And so forth. I find the computational aspects of most theories to be very rich, and it's really gratifying to code something up and "read off" the results.
Sage is a remarkable project. For Grobner bases it relies on the Singular computer algebra system. I cringe every time I'm at a talk where someone credits Sage when they should have cited Singular.
It's hard work writing a system like Singular, and not an obvious path to glory, so this stings.
Thanks! I'm studying this right now, after I got a taste of algebraic geometry (schemes) from Ravi Vakil's fantastic "algebraic geometry during the time of COVID" last year!
Good question. I mostly use it for algebra. It has varying levels of support for analysis. There is a SageManifolds project that allows for computations over differentiable and Riemannian manifolds, for example.
1. Grobner bases: http://bollu.github.io/computing-equivalent-gate-sets-using-...
2. Localization: https://github.com/bollu/bollu.github.io/blob/8cd335687ff3ef...
3. More broadly, an answer on math.stackexchange on how to debug math: https://math.stackexchange.com/questions/1769475/how-to-debu...
4. (WIP) continued fractions to compute pi: https://bollu.github.io/fractions/index.html
And so forth. I find the computational aspects of most theories to be very rich, and it's really gratifying to code something up and "read off" the results.