Mathematical methods from scattering amplitudes to modern scientific calculus

I recall generalised unitarity, integrand decomposition, integration-by-parts identities, differential equations, GKZ-systems and intersection theory, connecting Cauchy's residue theorem, Stokes' theorem and Gauss' linking numbers, and elaborate on the emerging crucial role played by De Rham co-homology theory, ruling the differential and algebraic properties of the integral functions that appear in the context of fundamental physics. I will recall recent applications to Feynman and Euler-Mellin integrals in particle physics and cosmology, as well as to quantum mechanics and Wick's theorem in QFT, concluding with the role that Physics Informed Neural Networks may play in the context of (twisted period) integrals evaluation.