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:
- NSF Graduate Research Fellowship Program
- NDSEG Fellowship
- Hertz Foundation Fellowship
- The Churchill Scholarship
- Ford Foundation Fellowship
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