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Feynman integrals play a central role in the modern scattering amplitudesresearch program. Advancing our methods for evaluating Feynman integrals will,therefore, strengthen our ability to compare theoretical predictions with datafrom particle accelerators such as the Large Hadron Collider. Motivated bythis, the present manuscript purports to study mathematical concepts related toFeynman integrals. In particular, we present both numerical and analyticalalgorithms for the evaluation of Feynman integrals.  The content is divided into three parts. Part I focuses on the method of DEQsfor evaluating Feynman integrals. An otherwise daunting integral expression isthereby traded for the comparatively simpler task of solving a system of DEQs.We use this technique to evaluate a family of two-loop Feynman integrals ofrelevance for dark matter detection. Part II situates the study of DEQs forFeynman integrals within the framework of D-modules, a natural language forstudying PDEs algebraically. Special emphasis is put on a particular D-modulecalled the GKZ system, a set of higher-order PDEs that annihilate a generalizedversion of a Feynman integral. In the course of matching the generalizedintegral to a Feynman integral proper, we discover an algorithm for evaluatingthe latter in terms of logarithmic series. Part III develops a numericalintegration algorithm. It combines Monte Carlo sampling with tropical geometry,a particular offspring of algebraic geometry that studies "piecewise-linear"polynomials. Feynman's i*epsilon-prescription is incorporated into thealgorithm via contour deformation. We present an open-source program namedFeyntrop that implements this algorithm, and use it to numerically evaluateFeynman integrals between 1-5 loops and 0-5 legs in physical regions of phasespace.

Evaluating Feynman Integrals Using D-modules and Tropical Geometry