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I would like to know math websites that are useful for students, PhD students and researchers (useful in the sense most of the students or researchers—of a particular area—are using it). Maybe you can share which math websites you sometime use and why you use it.

Let me give my websites and why I use them:

MathOverflow and Stack Exchange: Very good to look up questions about how to do mathematics, what mathematics is, how it evolves, and of course to have an exchange with smart people, who can help you if you are stuck at something.

arXiv: I don't use it now, but I think that's the main website to read papers and to publish (research level)

The Stacks Project: It is very useful for me to look algebraic things ups with more explanation than in some lectures...

MathSciNet: To search for papers that may help you.

Number Theory Web: Since I want to become a number theorist, this site helps me to see which kind of things exist.

Do you have more websites that help you to understand maybe mathematics, or just to connect to other mathematicians?

Everything is welcome. Just post it as an answer and not a comment.

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I have found these extremely useful, from time to time:

Online Encyclopedia of Integer Sequences

NIST Digital Library of Mathematical Functions

They both provide facts of a specialized nature, rather than understandings or arguments. That is, they are thin on explanations and proofs. The OEIS does have very thorough references, so this is not a serious problem. The DLMF also has many references, which I then skim with MathSciNet.

I have mixed feelings about Wikipedia's mathematics articles. Individual articles are often imprecise, more interested in giving the gist of a topic than in giving precise details; this is especially bad when it comes to precise statements of theorems or (worse) of proofs. There is usually only a loose connection between the cited references and the contents of the article: which (say) of the six cited sources contains details of this or that statement in the article? And there is poor coordination between collections of articles, which often use differing terminology and notation. On the other hand, it is a good source for understanding (say) current usage of mathematical terminology. I typically find 75 percent of MSE questions use some technical term in a way I do not understand, and Wikipedia often gives an indication of what the questions' posers had in mind.

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    $\begingroup$ I don't think Wikipedia on its own should be used to learn mathematics. It should e.g. be used to get a quick overview or to recall some definitions/theorems. $\endgroup$ Commented Feb 20, 2021 at 14:01
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Great question. It would probably also be interesting to think about what could be useful but is not yet out there. Here are a few picks that I can think of:

  • A useful website for ring theorists is the Database of Ring Theory. It is actually managed by rschwieb, a frequent user of MSE. You can look for examples or counterexamples of your favourite properties of rings and modules in the database.
  • In a similar vein, here is a Database for Number Fields.
  • Super useful for number theorists is also LMFDB. It's another database with which you can look for number-theoretic objects with certain desired properties.
  • For category lovers, the equivalent of the Stacks Project is Kerodon on Higher Category Theory. I think this one is still growing.
  • And of course, obvious ones like Wolfram Alpha and Wikipedia!

Maybe I should also mention certain social media meme groups in which you can apply your knowledge to understand the most recent homological algebra memes. But maybe this is not quite the right place for that... ;-)

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    $\begingroup$ Haha, thank you! I am searching for exactly these type of websites! $\endgroup$ Commented Feb 20, 2021 at 13:58
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There is a website devoted to counterexamples: http://www.mathcounterexamples.net/?fbclid=IwAR0dZKurrSUV1ft8AeW-Gh189x58BdjGZ6M2KzeeyiHJQ6grdCuCgLdGUgw

The Encyclopedia of Triangle Centers: https://faculty.evansville.edu/ck6/encyclopedia/ETC.html

and also the Periodic Table of Finite Elements: http://femtable.org/

For curiosities, there is a website that teaches (with audio files) the pronunciations of many mathematician's names: http://pronouncemath.blogspot.com/

Mathematical pictures: https://images.math.cnrs.fr/?lang=fr

Finally, there is the nice Halmos' album of pictures of mathematicians: https://www.maa.org/press/periodicals/convergence/whos-that-mathematician-images-from-the-paul-r-halmos-photograph-collection

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ProofWiki is an online collection of proofs that I have found to be useful at times. Each page contains links to definitions and lemmas that are used in the proof, as well as a list of external sources on the topic. More importantly, there is a list of math jokes ;)

Another useful site is Wolfram MathWorld. It contains articles on many topics in mathematics, with links to related topics and references. It too has a few jokes.

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I really enjoy the following schematic overview of various statistical, distributions, their relationships and properties. It's quite handy for giving students (and others!) a quick way of relating new distributions to distributions that they already know about.

Univariate Distribution Relationships

edit: typos

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On /r/math, there is A Compilation of Useful, Free, Online Math Resources. It is somewhat geared towards students in scope, but references many tools used by research mathematicians as well (see especially the comment on subject-specific resources). It contains most of the links already in this thread, and several not yet listed here - a selection of the latter group of links:

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(https://artofproblemsolving.com) is a very good math site. It has artofporblemsolving books and it also has a lot of math games to practice your math skills. Alcumus is the best game on there (https://artofproblemsolving.com/alcumus)

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http://www.algebra.com is a very good site for algebra problems and solutions.

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