Enumerative geometry, topological recursion, and the semi-infinite wedge

Recent decades have seen an explosion of activity in the symbiotic relationship between mathematics and physics. From this interplay has emerged the notion of topological recursion, a general theory that appears to govern seemingly disparate problems in geometry, algebra, and mathematical physics. This thesis considers four such problems lying at the confluence of these fields. The results include the resolution of outstanding conjectures from the literature as well as the introduction of new mathematical structures whose fascinating properties should inspire future discovery. This research forms part of a rich tapestry of ongoing work that connects diverse areas of mathematics.