- Advice to a Young Mathematician
- A Mathematician's Lament, by Paul Lockhart (see some commentary here)
- A Mathematician's Apology, by G. H. Hardy
- A Mathematician Comes of Age, by Steven Krantz
- What is it like to have an understanding of very advanced mathematics? (in a PDF here)
- What can one (such as myself) contribute to mathematics?
- From Open Science to Open Mathematics, by Felix Breuer
- On Contemplation in Mathematics, by F. Luke Wolcott (also look at the author's blog)
- Mathematics in the 20th Century, by Sir Michael Atiyah
- Deconstructing "Genius", by Robert Krzyzanowski
- The Wrong Way to Treat Child Geniuses, by Jordan Ellenberg (see some commentary here)
- There's more to mathematics than rigour and proofs, by Terence Tao
- I hate axioms, by Qiaochu Yuan
- Mathematics, morally, by Eugenia Cheng
- What is Good Mathematics?, by Terence Tao
- In Praise of Lectures, by Tom Körner
- Why lectures are a waste of time, by Eugenia Cheng
- The "Three Things" Exercise for getting things out of talks, by Ravi Vakil
- Aftermath, by E.C. Titchmarsh
- Physics, Topology, Logic and Computation: A Rosetta Stone, by John Baez and Mike Stay
- How to respond to "I was never much good at maths at school."
- A Mathematician's Survival Guide, by Peter Casazza
- The Two Cultures of Mathematics, by Timothy Gowers
- The Importance of Mathematics, by Timothy Gowers
- Does mathematics need a philosophy?, by Timothy Gowers
- On Proof and Progress in Mathematics, by Bill Thurston
- Missed Opportunities, by Freeman Dyson
- Birds and Frogs, by Freeman Dyson
- Mathematical Platonism and its Opposites, by Barry Mazur
- What should a professional mathematician know? by Barry Mazur
- When is one thing equal to some other thing? by Barry Mazur
- Is it plausible? by Barry Mazur
- Intuition and Logic in Mathematics, by Henry Poincaré
- Mathematics as a Creative Art, by Paul Halmos
- Ten Lessons I Wish I Had Been Taught, by Gian-Carlo Rota
- Three Lessons I Wish I Had Never Been Taught by Gian-Carlo Rota, by Paul Nevai
- Is the current system of math academia unsustainable?
- On structured theorems, by Stefano Maggiolo
- Top Ten Reasons To Not Share Your Code (and why you should anyway), by Randall LeVeque
- Finding and Solving Problems, by John Baez
- When is proof by contradiction necessary?, by Timothy Gowers
- Self-deprecation is also other-deprecation, by David Speyer
- What It Feels Like to Be Bad at Math
- Gender Bias 101 For Mathematicians, by Izabella Laba
- Gender Bias 102 For Mathematicians: Merit, by Izabella Laba

- A mathematician's survival guide: graduate school and early career development, by Steven Krantz
- Career advice, by Terence Tao
- How to escape the inclination to be a universalist or: How to learn to stop worrying and do some research.
- What do grad students in math do all day?, a humorous analogy (see the entire Quora thread)
Some fellowships to apply for:

Some advice on how to apply for those fellowships:

- Advice for applying for graduate science fellowships, by Philip Guo
- Applying for Graduate Research Fellowships, by Seth Sullivant
- Advice on applying for the NSF fellowship, by Kevin Sanft

Academia.SE, a Q&A forum for academics and those enrolled in higher education

- How should I deal with discouragement as a graduate student?
- How to avoid procrastination during the research phase of my PhD?
- Important factors for hiring a young mathematician
- Life as a math professor
- Do extracurricular research/activities in graduate school negatively affect your CV?
- What are the potential pitfalls of having a PhD?
- Should I stay in PhD program I dislike to have a shot at liberal arts teaching?
- How do advisors typically apply pressure on PhD students to get work done?

- Assistantships and Graduate Fellowships in the Mathematical Sciences, compiled by the AMS
- Preparing For Grad School, advice from MIT
- Some grad school advice from Harvard.
- Some grad school advice from Haverford.
- How to get a great letter of recommendation, by Matthew Might
- Recommendations, by Ravi Vakil
- Get into grad school for science, engineering, math and computer science, by Matthew Might
- Thoughts on graduate school, by Noah Snyder (an addendum by Ben Webster)
- More advice for prospective graduate students, by Noah Snyder
- Advice for potential graduate students in Arithmetic Geometry, by David Zureick-Brown
- Wiki collection of UC Berkeley qual exam questions
- A Graduate School Primer (MAA), by Doug Faires
- Advice on picking a Ph.D. program and advisor
- Choosing a graduate school, by Diana Davis
- Tips for Visiting Grad Schools, by Kimball Martin
- How to Be a Good Graduate Student, by Marie desJardins
- What Else Should Grad Students Be Learning?
- I've passed my quals, now what? A guide for Ph.D. candidates in Mathematics, by Mark Tomforde
- How to pick a thesis advisor?
- Graduate Study in the Computer and Mathematical Sciences: A Survival Manual, by Dianne O'Leary
- RPI's Grad Student Survival Guide in Math Sciences
- 10 easy ways to fail a Ph.D., by Matthew Might
- The Academic Job Search in Mathematics, by Thomas Rishel
- Postgraduate Toolbox
- You and Your Research, by Richard Hamming
- No, You're Not an Impostor, by Lucas Laursen, about the impostor syndrome
- The importance of stupidity in scientific research, by Martin Schwartz
- Why is my paper talking so long to review?
- Academic scattering, by Katie Mack

