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The percent-level precision achieved by experiments at the LargeHadron Collider demands theoretical predictions of comparable accuracy, necessitating higher-order perturbative corrections to scattering amplitudes. Traditionally, these corrections are computed by reducing the Feynman integralexpansion of an amplitude to a finite basis of linearly independent elements, followed by numerical evaluation. However, multi-scale processes posesignificant computational challenges, motivating the development ofcomplementary approaches. In this talk, I will review recent advances in Feynman integral calculus, highlighting mathematical techniques rooted in algebraic geometry, finite fieldreconstruction, and twisted cohomology theory. I will also discuss alternativereduction strategies based on the so-called "transverse integration identities"and explore how these innovations can enhance traditional computational methods to address the demands of modern precision physics.