I will give a survey about interpolation, evaluation, and

reconstruction methods for multivariate polynomial and rational problems with applications in linear algebra and Groebner basis. This will include both asymptotically fast as well as practical algorithms.

I'd like to present Rational Tracer, a solver of systems of Integration-By-Parts relations (and more) based on modular arithmetic methods, and explain how using new programming ideas we can achieve lower evaluation times and higher flexibility for practical IBP solving.

I will review a recent mathematical technique to efficiently compute scattering amplitudes in the forward scattering limit of N=4 SYM and explain how it can be used to fully determine the function space of scattering amplitudes in this limit to all orders in perturbation theory.

In this talk we will elaborate, among others, on root-valued alphabets and elliptic structures in Massive Operator Matrix Elements, together with concrete phenomenological results at O(a_s^3).

Elaborating on the recent developments concerning Feynman integrals' calculus, I shall introduce Intersection Theory for twisted de Rham co-homology, and mainly discuss the impact of Intersection Numbers on the Vector-Space structure of special functions appearing in Mathematics and Physics.