Tropical Homotopy Continuation and Laurent Phenomenon Algebras

Tropical geometry and the theory of Laurent phenomenon algebras (LPAs) both provide powerful frameworks for understanding algebro geometric objects in both pure and applied contexts. This thesis explores both of these topics. We extend Anders Jensen's technique of tropical homotopy continuation for computing stable intersections to the setting of Bergman fans, with applications to chemical reaction network theory and rigidity theory. On the other hand, we investigate the structure of LPAs arising from the configurations of lines on del Pezzo surfaces and explicitly describe a new finite-type LPA cluster structure on the homogeneous coordinate ring of the Cayley plane.