Mathematics one-pager

Arithmetic

• Do not write fractions as "mixed numbers" like in elementary school. For example, 3 followed by 1/2 means 3/2, not 3+1/2.
• Do not use an "x" to indicate multiplication of numbers, use parentheses or a dot instead.
• Do not use the division symbol with the line and two dots. Use a slash or fraction instead.
• Do not write ln instead of log. In mathematics, ln is antiquated and the base of log is assumed to be Euler's number.
• Algebraic simplifications typically do not matter, unless they somehow advance an argument or goal. The normal forms you were made to memorize in high school were basically meaningless. For instance, it is perfectly fine for a radical to appear in the denominator of a fraction.
• Do not do arithmetic with infinity. You are not ready yet.

Higher math (undergrad)

• Learn LaTeX. Maintain a LaTeX distribution (such as TeX Live) on your own computer. The LaTeX Wikibook is very helpful for learning. You can identify symbols with Detexify. Both asymptote and tikz are useful for writing graphics directly in your document. Use tikz-cd for commutative diagrams.
• Learn how to code. A popular first language is Python, but I now recommend Lua instead. Learn how to use a version control system, such as Git.
• Start writing papers as soon as possible, both expository and research. Like any other creative endeavor, your first efforts are likely to seem clumsy or awkward to you in the future. Better to make those mistakes early on. You aren't doomed if your first results aren't Earth-shattering.
• Email people whose work you are interested in. Tell them why you find it interesting. You don't have to understand all of it. People appreciate genuine interest, even if you aren't already an expert in their field. They do not like feigned interest, so don't insult them by pretending to understand or like something you don't.
• Get permission to post on the arXiv as soon as possible and post some stuff. Perhaps not your very first efforts, but you can post something of good quality even if you're not ready to publish in a journal.
• Start attending conferences and giving talks at them.
• Join or help run a math club, such as an AWM chapter.
• Participate in a Research Experience for Undergraduates (REU). Search for opportunities on mathprograms.org or ask a professor at your university.

Higher math (graduate)

• Get on mailing lists for topics which interest you. For example, I'm on ones for algebraic topology, category theory, and universal algebra. Note that you can join a Google-managed mailing list without a Google account.
• Mathstodon (or Mastodon in general) is a great place to discuss mathematics in a microblogging format.
• Choose an advisor you get along with. You don't have to be best friends, but you need to be able to work with this person for years. Don't prioritize your advisor's research area or prestige over compatibility. Many have regretted this kind of choice. Difficulty with your advisor can have very unpleasant consequences, even after graduate school.
• Create a website with your own domain name and hosting. You can share your research, code, and so forth there.
• Join or run a math graduate school organization, such as an AMS graduate student chapter.
• Try to publish a single-author paper as soon as possible. Ideally, this should not be part of your thesis research. It is understood that your advisor is overseeing your thesis, so this is a good way to demonstrate that you can work independently.
• Help run an REU. It is usually most convenient to do this at your home institution, but you can also check for opportunities for graduate students on mathprograms.org.
• Participate in the AMS Mathematics Research Communities program. This is somewhat like a graduate-level REU.
• Start mentoring undergraduates. This could take the form of a department-organized guided reading program or could be something less formal.