Special Functions and Geometries in Scattering Amplitudes: From Particle Physics to Gravity

This thesis focuses on the fields of scattering amplitudes and Feynman integrals, with an emphasis on the geometries and special functions that they involve, and is devoted to two distinct research directions. In the first half of the thesis, we explore elliptic Feynman integrals in maximally supersymmetric Yang-Mills (N = 4 SYM) theory. In particular, we study elliptic generalizations of ladder diagrams, identifying the first two families of Feynman integrals involving the same elliptic curve to all loop orders, which provide a great testing ground for developing mathematical tools that can facilitate the calculation of elliptic integrals. In this direction, we initiate the symbol bootstrap for elliptic Feynman integrals, for which we generalize the so-called Schubert analysis to predict elliptic symbol letters. As a proof of principle, we obtain for the first time the symbol of the two-loop twelve-point elliptic double-box integral, resulting in a compact, one-line formula. In the second half of the thesis, we pioneer a systematic investigation of the Feynman integral geometries that are relevant to the study of gravitational waves emitted during the inspiral phase of black-hole mergers within the post-Minkowskian expansion of classical gravity. Specifically, we classify the geometries and special functions appearing in the expansion up to four loops. Among other findings, we identify the first Calabi-Yau three-dimensional geometry relevant to gravitational-wave physics. Subsequently, we study the Feynman integral giving rise to this Calabi-Yau geometry in more detail, and solve it by bringing its differential equation into canonical form - a step that required developing a new method that accommodates for apparent singularities.