The advances in string theory revealed tantalizing connections between mathematics and physics, and has been source for groundbreaking collaborations between theoretical physicists and mathematicians over the three decades. String theory has emerged as a unifying framework to connect different fields of mathematics which previously were not known to be related. In this talk, I will present a portion of my past and present work which stands at the intersection of the conjectural refinement of Gromov-Witten theory and other branches of mathematics such as symplectic invariants of toric Calabi-Yau threefolds, knot theory, symmetric function theory, integrable systems.