Ouyang, Liya2022-09-132022-09-132022-03https://hdl.handle.net/11299/241606University of Minnesota Ph.D. dissertation.June 2022. Major: Mathematics. Advisor: Tianjun Li. 1 computer file (PDF); vi, 45 pages.We study the symplectic divisors corresponding to the Hamiltonian circle actions on symplectic surfaces. Li-Min-Ning showed that counting toric actions on a fixed symplectic rational surface is equivalent to counting toric log Calabi-Yau divisors. Inspired by a formula by Holm-Kessler, we introduce generalized symplectic log Calabi-Yau divisors on symplectic irrationalruled surfaces. Using the language of marked divisors we prove a version of Torelli theorem, stating that the symplectic deformation classes of these divisors (with a few extra conditions) are determined by their homological information. We show that there is an one-to-one correspondence between Hamiltonian circle actions and S1-generalied symplectic log Calabi-Yau divisors in a fixed symplectic irrational ruled surface. As an application, we give a new proof of the finiteness of inequivalent Hamiltonian circle actions.enThesis or Dissertation