Ask HN: Mathematicians, what textbooks are best for learning these math topics?

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 Mathematics

1).Group Theory(rubics cube,e.t.c)

2).Order Theory

3).Combinatorics(trees,graphs,e.t.c)

4).Fractal Geometry

5).Topology(cup = donut)

6).Measure Theory

7).Differential Geometry

8).Vector Calculus

9).Dynamical Systems(Fliud flow,ecosysytems,Control Theory)

10).Chaos Theory(Butterfly effect)

11).Complex Analysis(Functions with complex numbers)

Applied Mathematics

1). Numerical Analysis

2). Game Theory

3). Probability

4). Statistics

5). Optimization

6). Cryptography

7). Computer Science

Foundations

1).Mathematical Logic

2).Set Theory

3).Category Theory

4).Godel Incompleteness Theorems

P.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 - Milne

5)Topology - Mukres

6)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 Rice

9, 10 ) Non linear dynamics and chaos - Strogatz

11) Complex analysis - There exists a set of 4 books covering real, complex and functional analysis by Stein and Shakarchi, which should serve your purpose

Applied Math :

3) Probability - Grinstead and Snell, Durrett both have good books.

4) All of Statistics by Larry Wasserman for a more ML bent to it

5) Optimization by Boyd and Vanderberghe

7) 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.

Textbooks which people on the internet consider very good:

>Group Theory

Carter - Visual Group Theory

>Complex Analysis

Needham - Visual Complex Analysis

Less confident recommendations:

>Probability

https://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 Theorems

Start 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 Theory

Hrbacek - Introduction to Set Theory

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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.

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.

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.