- A primer of mathematical writing, by Steven Krantz
- How to Write Your First Paper, by Steven Krantz
- How to Give a Good Colloquium, by John McCarthy
- How to Write Mathematics, by Paul Halmos
- Common English errors in mathematical papers, by Jerzy Trzeciak
- Writing Mathematical Papers — a Few Tips, by Jerzy Trzeciak
- Writing advice, by Terence Tao
- What's wrong with these equations?, by David Mermin
- Periods and commas in mathematical writing
- Mathematical Writing, by Donald Knuth, Tracy Larrabee, and Paul Roberts
- Some writing and presenting advice, by Chris Godsil
- Tips on giving talks, by Jordan Ellenberg
- Technically Speaking, advice put out by the NSF
- 10 tips for academic talks, by Matthew Might
- How to give a good research talk, by Simon Jones, John Hughes, and John Launchbury
- The nuts and bolts of writing mathematics, by Dave Richeson
- A Scrutiny of the Abstract, by Kenneth Landes
- Tips for Authors and Misused Words, by J. S. Milne

- Time management advice, by Terence Tao
- Productivity tips, tricks and hacks for academics, by Matthew Might
- Boost your productivity: Cripple your technology, by Matthew Might
- Good and Bad Procrastination, by Paul Graham
- How to survive creative burnout, by Scott Berkun
- Burnout Is Real: How to Identify and Address Your Burnout Problem, by Adam Dachis

- MathOverflow, a Q&A forum for research-level mathematics
- Math.SE, a Q&A forum for math of any level
- The Unapologetic Mathematician, John Armstrong's blog
- What's new, Terence Tao's blog
- Annoying Precision, Qiaochu Yuan's blog
- Climbing Mount Bourbaki, Akhil Mathew's blog
- Rigorous Trivialities, Charles Siegel's blog
- Ars Mathematica
- MIT's OpenCourseWare
- The nLab, a wiki / collaborative notebook focusing on higher category theory
- TheCatsters, a YouTube channel with many excellent, short lectures on category theory
- Tricki, a wiki-style collection of mathematicians' favorite problem-solving tricks and techniques
- Robert Ash's website
- Math Reference Project, maintained by Karl Dahlke
- The Mathematical Atlas, a site outlining the main areas of mathematics, sorted by MSC code
- Online number theory lecture notes and teaching materials

- LaTeX Wikibook
- Detexify, a website to help you find the LaTeX symbol you're looking for
TeX.SE, a Q&A forum for all things TeX-related

Some advice on live-TeXing:

- Advice on real-time TeXing, by Anton Geraschenko
- Taking lecture notes in lectures (a MathOverflow thread)
- Chromatic Homotopy II or how I learned to stop worrying and love LaTeXing in Real Time, by Chris Schommer-Pries
- Commu, a program that helps you make commutative diagrams in tikzcd. It is currently only available on Linux, so I haven't even tried it myself yet, but it seems like a great idea.

- How to TeX
- TeX Resources on the Web, organized by the TeX Users Group
- User's Guide for the amsmath Package
- Toward LaTeX coding standards, a talk by Didier Verna
- The Comprehensive LaTeX Symbol List
- The Beamer Theme Matrix
- Using Computer Modern on the web
- ShareLaTeX, an online collaborative LaTeX editor.
- WriteLaTeX, another online collaborative LaTeX editor.
- verbosus, another web platform for LaTeX (also available for Android)
- GmailTeX, a plugin for Gmail that compiles TeX / LaTeX code in your emails

- Journal publishing reform, a wiki site created by Michael Nielsen to collect information, opinion, and activism on this issue
- Math2.0, a forum on mathematical publishing created by Scott Morrison and Andrew Stacey
Fields Medal winner Timothy Gowers has blogged about this extensively. Some of his posts:

- Elsevier journals — some facts
- Why I've Joined the Bad Guys, and Why I've Also Joined the Good Guys
- A new open-access venture from Cambridge University Press
- Elsevier's open letter point by point, and some further arguments
- Abstract thoughts about online review systems
- Elsevier - my part in its downfall (follow-up post here)
- How might we get to a new model of mathematical publishing?

