Ask HN: Mathematicians, what textbooks are best for learning these math topics? 125 points by smithmayowa on April 9, 2019 | hide | past | favorite | 33 comments So I have just decide to self-learn mathematics up to undergraduate level, and after researching I decided that learning these topics will do the trick for learning to the undergraduate level, but sadly I don't know the best resources(textbooks) to use for easily self-learning them.Pure Mathematics1).Group Theory(rubics cube,e.t.c)2).Order Theory3).Combinatorics(trees,graphs,e.t.c)4).Fractal Geometry5).Topology(cup = donut)6).Measure Theory7).Differential Geometry8).Vector Calculus9).Dynamical Systems(Fliud flow,ecosysytems,Control Theory)10).Chaos Theory(Butterfly effect)11).Complex Analysis(Functions with complex numbers)Applied Mathematics1). Numerical Analysis2). Game Theory3). Probability4). Statistics5). Optimization6). Cryptography7). Computer ScienceFoundations1).Mathematical Logic2).Set Theory3).Category Theory4).Godel Incompleteness TheoremsP.s: I have a diploma in Marine Engineering and so I am not a total noob to math in general.

 I'm not really a mathematician, my experience comes from machine learning / statistics. Note that most math books tend to be known by the name of the author, as opposed to a fixed title.Pure math :1)Group Theory - Milne5)Topology - Mukres6)Measure Theory - Terrence Tao has a course, Robert Ash has a book on probability theory, and it is recommended that you study a bit of real analysis before you do this.7) Differential Geometry and Statistics - Murray and Rice9, 10 ) Non linear dynamics and chaos - Strogatz11) Complex analysis - There exists a set of 4 books covering real, complex and functional analysis by Stein and Shakarchi, which should serve your purposeApplied Math :3) Probability - Grinstead and Snell, Durrett both have good books.4) All of Statistics by Larry Wasserman for a more ML bent to it5) Optimization by Boyd and Vanderberghe7) There's no single topic called "Computer Science", but going with the theme of the topics you are looking at, Algorithms by Cormen, Leiserson et. al, Theory of Computation by Michael Sipser should be good starting points.Note that for a lot of these, you can find high quality material online (Both videos as well as course material). Just do a search for "Topic MIT OCW", replacing "Topic" with your choice.
 Thanks for taking the time to write this list, I will definitely get them and MIT OCW could really come in handy in providing a kind of lecturing feel to this goal.
 Remark: Optimization by Boyd and Vanderberghe is actually Convex optimization. It doesn't cover non-convex stuff.
 Of course, you're correct. I chose to ignore that aspect since Boyd's book covers a lot of basic material, and I'm not actually aware of any such text covering non-convex methods in general ( and considering how fast research is in this space, I'm not sure of any general work, in contrast to work like [1] ).[1] - Non Convex Optimization for Machine Learning - Jain and Kar (https://arxiv.org/abs/1712.07897)
 Textbooks which people on the internet consider very good:>Group TheoryCarter - Visual Group Theory>Complex AnalysisNeedham - Visual Complex AnalysisLess confident recommendations:>Probabilityhttps://ocw.mit.edu/courses/mathematics/18-05-introduction-t... as a very entry level introduction. After that you might still need some textbook for more depth.>Mathematical Logic and Godel Incompleteness TheoremsStart with ForallX by Magnus. Then continue with Computability and Logic by Boolos (don't read all chapters, check out preface to see what you need for Godel Incompleteness Theorems).>Set TheoryHrbacek - Introduction to Set Theory---I am currently developing an online course which teaches logic, set theory, and computability theory (this includes Godel Incompleteness Theorem). It uses the textbooks I've mentioned above. You can check it out at https://app.grasple.com/#/course/141?access_token=3HCK4oRipe.... Use "Fundamentals of formalization" and ignore other tracks.
 Currently going through your course, thanks for taking the time to make something like that.
 I skimmed all the replies below - this is years of learning and study - so I ask:Is there some dependency order someone could quickly sketch out for some of these topics? Eg, linear algebra comes before X?HN is an incredibly useful crowdsourcing resource for the self-motivated!
