This page is split up into two sections. The first, and most important, is a set of resources I myself have found useful in understanding concepts. This includes, books, papers, notes, slides, problem sets, etc. The rest is a more general list of essays, articles, comments, videos, and questions that are interesting and useful to consider.
This was heavily inspired by the student website of Zev Chonoles.
$\S$ Mathematics (Resources)
Category Theory
- Kan Extensions: An Exploration by Matt Capetola | archive
- All Concepts are Kan Extensions by Marina Lehner | archive
- Applied Category Theory Lectures
- Category Theory by Randall Holmes | archive
Simplicial Constructions
- Simplicial Sets vs. Simplicial Complexes | archive
- CW Complexes with Simplicial Structures - Appendix by Allen Hatcher | archive
- Definition of “simplicial complex” | archive
- An elementary illustrated introduction to simplicial sets by Greg Friedman | archive
- Introduction to Combinatorial Homotopy Theory by Francis Sergeraert | archive
- A Leisurely Introduction to Simplicial Sets by Emily Riehl | archive
- The Geometric Realization of a Simi-Simplicial Complex by John Milnor | archive
- $\Delta$-sets I: Homotopy Theory by Rourke, Sanderson | archive
Model Categories
- Homotopy Theories and Model Categories by Dwyer, Spalinski | archive
- Higher Dimensional Categories by Emily Riehl | archive
Homotopy Theory (The basics)
- Fifty Years of Homotopy Theory by G. W. Whitehead
- What Every Young Topologist Should Know by Davis, Kirk | archive | v1
- Lectures on Derived and Triangulated Categories by Behrang Noohi | archive
- Lectures on Derived Categories by Dragan Milicic | archive
- The Stable Homotopy Category by Cary Malkiewich | archive
- An Introduction to Stable Homotopy Theory | archive
- Introduction to Stable Homotopy Theory | archive
- Future Directions by Tyler Lawson | archive
- The Future of Homotopy Theory by Clark Barwick | archive
Homotopy (Co)limits
- A Primer on Homotopy Colimits by Daniel Dugger | archive
- Categorical Homotopy Theory by Emily Riehl | archive
- Notes on Homotopy (co)limits by Philip Hirschhorn | archive
- Homotopy (Limits and) Colimits by Emily Riehl | archive
- Exercises on Homotopy Colimits by Samuel Baruch Isaacson | archive
Motivic Homotopy Theory
- $\mathbb A^1$ Homotopy Theory by Vladimir Voevodsky | archive
- Prerequisites in Algebraic Topology by Bjørn Dundas | archive
- Prerequisites in Algebraic Geometry by Marc Levine | archive
- Nordfjordeid Lectures by Vladimir Voevodsky | archive
- Minicourse on Abstract and Motivic Homotopy Theory by Peter Arndt
- An Intuitive Introduction to Motivic Homotopy Theory by Vladimir Voevodsky
- An Introduction to Motivic Homotopy Theory by Oliver Röndigs
- A Primer for Unstable Motivic Homotopy Theory by Antieau, Elmanto | archive
- Lecture Notes on Motivic Cohomology by Mazza, Voevodsky, Weibel | archive
Topos Theory
- Rethinking Set Theory by Tom Leinster | archive
- Nonetheless One Should Learn the Language of Topos by Colin McLarty
- How Grothendieck Simplified Algebraic Geometry by Colin McLarty | archive
- ZFC and ETCS: Elementary Theory of the Category of Sets by Todd Trimble | archive
- ETCS: Internalizing the logic by Todd Trimble | archive
- [ETCS: Building joins and coproducts](ETCS: Building joins and coproducts) by Todd Trimble | archive
- Constructing Material and Structural Set Theories by Mike Shulman | archive
- Set Theory for Category Theory by Mike Shulman | archive
- An Introduction to Topos Theory by Ryszard Paweł Kostecki | archive
Higher Category Theory
- The algebra and geometry of $\infty$-categories by Emily Riehl | archive
- Infinity category theory from scratch by Emily Riehl | archive
- Elements of $\infty$-Category Theory by Riehl, Verity | archive
- Notes and exercises on $\infty$-categories by Mitya Boyarchenko | archive
- An Introduction to $\infty$-categories by Hiro Tanaka | archive
- Intrinsic Naturalness by Urs Schreiber | archive
- A Perspective on Higher Category Theory by Tom Leinster | archive
- Confessions of a Higher Category Theorist by Mike Shulman
(Co)Sheaf Theory
- Sheaves Do Not Belong to Algebraic Geometry by Tom Leinster (pdf) | archive
