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Also, use computers! I use SAGE extensively to get a feeling for lots of mathematical objects! 1. Grobner bases: http://bollu.github.io/computing-equivalent-ga
by bollu 6y ago
Also, use computers! I use SAGE extensively to get a feeling for lots of mathematical objects!
1. Grobner bases: http://bollu.github.io/computing-equivalent-gate-sets-using-grobner-bases.html http://bollu.github.io/computing-equivalent-gate-sets-using-...
2. Localization: https://github.com/bollu/bollu.github.io/blob/8cd335687ff3ef6ab9c62c7ebf17bdfa1a092f71/localization.md 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-debug-math/1769492#1769492 https://math.stackexchange.com/questions/1769475/how-to-debu...
4. (WIP) continued fractions to compute pi: https://bollu.github.io/fractions/index.html 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.
- spekcular 6y agoYou might enjoy Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by David Cox.
- bollu 6y agoThanks! 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!
- Karrot_Kream 6y agoDoes SAGE work well for analysis as well, or is it mostly for algebras/objects?
- bollu 6y agoGood 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.
- Syzygies 6y agoSage 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.
- bollu 6y agoI am sorry. I shall reference Singular in the future whenever I discuss SAGE. Thank you for your fantastic work on Singular :)