Efficient manipulation of tensor expressions

This talk introduces the field of Tensor Computer Algebra, for large scale computations with tensor expressions. Tensor manipulations are essential in Algebra and Differential Geometry, and hence play a central role in most physical theories, including fluid dynamics, electromagnetism or general relativity. After a brief description of the field, we present the system xAct, for Mathematica, which implements highly efficient algorithms for simplification of indexed expressions, based on well known techniques of computational group theory. This system has been already used to solve a number of mathema:cal problems in general relativity, like the analysis of the hyperbolicity properties of formulations of the Einstein equations, the theory of curvature invariants or high‐order perturbation theory.