- A Gentle Introduction to (Co)homology and Sheaf Cohomology | archive
- Canonical Example of a Cosheaf | archive
- What would be the equivalent of the “gluing axiom” for a cosheaf | archive
Homological Algebra
Homotopy Type Theory
- HoTT-Wiki:References | archive
- Homotopy Type Theory : Project Description by Steve Awodey | archive
- Type Theory and Homotopy by Steve Awodey | archive
- An Introduction to Univalent Foundations of Mathematics by Daniel Grayson | archive
- Homotopy Type Theory: What is it? | archive
- A Taste of Type Theory by Bartosz Milewski, 2019
- Exploring Type Theory by Bartosz Milewski
- Is it possible to learn homotopy theory from the perspective of HoTT? | archive
- HoTT is not an interpretation of MLTT into abstract homotopy theory | archive
- Are simplicial sets the intended model of HoTT? | archive
- Prospects for reverse mathematics in Homotopy Type Theory | archive
- James Ladyman (Bristol): " The Hole Argument and Homotopy Type Theory"
- The Origins and Motivations of Univalent Foundations by Vladimir Voevodsky | archive
- Homotopical Trinitarianism by Mike Shulman | archive
- Induction on Equality by Mike Shulman | archive
Stacks
- Homotopical Interpretation of Stacks by Fosco Loregian | archive
Frames and Locales
- The Point of Pointless Topology by Peter Johnstone
Logic
- A Beginner’s Guide to Forcing | archive
- Proof of negation and proof by contradiction | archive
- Diagonal Argument
Type Theory
Constructive Mathematics
- The weird and wonderful world of constructive mathematics By Mike Shulman | archive
Commutative Algebra
- Algebra in Bishop’s style by Henri Lombardi | archive
Algebraic Geometry
- Sur quelques points d’algèbre homologique [English] a.k.a., “the Tôhoku paper” by Alexander Grothendieck | archive
- Grothendieck’s yoga of six operations - in relatively basic terms? | archive
Differential Geometry
Algebraic K-Theory
- Attitudes of K-Theory by Inna Zakharevich | archive
Cellular Automata
- Global Properties of Cellular Automata by Ed Powley | archive
- Cellular Automata
- Elementary Cellular Automata (in Python)
- Game of Life
- The Game of Life : A Beginner’s Guide | archive
- Smooth Game of Life | Smooth Life
- Conway’s Game of Life for Curved Surfaces (Part 1)
- Conway’s Game of Life for Curved Surfaces (Part 2) | There is an interesting documentary about deep sea life; their luminosity seems to resemble smooth curved game of life on surfaces.
- How to calculate CellularAutomaton rule numbers in higher dimensions? | archive
- Cellular Automata Theory | archive
- Life Lexicon | archive
$\S$ Mathematics (General)
Mathematics (Essays)
- “Theoretical Mathematics” by Jaffe, Quinn | archive
- On Proof and Progress in Mathematics by Bill Thurston | archive
- Response to Comments on “Theoretical Mathematics” | archive
- On Contemplation in Mathematics by Frank Lucas Wolcott| archive
- What is good mathematics? by Terence Tao | archive
- A Mathematician’s Survival Guide by Peter G. Casazza | archive
- Missed Opportunities by Freeman Dyson | archive
- The Unreasonable Effectiveness of Mathematics in the Natural Sciences By Eugene Wigner | archive
- Does mathematics need a philosophy? | archive
Mathematics (General)
- What’s a mathematician to do? | archive
- How to study math to really understand it and have a healthy lifestyle with free time? | archive
- Do you prove all theorems whilst studying? | archive
- On “familiarity” (or How to avoid “going down the Math Rabbit Hole”?) | archive
- Math and mental fatigue | archive
- Finding and Solving Problems by John Baez | archive
- What books must every math undergraduate read? | archive
- What remains in a student’s mind | archive
- Best Sets of Lecture Notes and Articles | archive
- Effective Research Notes | archive
- Ways to study mathematics while commuting | archive
- Mathematical literature to lose yourself in | archive
- I Hate Axioms by Qiaochu Yuan | archive
- There’s more to mathematics than rigour and proofs | archive
- It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that “fuzzier” or “intuitive” thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as “non-rigorous”. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education.
