List of free mathematical software

The list below presents an absolutely arbitrary selection of software I know about that can be used for mathematical (mostly algebraic) computation. There are a lot of omissions since either I don't know about some apps or I don't consider them worth mentioning. As mentioned above, the selection is purely arbitrary, so please don't mail me about a java-based calculator you have happened to write that you wish to be included. I won't. Period.

General purpose

Commutative algebra and algebraic geometry

Group theory

Number theory

Numerical methods and statistics