 The minimal classical way to bootstrap your math knowledge is algebra (mostly linear) + analysis. This is the approach used by e.g. Harvard Math 55, which is a really famous course.Harvard Math 55 employed Halmos + baby Rudin as main textbooks. Halmos has now been replaced by Axler, which is an excellent textbook (but the typographic changes in the last edition are very distracting).Rudin is probably too synthetic and dry for a beginner. You could easily replace it by Hubbard & Hubbard (which was the sole main textbook in one Harvard Math 55 edition), or use an aid text like Gelbaum & Olmsted. You can also skip Axler if you go the Hubbard way.A favorite open question of mine is what would a math bootcamp look like if you went up one level of abstraction and focus more on logic and abstract algebra.
 I would love a bootcamp like that. Or even one that helped you with proofs. I've found that's my biggest issue trying to teach myself pure mathematics. I just can't start the damn proof. Once I get the start, I can usually finish it, but starting the proof I just feel so clueless how to do it.
 You can learn to do basic proofs the way old school Tsarist and Soviet Russia students did, by studying geometry from Kiselev. It should be pretty straightforward for any adult, and a nice stepping stone to build your knowledge on.A more advanced approach would be to go through a logic book like Velleman that teaches proofs as structured programming.
 This is what they did at my undergraduate universityDo all of these in order first:Calculus 1 and 2Linear algebra and multivariable calculus and an introduction to proofs / logic course (you are ready for some electives at this point)Ordinary differential equationsAny of these can be done concurrently, choose one Analysis and one algebra :Advanced calculus (eg “understanding analysis” by Abbott)Linear algebra in the sense of finite dimensional vector spacesEasier abstract algebra (senior level classes are eg Artin and rudin, these ones are more elementary textbooks)Core Senior level courses that you take if you want to get good at math:Analysis sequence (1 year on baby Rudin)Algebra sequence (1 year on artin)Topology (munkres)Electives:Probability (can be done after multivariable calc)Linear optimization (after linear algebra + multivariable calc)Logic (compactness completeness godel etc whatever, can be done after intro to proofs course but will probably make less sense if you didn’t study some more stuff first)Numerical analysis (after ODEs I guess or calculus + linear algebra if you want to skip tht stuff)Statistics (after probability)Combinatorics - after calc 2 and linear algebraGeometry - after multivariable calc, linear algebra, proofsIntro Differential geometry: after advanced calculusDon’t really have much more knowledge for graduate courses etc. or even some common ones like complex analysis. if you know the senior level core stuff you’re probably “good enough” to make some progress on a lot of things. Each of these classes is 100-200 hours of total study so it seems odd to me that someone will just try to study it on their own by there you go I guess
 Not a answer to your question but this query about dependency graph off skills comes up so often that we're trying to build this once and for all at https://github.com/learn-awesome/learn-awesomeIt's open-source and a community effort so you're welcome to join & contribute.
 Yes, I sketched out textbook suggestions for almost all of OP's topics in a preferred order for reading them. You can roughly segregate material (at the early undergraduate level) into algebra and analysis sequences. See my other comment in this thread.
 For me, I learn mostly for practical use, so for instance I want to build a markov chain generator:- I refresh up on probabilities in general- I look at the markov chain wiki article- I look at related pages to see what else I could do- etc. (goto step 2/3)
 Fellow Marine Engineer (KP) here, and just wrapping up my Master's in CS Data Science. If you're like me, ou got rushed through a number of higher level match (between bar crawls), passing tests, but not digesting.So, the feer of a math intensive grad program (and admission had me worried). My CS program offered some math refresher, but I ended up just jumpting in without it. If the program is decent, you will be guided along at a digestable pace. As a responsible adult with a hunger to learn, and you will enjoy and digest more. In a curriculum, you'll also have peers/teachers to help you when absolutely stumped - which will happen.For books - I just have one that I saw referenced here on HN, which is a little odd, but highly recommend for understanding Fourier transforms [1].1. Who Is Fourier? A Mathematical Adventure 2nd Edition... Amazon link: https://www.amazon.com/dp/0964350432/ref=cm_sw_r_other_apa_i...
 I noticed you left out linear algebra. You may struggle in topics such as topology and functional analysis without a strong background in linear algebra. You mentioned studying marine engineering, so I'm not sure, but in my experience engineering courses tend to cover only the basics up to a first year level. There is so much more than that.
 Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard is good book, written somewhat from the point of view that these subjects are better presented interleaved than as a self-contained book.
