Positive Singularities and Volumes in Scattering Amplitudes
Recent advances have revealed that scattering amplitudes in certain quantum field theories admit a geometric formulation in terms of positive geometries. In this framework, amplitudes are encoded by differential forms whose boundary structure reflects physical principles such as locality, unitarity, and factorization. This thesis gives a self-contained introduction to positive geometries, with particular emphasis on the Amplituhedron, which describes amplitudes in planar maximally supersymmetric Yang--Mills theory through its canonical form. We develop three related directions connecting positivity, volumes, and singularities. First, we study positivity properties of canonical forms through dual volume representations. For polytopes, canonical functions compute volumes of dual polytopes; we extend this picture to nonlinear positive geometries, uncovering non-negative transcendental measures and links with complete monotonicity. Second, we investigate loop-level amplitudes via their singularity structure. Combining Amplituhedron geometry with Landau analysis, we classify leading singularities of the Wilson loop with Lagrangian insertion and constrain the possible singular loci. Third, we examine conjectural relations between Landau singularities, positivity, and cluster algebras. Using momentum-twistor and Grassmannian methods, we identify recursive structures across loop orders, prove several infinite families of cases, and propose a strategy toward the general conjectures. Overall, the thesis develops the perspective that positive geometry provides a unifying language for the analytic and geometric organization of scattering amplitudes, from canonical forms and volumes to the singularities that emerge after loop integration.