Moduli spaces of curves and integrable hierarchies

This work is structured to explore two distinct facets of the moduli space of
curves, each inquiring into different aspects: one focuses on topological recursion relations in open Gromov-Witten theory, and the other on the bihamiltonian aspects of the double ramification hierarchy.
The first topic investigates open topological recursion relations in genus 1, represented by a set of partial differential equations (PDEs) that are conjectured
to control open Gromov-Witten invariants in genus 1. In this segment, an
explicit formula is derived, serving as an analog to the Dijkgraaf-Witten formula, specifically for a descendent Gromov-Witten potential in genus 1. This
formula stands as a solution to the recursion relations, and is proven that the
exponent of an open descendent potential, when approximated up to genus 1,
satisfies a system of linear evolutionary PDEs, explicitly constructed with a
single spatial variable.
The second facet of the thesis is based on a conjecture of Buryak, Rossi, and
Shadrin [BRS21]. This conjecture proposes a formula for a Poisson bracket
associated with any given homogeneous cohomological field theory (CohFT). It
is hypothesized that this bracket defines a second Hamiltonian structure for the
double ramification hierarchy for the given CohFT. This part of the research
validates the conjecture at an approximation up to genus 1 and establishes
a relationship between this bracket and the second Poisson bracket of the
Dubrovin-Zhang hierarchy by an explicit Miura transformation.