps constructed by Jun Li (2001, 2002) alongside virtual localization techniques, a simpler proof of the ELSV formula is obtained. To achieve this, we first develop the necessary mathematical machinery, including the theory of algebraic stacks, equivariant cohomology, and virtual localization. By synthesizing these foundational tools, the thesis provides a rigorous and self-contained exposition of the proof.