​Lean, Univalent foundations and Type Theory are interesting. And 3Blue1Brown has incredible explanations for various math topics.

Notes

• Math can be learned, it's just a bit silly to dive into a homotopy type theory book and expect to get a firm grasp of things. It's like picking up a Chinese news paper and expecting to get an idea of what its content is about.

• Without understanding the domain, you cant read a text well. Without understanding the grammar you cant read a text. Without understanding the symbols you cant even figure out the grammar.

• The prevailing view in modern mathematics is that structure is all that matters. i.e. isomorphic objects are really the same, they don't just look the same or act the same, they really are the same thing.

• There's a set of basics that you will want no matter which direction you go: calculus/real analysis, linear algebra, differential equations/dynamical systems, and sets, groups, rings, and lattices.

  • Calculus: learn to extract qualitative information about a function (it goes up here, has a maximum there, goes down there, oscillates with an increasing period, goes to this value at infinity...) and to numerically compute quantitative information about it (its value at 3 is blah, its integral over this interval is blah, its maximum value is blah).

  • Linear algebra: vector spaces and linear operators, and their representation as vectors and matrices. Functions as forming infinite dimensional spaces, and Banach and inner product spaces.

  • Differential equations and dynamical systems: extending what we did for calculus to differential equations. Phase space, orbits, Fourier and Laplace transforms, sets of linear differential equations, numerical integration, some partial differential equations. You do not need all the little tricks for special kinds of equations that you will find in, say, Boyce and dePrima. They're not helpful.

  • Sets, groups, rings, and lattices. Mathematics today is written in terms of set theory. You need to understand the basics of manipulating sets and functions between them. Then you should know something about groups, rings, and lattices, which are the most ubiquitously useful algebraic structures besides vector spaces.

  • After that, where you go is going to vary enormously. Based on what you're aiming to do.

• ​Some topics and books to learn from:

  1. Topology (book by Munkres)

  2. Real Analysis and Measure Theory (book series by Stein Shakarchi)

  3. Algebra (book by Aluffi)

  4. Linear Algebra (book by Friedberg Insel)

  5. Measure Theoretic Probability (book by Cinlar)

  6. Differential Geometry (book Smooth Manifolds by Lee)

  7. Numerical Analysis (book by Quarteroni)

  8. Set Theory and Propositional Logic (books by Goldrei)

• Or these books for more fundamental basic knowledge:

  1. Higher algebra - Hall and Knight

  2. Trigonometry I - Loney

  3. Coordinate geometry - Loney

  4. Calculus of one variable - Maron

• ​If you can, find someone else to work with, and someone else who already knows some math to ask occasional questions would probably help you a lot.​

Links

• ​What does it feel like to invent math?​

• ​How to read Mathematics​

• ​The Banach–Tarski Paradox​

• ​John Conway: Genius Behind The Numbers​

• ​Moiré pattern in JavaScript - Beautiful.

• ​Tools for thought​

• ​Making things with maths​

• ​The Map of Mathematics​

• ​How to self learn math​

• ​Math books​

• ​Mathematics for Computer Science: Readings​

• ​mathematicalmonk YouTube playlists - A lot of stuff covered.

• ​How Grothendieck Simplified Algebraic Geometry​

• ​A new “Mathematician’s Apology”​

• ​Interesting theorems​

• ​Ask HN: How do I learn math/physics in my thirties?​

• ​Napkin Project​

• ​Learn Math Fast - Curated list of awesome mathematics resources.

• ​Russian wiki on math & finance​

• ​Math for CS MIT Course​

• ​MSC2010 - Huge catalog of math topics.

• ​Cambridge math notes​

• ​Mathematics as thought​

• ​Penrose - Language-based platform for creating beautiful mathematical diagrams.

• ​Domain-Specific Languages of Mathematics​

• ​A Programmer's Introduction to Mathematics book​

• ​How to learn Math and Physics (2018)​

• ​Ask HN: How to self-study mathematics from the undergrad through graduate level? (2019)​

• ​How to Science [Part 2: Our Universe = Math?] - Amazing series.