 > Group Theoryi still have a copy of Fraleigh -- a first course in abstract algebra. not sure if it is the best, but it did the job.I really enjoyed the two real analysis & functional analysis courses when i attended university, but alas, the reading material for these courses were notes produced by each lecturer, they're not available as published books.> 5).Topology(cup = donut)i recall covering the material for "cup = donut" style results in an algebraic topology course, i think in 3rd or 4th year, after first being drilled with 2-3 years of pure math including real analysis, including lots of basic stuff about topological spaces, continuous and smooth functions, measure theory, some abstract algebra, etc.`````` 3). Probability 4). Statistics `````` MacKay's "Information Theory, Inference, and Learning Algorithms" is a great read: http://www.inference.org.uk/itila/book.html`````` 5). Optimization `````` Sign up for this course: https://www.coursera.org/learn/discrete-optimizationedit:you didn't mention PDE, but i still have a copy of Evans -- partial differential equations serving as a monitor stand. you probably want to have 3 years of pure math including linear algebra, lots of real analysis & some differential equations under your belt first.
 Not a mathematician, but have spent some time self-learning math. I have not gone through the following books entirely, however I've used them when needed to learn or clarify a concept:> Group Theory - I would look for a good abstract algebra book that also covers groups, such as A First Course in Abstract Algebra, Fraleigh or Basic Algebra, Knapp. Another popular recommendation is Abstract Algebra, Dummit and Foote.> Topology I think you should first look at a real analysis course and understand metric space topology for Euclidean spaces. If you have taken calculus, you can try and go through Baby Rudin. For topology specifically you can look into Munkres or Introduction to Topological Manifolds, J Lee.> Measure Theory T. Tao has a good book on this topic.> Differential Geometry Introduction to Smooth Manifolds by J Lee> Probability Probability Theory, Achim Klenke> Game Theory Game Theory by Maschler, Solan & Zamir
 For complex analysis the book by Churchill -written for engineers- is a very down to earth introduction with lots of examples and applications to resolution of integrals and PDEs.In the case of set theory I cannot recommend enough Halmos' "Naive set theory". It's completely rigorous yet easy to read, and lays out all the basics of ZF, constructing both ordinals and cardinals.For probability Khinchin has some very thin fun books, written for some classes he gave in a military school. The books do not get into deep topics but are a good introduction.For group theory I would try Isaacs' Algebra book. Although the book is one of those books which contain too many topics, he was really enthusiastic of group theory and I have been told the corresponding chapters are very good.
 I will say it: Wikipedia. You do need to have the common knowledge used to describe the topics but Wikipedia is an amazing source.(out of your list, I mainly looked at Game Theory, Probability, and Statistics there).
 There is a good set of recommendations from mathwonk in this physicsforums thread (pretty detailed) https://www.physicsforums.com/threads/the-should-i-become-a-...
 You can study Group Theory and Category Theory simultaneously using Chapter 0 by Aluffi: https://www.goodreads.com/book/show/6829004-algebra
 > Mathematical LogicA Mathematical Introduction to Logic, by Herbert Enderton.
 Thanks man, will definitely get this book.
 I'll add, that I strongly advise against beelining for Gödel's Incompleteness theorems.Learn the formal language aspects, like grammar and parsing and structural induction. Learn semantics. Learn a few deductive calculi (say, natural deduction and Hilbert style) and how they interrelate, and actually use them to prove some (very simple) results, ideally from some important axiom systems like Peano arithmetic and ZF set theory. Learn model theory, Gödel's completeness theorem, the compactness theorem, and their more immediate implications.You should also learn how logic interrelates with* Computation, both in the sense of enumerability of deductive proof systems, and in the sense in which expressability in certain logical theories is Turing-complete; and* Set theory, both to grasp the sense in which a first-order set theory like ZF seems to suffice to supply an ontology for the rest of mathematics, and to understand the role of cardinality in e.g. model theory.There isn't really a correct order in which to approach these fields. You'll find that for a proper understanding of any of these topics, you'll have to move back and forth between them frequently.I would put off Gödel's incompleteness theorems until you've done most of this. In particular, learn the completeness theorem, up to a point of confidently being able to apply the compactness theorem, first. Many of the least-informed abuses of the incompleteness theorems come from people who can't distinguish different notions of entailment, and are unfamiliar with the successes of deductive calculi and the categoricity shortcomings of first-order logic.
 i'm not totally sure what 'fractal geometry' is, although there is something called 'geometric measure theory'..the canonical text here is Federer, although it's supposedly a tome. Krantz's _the geometry of domains in space_ appears more approachable....also, note that these topics aren't totally separable.
 You know,I never knew Computer Science was an applied math. But I always went with statistics.