What Scattering amplitudes are one of the central quantities in quantum field theory, where they encode particle masses, decay rates, and the probability of different configurations of particles scattering into each other. This makes them a key ingredient for making predictions in particle physics experiments that probe the basic microscopic laws of the universe. In addition to their experimental import, scattering amplitudes exhibit a great deal of mathematical structure, much of which remains unexplained in terms of physical principles. My project seeks to understand some of this structure-in particular, specific features of these objects only recently discovered using the tools of algebraic geometry-and to develop efficient computational techniques that harness these features. Why Despite the central importance of scattering amplitudes to particle physics, these quantities remain extremely difficult to calculate in general. This is partially due to the fact that the special (and seemingly universal) algebraic properties that amplitudes emerge with at the end of these computations are obscured in intermediate steps. By developing a proper understanding of why scattering amplitudes have the algebraic properties they do, we can develop computational methods that make these properties manifest, and correspondingly more efficient. Understanding the physical principles these algebraic properties encode also promises to fundamentally change how we think about scattering amplitudes, and about quantum field theory more generally. How This project identifies three facets of algebraic structure in scattering amplitudes that especially call out to be better understood: (i) the emergence of a novel symmetry respected by amplitudes in the context of cosmic Galois theory, (ii) the (seeming) appearance of only Calabi-Yau geometries in the integration contours contributing to amplitudes, and (iii) the connection between cluster algebras and kinematics in supersymmetric amplitudes. By systematically studying each of these properties using techniques from algebraic geometry, their implications can be better understood and used to refine current computational methods. This will then facilitate the calculation of more complicated amplitudes, providing further data that can be used to develop even better computational techniques. SSR This project endeavors to connect people and ideas in theoretical physics to those in algebraic geometry. As with any interdisciplinary collaboration, the potential benefits to both fields far exceed the specific initial goals for establishing this connection (and indeed, certain algebraic geometers have already expressed enthusiasm for amplitudes as an interesting class of objects for them to study). It also encourages the development of texts that make the relevant ideas more broadly accessible to other fields. The long-term results of this research are likely to enter predictions at particle colliders such as the Large Hadron Collider, thus contributing to massively international collaborations and our fundamental understanding of the universe.