• ​University of Oxford notes and exercises for math undergraduate courses​

• ​MacTutor History of Mathematics archive​

• ​Formalizing 100 Theorems​

• ​Some Fundamental Theorems in Mathematics (2018) (HN)

• ​Some quotes from Robert Harper​

• ​Hessix - HN-like math research discussion for Ph.D.s (HN)

• ​Curv - Language for making art using mathematics.

• ​Mathematics For Machine Learning book (2019)​

• ​All in One Mathematics Cheat Sheet (HN)

• ​too simple to be simple​

• ​Self Studying the MIT Applied Math Curriculum (2019) (HN)

• ​Timeline of Mathematics​

• ​Ask HN: Where can I do a self-study math degree online? (2019)​

• ​Beautiful Math - New book shows off spectacular works of art inspired by mathematical principles.

• ​Cardano's formula (2019) (Lobsters)

• ​Collection of computational playthings (OSS)

• ​Approach0 - Math-aware search engine. (OSS)

• ​Think in Math. Write in Code. (2019) (HN)

• ​Analysing Mathematical Reasoning Abilities of Neural Models (2019)​

• ​Cambridge Maths Outline (2019)​

• ​Dexter Chua's math notes​

• ​Math notes by Gareth Taylor​

• ​How to Become a Pure Mathematician (or Statistician)​

• ​Chicago undergraduate mathematics bibliography​

• ​Ask HN: Best resources to gain math intuition? (2019)​

• ​The Riemann Hypothesis (2019) (HN)

• ​Nonlinear Dynamics and Chaos lectures​

• ​The Future of Mathematics? (2019)​

• ​CoCalc - Collaborative Calculation in the Cloud.

• ​Prime Numbers and the Riemann Hypothesis​

• ​nLab - Wiki-lab for collaborative work on Mathematics, Physics and Philosophy — especially from the n-point of view: in so far as these subjects are usefully treated with tools and notions of category theory or higher category theory.

• ​Understanding Mathematics​

• ​Deep learning for symbolic mathematics (2019)​

• ​Eigenvectors from eigenvalues (2019)​

• ​Cantor's Paradise Math Blog​

• ​Better intuition for information theory (2019)​

• ​Mathematics Free and Open-Source Textbooks (HN)

• ​Stuff I learnt in 2019 (Reddit)

• ​T * sin (t) ≈ Christmas tree (2013) (HN)

• ​What was the hardest area of math for you? (2020)​

• ​Expository papers by K. Conrad​

• ​Math for the Layman​

• ​Leonhard Euler | The Greatest Mathematician To Ever Live (2020)​

• ​MIT Mathematics for Computer Science course (2015) - Serves as an introduction to discrete mathematics, probability, and mathematical thinking for computer scientists.

• ​Who are some lesser known mathematicians, and what are some of their accomplishments, or interesting facts about them​

• ​Information theory​

• ​Timothy Gowers: The Importance of Mathematics​

• ​Oxford Undergraduate Courses in Math​

• ​Oxford Postgraduate Courses in Math​

• ​An inquiry into the Foundations of Mathematics (2020)​

• ​Anthony Várilly-Alvarado, "The Geometric Disposition of Diophantine Equations." (2020)​

• ​Mathpendium - Discover the world of mathematics.

• ​Where is the fashionable mathematics? (2020)​

• ​The Map of Mathematics (2020)​

• ​Introduction to Clifford Algebra (HN)

• ​Grant Sanderson's TED talk on effective teaching of math (2020)​

• ​Fields Institute - International centre for scientific research in mathematical sciences at the University of Toronto.

• ​HN: Maryam Mirzakhani​

• ​Mathematics for the adventurous self-learner (2020) (HN)

• ​Math is your insurance policy (2020) (HN)

• ​Discrete MathematicsAn Open Introduction​

• ​Lecture notes from Cambridge maths (Code)

• ​Cambridge Maths Tripos notes​

• ​Real Not Complex: Free Math Resources (HN)