- nLab for abstract nonsense (and mLab for actual nonsense).
- Overleaf for collaborative LaTeX projects.
- Math Stackexchange and MathOverflow for questions.
- TeX Stackexchange for TeX support and Detexify for looking up symbols.
- TikZ-cd for diagrams, and Yichuan Shen’s editor or quiver for help getting started.
- SageMath, GAP, Singular, OSCAR and Macaulay2 for computations.
- The Youtube channel 3blue1brown.
- The blog Math3ma.
- The podcast My Favorite Theorem.
- The blog Math ∩ Programming.
- The magazine Quanta Magazine.
Stories and Inclusion
- Spectra for LGBTQ+ mathematicians and allies.
- The ebook Living Proof.
- The forall Instagram account.
- Federico Ardila’s four axioms.
- The Math-Life Balance youtube channel.
- Ravi Vakil’s advise on seminars.
- Dan Margalit’s resource pages.
- Matt Might's blog.
Commutative Algebra and Algebraic Geometry
- A webinar with Ravi Vakil and Bernd Sturmfels about what algebraic geometry (and mathematics in general) is.
- Lecture notes on Commutative Algebra and Algebraic Geometry by Andreas Gathmann.
- Lecture notes on varieties by Geir Ellingsrud and John Christian Ottem.
- A visualization of 27 lines on a cublic by Paul Masson.
- A Brief Introduction to Schemes and Sheaves by David Urbanik.
- Lectures on commutative algebra and algebraic geometry by Seidon Alsaody.
- Introduction to Schemes by Geir Ellingsrud and John Christian Ottem.
- List of algebraic geometry definitions and theorems by Christopher Keyes.
- The historical development of Algebraic Geometry by Jean Dieudonné.
- The Rising Sea by Ravi Vakil.
- The Stacks Project.
- “Introduction to Tropical Geometry” by Diane Maclagan and Bernd Sturmfels.
- Twelve Lectures on Tropical Geometry by Bernd Sturmfels.
- An introduction to tropical geometry: theory and applications by Fatemeh Mohammadi.
- Lectures from the Algebraic and Tropical Online Meetings (ATOM).
- Tropical Geometry of Curves by Madeline Brandt.
- Essentials of Tropical Combinatorics by Michael Joswig.
- Introduction to Toric Geometry by Simon Telen.
- Ibadan Lectures on Toric Varieties by Frank Sotille.
- A Video Course on Toric Varieties by Jürgen Hausen.
- “Toric Varieties” by David Cox, John Little and Hal Schenck.
Applied and Computational Algebraic Geometry
- Solving polynomial equations and applications by Simon Telen.
- “Ideals, Varieties, and Algorithms” and “Using Algebraic Geometry” by David A. Cox, John Little, and Donal O’shea.
- Introduction to Non-Linear Algebra by Mateusz Michalek and Bernd Sturmfels.
- Applications of Polynomial Systems (a lecture series by David Cox and invited speakers).
- Nonlinear algebra and applications by Paul Breiding et al.
- Biochemical reaction networks: An invitation to algebraic geometers by Alicia Dickenstein.
- The Mathematics of Reaction Networks seminar.
Numerical Algebraic Geometry
- The HomotopyContinuation.jl package.
- Lectures from the SANNA 2021 workshop at MPI Leipzig.
- “The numerical solution of systems of polynomials arising in engineering and science” by Andrew J. Sommese and Charles W. Wampler, II.
- “Contemporary Linear Algebra” by Howard Anton and Robert C. Busby.
- Essense of Linear Algebra by 3blue1brown.
- “Numerical Linear Algebra” by Lloyd N. Trefethen and David Bau, III.
- “Abstract Algebra: Theory and Applications” by Thomas W. Judson.
- “Algebra: Chapter 0” by Paolo Aluffi.
- “Representation Theory” by William Fulton and Joe Harris.
- Lecture notes by Thomas Krämer.
- “Introduction to Representation Theory” by Pavel Etingof et al.
- “Quantum Groups” by Christian Kassel.
- “Naive Lie Theory” by John Stillwell.
- “Lie Groups, Lie Algebras, and Representations” by Brian C. Hall.
- “Representations of Compact Lie Groups” by Theodor Bröcker and Tammo tom Dieck.
- “Lie Groups Beyond an Introduction” by Anthony Knapp.
- “An Introduction to Homological Algebra” by Joseph J. Rotman.
- An 80 minutes introduction to homological algebra by Rishi Vyas.
- The snake lemma makes a short cameo in It’s My Turn (1980).
- “A User’s Guide to Spectral Sequences” by John McCleary.
- “Topology” by James Munkres.
- “Introduction to Topological Manifolds” by John M. Lee.
- “Counterexamples in Topology” by Arthur Seebach Jr. och Lynn Steen.
- Topology: A Categorical Approach by Tai-Danae Bradley, Tyler Bryson, and John Terilla.
- “Topology from a Differentiable Viewpoint” by John W. Milnor.
- “Algebraic Topology” by Allen Hatcher.
- Lectures on Introductory Algebraic Topology by Pierre Albin.
- Lecture notes by James F. Davis and Paul Kirk.
- “A Concise Course in Algebraic Topology” by Peter May.
- “The Knot Book” by Colin C. Adams.
- “Knot Theory” by Charles Livingston.
- “Knots Knotes” by Justin Roberts.
- “An Introduction to Knot Theory” by Raymond Lickorish.
- The Knot Atlas.
- Videos from Roger Fenn’s and Louis Kauffman’s 2020 knot theory course.
- Virtual Low-Dimensional Topology.
- “Differential Geometry of Curves and Surfaces” by Kristopher Tapp.
- “Introduction to Smooth Manifolds” by John M. Lee.
- “Riemannian Geometry” by Sigmundur Gudmundsson.
- Lectures on the Geometrical Anatomy of Theoretical Physics by Fredric Schuller.
- “Riemannian Manifolds: An Introduction to Curvature” by John M. Lee.
- “Riemannian Geometry” by Manfredo do Carmo.
- “A Comprehensive Introduction to Differential Geometry” by Michael Spivak.
- “Introductory Functional Analysis with Applications” by Erwin Kryszig.
- “A Course in Functional Analysis” by John B. Conway.
- Lecture notes by Alan Sokal.
- A Mathematician’s Lament by Paul Lockhart.
- The Ideal Mathematician by Phillip J. David and Reuben Hersh.
- Practical suggestions for mathematical writing by Bjorn Poonen
- A TeX Stackexchange thread about things to check in software-generated bibliographic entries.
- Mathematical Writing by by Donald Knuth, Tracy Larrabee, and Paul Roberts.
- “Mathematics++” by Ida Kantor, Jiří Matousek and Robert Šáma.
- Keith Conrad’s notes collection on a wide array of subjects.
- An Infinitely Large Napkin by Evan Chen.
- “Introduction to Experimental Mathematics” by Søren Eilers and Rune Johansen.
Just for fun
- Finite Simple Group (of Order Two) by The Klein Four (and everything else on the album Musical Fruit Cake).
- The Grothendieck Song by Richard Elwes.
- A lecture on finite fields in the twelve halves tounge with Aaron Landesman.
- That’s a Leray!, The Peano Man, Shut Up and Calculate and other alternative song lyrics by Arun Debray.
- Hello, Grad School, No Time and Work Alone by the Computer Science department at UCSD.
- Acme Klein Bottles.