Mathematical links and resources

General Tools

• nLab for abstract nonsense (and mLab for actual nonsense).
• Detexify for looking up symbols.

Popular Math

• The magazine Quanta Magazine.
• The podcast My Favorite Theorem.
• The blog Math ∩ Programming.
• The Youtube channel 3blue1brown.
• The blog Math3ma.

Stories and Inclusion

• Spectra for LGBTQ+ mathematicians and allies.
• The ebook Living Proof.
• The forall Instagram account.
• Federico Ardila’s four axioms.

Advice

• 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 algebraic geometry (and mathematics in general).
• Lecture notes on Plane Curves, 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.

Tropical Geometry

• “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.

Toric Geometry

• 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

• OSCAR and Macaulay2 for symbolic computations.
• 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.

Category Theory

• Category Theory in Context by Emily Riehl.
• The Catsters on Youtube.

Linear Algebra

• “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

• “Abstract Algebra: Theory and Applications” by Thomas W. Judson.
• “Algebra: Chapter 0” by Paolo Aluffi.

Representation Theory

• “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.

Lie Theory

• “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.

Homological Algebra

• “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

• “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.

Knot Theory

• “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

• “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.

Functional Analysis

• “Introductory Functional Analysis with Applications” by Erwin Kryszig.
• “A Course in Functional Analysis” by John B. Conway.
• Lecture notes by Alan Sokal.

About Mathematics

• A Mathematician’s Lament by Paul Lockhart.
• The Ideal Mathematician by Phillip J. David and Reuben Hersh.

Mathematical Writing

• Practical suggestions for mathematical writing by Bjorn Poonen
• A TeX Stackexchange thread about things to check in software-generated bibliographic entries.
• Mathematical Writing by Donald Knuth, Tracy Larrabee, and Paul Roberts.

Miscellaneous

• “Mathematics++” by Ida Kantor, Jiří Matousek and Robert Šáma.
• Richard Borcherd's Youtube lectures.
• Keith Conrad's notes collection on a wide array of subjects.
• An Infinitely Large Napkin by Evan Chen.

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.
• The TeXnique typesetting game.
• Acme Klein Bottles.