​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.​

• ​Monad on Set is a way to describe some sort of gadget that's a set equipped with a bunch of operations obeying a bunch of equations​

• ​There's no closed-form solution for the motion of 3 gravitational masses in space, because solving that would tell us whether some guy's pure math construct halts or not.​

• ​To learn math/anything: 1. Grab a book/paper (try to understand what it's teaching). 2. Do exercises to understand (or build something). 3. Repeat.​

Links

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

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• ​Understanding Mathematics​

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• ​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​

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• ​HN: Maryam Mirzakhani​

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• ​Short book recommendations on advanced topics? (2020)​

• ​Are there any notable mathematicians/scientists that struggled with math but became prominent in their fields? (2020)​

• ​Intuitive Explanations - Understand, don't memorize. (Code)

• ​Books to study math from first principles (2020)​

• ​The formula for pi buried in a hydrogen atom (HN)

• ​Math Seminars​

• ​Abstract Algebra Cheatsheet​

• ​HN: Ladder of Algebraic Structures (2020)​

• ​Math3ma - Math blog by Tai-Danae Bradley.

• ​Princeton Companion to Mathematics book (2007)​

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• ​What was the math that blew your mind the most, at any point in your physics career? (2020)​

• ​Springer Math Books​

• ​Advanced Math resources​

• ​Math & Programming blog main content​

• ​Project MOSAIC - Community of educators working to develop a new way to introduce mathematics, statistics, computation and modeling to students in colleges and universities.

• ​The Psychology of Advanced Mathematical Thinking​

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• ​Graduate Student Solves Decades-Old Conway Knot Problem (2020) (HN)

• ​What Made me a Mathematician: Off-the-Cuff Story Telling for a Story-Telling Class (2020)​

• ​Mathematical Notes by Prof. Steven G. Johnson​

• ​The Definitive Glossary of Higher Math Jargon​

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• ​karasu - Self-hosted remote communication system designed for math discussions. (Web)

• ​MathPages​

• ​Bill Thurston's answer to “What's a mathematician to do?” (HN)

• ​Visual Math Tools​

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• ​Ask a Mathematician / Ask a Physicist​

• ​Important concepts in numerical analysis and related areas​

• ​Bi-whatever transformations​

• ​The Mathematical Hacker (2012) (Lobsters)

• ​Ask HN: Good articles/courses for understanding mathematical notation? (2020)​

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• ​Semigroups and Monoids (Lobsters)

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• ​Ask HN: Which general programming language is good for learning/exploring math? (2020)​

• ​Theta - Web app that graphs an equation and allows you to pan and zoom around the coordinate grid. (Code)

• ​Algebraic Stacks Project - Collaborative web-based project writing a text book on algebraic stacks and the algebraic geometry that is needed to define them. (Code)

• ​Why is there no equation for the perimeter of an ellipse‽ (2020)​

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• ​The Map of Mathematics (HN)

• ​Graphtoy (HN)

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• ​CalcuLaTeX - Pretty printing calculator language. (Code) (HN)

• ​Defining Visual Narratives for Mathematics Declaratively (2019)​

• ​Byrne’s Euclid - First Six Books of The Elements of Euclid With Coloured Diagrams and Symbols.

• ​Can One Hear the Shape of a Drum? (HN)

• ​Complete Proof of the Collatz Conjecture (2021)​

• ​A Guide to Writing Mathematics (HN)

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• ​How to Solve It Book by George Pólya (1945)​

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• ​How entropy can be seen as a volume - the geometric interpretation of information theory (2021) (Reddit)

• ​How (and why) to raise e to the power of a matrix (2021)​

• ​ELI5 what is chaos theory, and what is it used for? (2021)​

• ​What are the best/most interesting Springer Math Books? (2021)​

• ​What Became of Moses Schönfinkel, Inventor of Combinators (2021)​

• ​Mathematical Markup Language (MathML) (Code)

• ​Map of Mathematical Structures (MIT)​

• ​Why do you like math? (2021)​

• ​Oliver Heaviside (1850-1925) – Physical Mathematician (1983) (HN)

• ​Galois Theory (2021)​

• ​Useful inequalities cheat sheet​

• ​The Two Cultures of Mathematics (2000) (HN)

• ​Good books to relearn math (2021)​

• ​Intuitive Guide to Hyperbolic Functions​

• ​Cognicull - Learning site that pursues "ease of understanding" and "comfort".

• ​Which was the first book that gave you the taste of real mathematics? (2021)​

• ​How to Code a Math Formula (2020) (Reddit)

• ​What's your favorite "strange" function? (2021)​

• ​The Textbook That Unleashed Ramanujan's Genius (HN)

• ​Abakcus - Collection of the Best Mathematics and Science Stuff. (Twitter)

• ​The Essence of Mathematics Through Elementary Problems (HN)

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• ​What's your favorite proof? (2021)​

• ​Math Memes Reddit​

• ​Dr Peyam - YouTube​

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• ​The Utility of Mathematics (HN)

• ​Convergence of Infinite Products (2008)​

• ​The Poincaré Conjecture, Explained (2021) (HN)

• ​Discrete Optimization Course​

• ​A Bestiary of Functions for Systems Designers (2021)​

• ​Steven Strogatz: In and out of love with math | 3b1b podcast (2021)​

• ​Math texts, pi creatures, problem solving, etc. | 3blue1brown Q&A (2021)​

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• ​SageMath - Open-Source Mathematical Software System. (Code)

• ​Banach-Tarski and the Paradox of Infinite Cloning (HN)

• ​A Layman’s Guide to Recreational Mathematics Videos (2021) (HN)

• ​MIT Mathlets (HN)

• ​Elliptic Curve Explorer​

• ​Visually stunning math concepts which are easy to explain (HN)

• ​Cheat-sheet for mathematical notation in code form​

• ​Eleven proofs of the Gaussian integral (HN)

• ​A practical guide to radial basis functions​

• ​Introduction to Higher Mathematics​

• ​Helmhurts (2014)​

• ​A mathematician's guided tour through higher dimensions (2021) (HN)

• ​The Dedekind tessellation​

• ​Imagination and the Impossible (2021)​

• ​Designer Maths (2021)​

• ​Looking through Objects - How Tomography Works! (2021) (Tweet)