After giving a brief survey of the study of derived categories in algebraic geometry, I present a pair of research papers. The first, joint with Daniel Halpern-Leistner and Jeffrey Jiang, introduces the notion of quasi-convergent paths in the space of stability conditions. We prove that quasi-convergent paths give rise to decompositions of triangulated categories (e.g. derived categories of coherent sheaves on a variety) and that conversely for a smooth and proper dg-category all polarized semiorthogonal decompositions arise in this fashion. In the second paper, I study the geometry of certain moduli spaces of genus 0 curves with differentials called multiscale lines which were introduced in joint work with Daniel Halpern-Leistner. I prove that these spaces are complex projective varieties by giving an explicit isomorphism with a blow-up of a linear subspace arrangement of projective space. I use this isomorphism to connect these spaces of multiscale lines with other spaces in the literature studied by Zahariuc.