- The Economics of JSTOR, by John Levin
- Locked Out
- More on commenting and the publishing reform, by Izabella Laba
- The Future of Mathematical Publishing, by Michael Cowling
- The High Cost of Scholarly Journals (And What to Do About It), by Richard Edwards and David Shulenburger. Ironically, you must have JSTOR access to read the article.
- Why scientists are boycotting a publisher, by Gareth Cook
- Value removed by journals, by Noah Synder
- Is Scientific Publishing About to be Disrupted? by Michael Nielsen
- Want to Change Academic Publishing? Just Say No, by Hugh Gusterson
- Jumping Ship:
*Topology*Board Resigns, by Allyn Jackson (see their resignation letter) - Review of
*The Access Principle*by John Willinsky, by Scott Aaronson - Declaring Independence: Returning Scientific Publishing to Scientists, by Alison Buckholtz
- In Win for Libraries Over Publishers, Supreme Court Upholds Reselling of Foreign Books, by Jennifer Howard
- The Scholarly Publishing and Academic Resources Coalition (SPARC)
- PR's 'pit bull' takes on open access, an article in
*Nature*by Jim Giles - The International Mathematical Union's Blog on Mathematical Journals
- I confess, I wrote the Arsenic DNA paper to expose flaws in peer-review at subscription based journals, by Michael Eisen
- An editable database tracking freely accessible mathematics literature
- Wake Up!, a blog post by David Mumford

- The AMS's collection of links for math students
- The MAA's collection of links for math students
- The Mathematics Genealogy Project
- As If Summoned from the Void: The Life of Alexandre Grothendieck, by Allyn Jackson (part 2 is here)
- Obvious But Underappreciated Truths, by Alan Stein
- The Tau Manifesto, by Michael Hartl
- Content creation: the organization and dissemination of knowledge, by Rice Issa
- Diana Davis' submission for the "Dance Your Ph.D." contest"
- Mathematica.SE, a Q&A forum for all things Mathematica-related
- Learn your Greek letters! See here and here.
- Make a randomly generated theorem, by David Simmons-Duffin
- Make a randomly generated mathematics paper, by Nate Eldredge
- Dror Bar-Natan's Academic Penseive, an excellent example of open notebook science.
- Mathematical Fiction, a list compiled by Alex Kasman
- The Feeling of Power, a somewhat dated but still excellent short story by Isaac Asimov
- Logic and Terror by Adrian Mathias, about the politicization and suppression of logic in the Soviet Union. I learned about this article from this MathOverflow thread.
- The Rising Sea: Grothendieck on simplicity and generality I, by Colin McLarty
- An article (and a follow-up article) by Keith Devlin debunking most of the popular mythology about the "Golden Ratio"
- What is a Closed-Form Number?, by Timothy Chow
- A neat video (part 1 and part 2) giving an excellent explanation of the sphere eversion.
- Möbius Transformations Revealed
- An interview with Sir Michael Atiyah about beauty in mathematics
- A documentary about the proof of Fermat's Last Theorem
- Matrices and Their Singular Values, a short film about SVD made at Los Alamos National Laboratory
- Integration: The Feyman Way, a brief explanation of differentiation under the integral sign
- The History of Mathematical Symbols, by Douglas Weaver
- Notating Discovery - Discovering Notation, by John Stell
- The artist Bernar Venet uses diagrams and text from mathematics texts in his artwork. See also this catalog and this article in the Notices of the AMS.
- Here is a detailed guide of what you need to learn algebraic number theory by Sam Ruth.
- How might I improve my math department as an undergrad?
- Some disasters caused by numerical errors, by Kees Vuik
- Verizon representatives think 0.002 cents is equal to 0.002 dollars
- A nice, simple explanation of WZ theory.
- A review of Stephen Wolfram's
*A New Kind of Science*. - A Sampling of Mathematical Folk Humor, by Paul Renteln and Alan Dundes
- A Contribution to the Mathematical Theory of Big Game Hunting, by Ralph Boas, Jr.
- Here are some stories collected by J. S. Milne.
- Mathematical Miniatures, by Beno Eckmann
- The Crackpot Index, by John Baez
- Ten Signs a Claimed Mathematical Breakthrough is Wrong, by Scott Aaronson
- Advice for amateur mathematicians on writing and publishing papers, by Henry Cohn
- Beyond Papers - GitWikXiv, by Jess Riedel
- The Shadow Scholar, and follow-up article An Academic Ghostwriter Comes Clean
- Sokal's Hoax, by Steven Weinberg
- What the Sokal Hoax Ought to Teach Us, by Paul A. Boghossian
- How to Deconstruct Almost Anything - My Postmodern Adventure, by Chip Morningstar
- Some fascinating research done by Doug Zongker.
- This is a news website article about a scientific paper, by Martin Robbins
- Five Math Experts Split the Check
- Can up to 70% of scientific studies not be reproduced?
- Why Most Published Research Findings Are False, by John P. A. Ioannidis
- Retraction Watch, a blog that tracks retracted scientific papers and related scandals