Thesis DefenseFriday August 2nd, 2019 Time: 2:00 PM Title: Variational Formulas and Strata of Abelian Differentials Speaker: Xuntao Hu, Stony Brook University Location: Math Tower 5-127 Abstract: In this dissertation we study the degeneration of abelian differentials. We use the jump problem technique developed in a recent paper by Grushevsky, Krichever and Norton to compute the variational formulas of any stable differential and its periods to arbitrary precision in plumbing coordinates. In particular, we give explicit variational formula for period matrices, easily reproving results of Yamada for nodal curves with one node and generalizing them to arbitrary stable curves. We apply the same technique to give an alternative proof of the sufficiency part of the main theorem in a paper by Bainbridge, Chen, Gendron, Grushevsky and Moeller on the closures of strata of differentials with prescribed multiplicities of zeroes and poles. We also give an explicit modular form defining the loci of quartics with a hyperflex, also known as the minimal stratum of abelian differentials in genus 3. Using our variational formulas for the period matrices and the modular form we obtained, we provide a direct way to compute the divisor class of such locus in the Deligne-Mumford compactification of moduli of curves of genus 3.