Resources

References for mathematics:

The “periodic table of finite simple groups”, by Ivan Andrus: https://irandrus.wordpress.com/2012/06/17/the-periodic-table-of-finite-simple-groups/ To do: make an equally useful and silly picture, but with improved organization (no reason to split the classical groups across both sides, except to make it look more like the chemical periodic table).
Spacebook: https://topology.jdabbs.com/ (“Does first countable and separable imply second countable?”, etc)
Sage quickref: https://wiki.sagemath.org/quickref
Design database: designtheory.org

References for writing:

The grammar according to West (guide to writing in mathematics, particularly for non-native speakers): https://faculty.math.illinois.edu/~west/grammar.html
Google ngrams: “Is it more usual to say ‘well known’ or well-known’?”, etc. https://books.google.com/ngrams
TeXample: http://www.texample.net/. Treat somebody else’s work as a template.
Detexify: “What’s the latex command for the symbol [some scribble]?” http://detexify.kirelabs.org/classify.html
latex2wp: Compile basic latex –> wordpress. https://github.com/seaneberhard/latex2wp
MRLookup: “Give me a bibtex-formatted citation, please”: https://mathscinet.ams.org/mrlookup (or maybe even better: see https://mathoverflow.net/a/319982/20598)
Digital timestamp: Need to certify that you have a proof of the Riemann hypothesis (or whatever) today, but not yet ready to go public? http://www.itconsult.co.uk/stamper.htm (and then don’t email it to me)
mathgen: In case of writer’s block, use this to generate a random math paper: https://thatsmathematics.com/mathgen/

References for speaking:

General advice:

Ten lessons I wish I had been taught, by Gian-Carlo Rota: https://www.ams.org/notices/199701/comm-rota.pdf
Similar: advice on being a mathematician from Paul Halmos: http://www.ams.org/notices/200709/tx070901136p.pdf