Landau Singularities and Positive GeometriesStanojevic, StefancreatorSpradlin, MarcusAdvisorJevicki, AntalReaderAlexander, StephonReaderBrown University. Department of Physicssponsor20209, 109 p.born digitalThesis (Ph. D.)--Brown University, 2020thesesThis work studies the analytical structure of scattering amplitudes in a class of toy models, from the perspective of novel geometrical structures underlying scattering amplitudes, and utilizing discrete mathematics. We find the set of all-loop n - point Landau singularities of the first type arising in any massless planar theory using graph operations inspired by the theory of electrical circuits. We also describe how to find the set of possible branch points of amplitudes at any fixed loop order and all particle multiplicities in planar N = 4 SYM theory, using the information contained in the boundaries of the amplituhedra. Along the way, we devise a method for classifying the boundaries of amplituhedra for any loop order L and helicity k. We determine the branch points of all two-loop NMHV amplitudes by solving the Landau equations for the relevant configurations and are led thereby to a conjecture for the symbol alphabets of all such amplitudes. The study of amplituhedra is then pushed beyond the realm of N = 4 SYM theory. We provide an efficient recursive formula to compute the canonical forms of arbitrary d-dimensional simple polytopes, and illustrate our method via an application to Stokes polytopes, related to scalar theory with quartic coupling.Quantum field theoryGraph theoryPolytopesEnglishRPB20200720In CopyrightCollection is open for research.dissertations