- Is there any secret in being good at math?
- Ignore everything you learned in school. There is no such thing as subjects. Learn everything at the same time, and continuously. There is also no such thing as books, so don’t worry about finishing them; all of the math out there is a giant book you will never finish reading.
- Going beyond your comfort zone by Qiaochu Yuan | archive
- On “compilation errors” in mathematical reading, and how to resolve them | archive
- Why doesn’t mathematics collapse even though humans quite often make mistakes in their proofs? | archive
- Some thoughts on doing a PhD in topology/geometry | archive
- Ten Lessons I wish I had been Taught | archive
Everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned.
- Does mathematics need a philosophy? | archive
- Source for analysis of identification of structures in learner’s mind and mathematical structures? | archive
- Good papers/books/essays about the thought process behind mathematical research | archive
Mathematics (Roadmaps)
- Path to Basics in Algebraic Geometry from HS Algebra and Calculus? | archive
- Roadmap for learning Topological Data Analysis?
- Learning roadmap for Knot theory
- Learning Roadmap for Algebraic Topology
- Roadmap for Mirror Symmetry | archive
Mathematics (Specific)
- Category Theory
- Algebraic Topology
- Commutative Algebra
- Algebraic Geometry
- Advice for potential graduate students in Arithmetic Geometry by David Zureick-Brown | archive
- For potential Ph.D. students by Ravi Vakil | archive
- What should be learned in a first serious schemes course? | archive
- Hartshorne or Vakil’s notes | archive
- Why learning modern algebraic geometry is so complicated? | archive
- What are the required backgrounds of Robin Hartshorne’s Algebraic Geometry book? | archive
- Best Algebraic Geometry text book? (other than Hartshorne) | archive
- Material in a first course in algebraic geometry? | archive
- Analysis
Topology (Applied)
- Studying the Shape of Data Using Topology | archive
- From Topological Data Analysis to Deep Learning: No Pain No Gain | archive
- Measuring Shape With Homology by Robert MacPherson
Historical
- A Country Known Only By Name by Pierre Cartier | archive
Philosophy of Mathematics
- Filosofía sintética de las matemáticas contemporáneas by Fernando Zalamea | archive
- Democracy and Mathematics by André Joyal | archive
$\LaTeX$
- Help:Displaying a formula
- What are good advanced LaTeX books? | archive
- How to write diagram chases | archive
$\S$ Academia
General
- Science as a Vocation by Max Weber | archive
- The Scholar’s Vocation
Productivity
- Productivity tips, tricks and hacks for academics (2015 edition) by Matt Wright | archive
- Boost your productivity: Cripple your technology by Matt Wright | archive
- 12 resolutions for grad students by Matt Might | archive
- 12 resolutions for programmers by Matt Might | archive
- End artificial scarcities to increase productivity by Matt Might | archive
- Tips for work-life balance by Matt Might | archive
- How do advisors typically apply pressure on PhD students to get work done? | archive
Motivation
- How should I deal with discouragement as a graduate student? | archive
- How to avoid procrastination during the research phase of my PhD? | archive
- How to stop feeling guilty about unfinished work? | archive
- What Else Should Grad Students Be Learning? | archive
Open Access
- As More Universities ‘Ditch’ Elsevier, Sci-Hub Blossoms | archive
- The Cost of Knowledge by John Baez | archive
- Nature Announces Support for Preprint Papers, Drops Ingelfinger Rule | archive