  Feynman integrals play a central role in the modern scattering amplitudes
research program. Advancing our methods for evaluating Feynman integrals will,
therefore, strengthen our ability to compare theoretical predictions with data
from particle accelerators such as the Large Hadron Collider. Motivated by
this, the present manuscript purports to study mathematical concepts related to
Feynman integrals. In particular, we present both numerical and analytical
algorithms for the evaluation of Feynman integrals.
  The content is divided into three parts. Part I focuses on the method of DEQs
for evaluating Feynman integrals. An otherwise daunting integral expression is
thereby 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 of
relevance for dark matter detection. Part II situates the study of DEQs for
Feynman integrals within the framework of D-modules, a natural language for
studying PDEs algebraically. Special emphasis is put on a particular D-module
called the GKZ system, a set of higher-order PDEs that annihilate a generalized
version of a Feynman integral. In the course of matching the generalized
integral to a Feynman integral proper, we discover an algorithm for evaluating
the latter in terms of logarithmic series. Part III develops a numerical
integration 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 the
algorithm via contour deformation. We present an open-source program named
Feyntrop that implements this algorithm, and use it to numerically evaluate
Feynman integrals between 1-5 loops and 0-5 legs in physical regions of phase
space.
