books as an alternative for the lack of adequate courses in Advanced Topics

[ArcticBatcat]

would it be possible for people with deep knowledge of mathematics to at least suggest a few books as an alternative to courses in Advanced Topics that are lacking adequate courses?

it would be of great help, I assume people will just google for courses or books and get guidance elsewhere but I thought it would be helpful given the trust we have for ossu for the book recommendations to come from this repo

[waciumawanjohi]

Yes. One factor that OSSU considers when comparing resources is feedback for the learner. Books often (but not always) do poorly in providing feedback. But in a situation that an excellent book exists for a topic and no excellent course exists, OSSU is happy to point learners to that book resource!

[spamegg1]

This is where we hit the limits of self study. It's quite unlikely that you can learn these subjects alone from a book. You need a teacher and fellow students. Often, these courses will use lecture notes written by a professor rather than a textbook, or only use the textbook as reference.

It's difficult to find solutions to exercises. Even if you do, it'll be hard to understand them. The proof techniques needed to solve them are often not found in previous courses, and the books themselves don't always make them clear (these ideas are "transmitted in person" because it's too difficult or too long to properly explain in text).

But... here you go. These are some of the books used in higher undergraduate / graduate math courses (I refrained from commenting on subjects I'm not too familiar with):

A few things to note: Real / Complex / Functional analysis (not mentioned here) and topology are deeply connected, some texts will cover some of each subject. Similarly, algebra / topology / geometry are very connected too.

Probability / Statistics, numerical analysis etc. are consumers of analysis theory (probability is based on measure theory and integration). Discrete subjects like Game Theory, Combinatorics etc. tend to stand on their own rather separately.

Another important thing to keep in mind is the various schools of math. Russians tended to focus on analysis, while Hungarians dominated combinatorics, and so on. So these schools and their traditions often ended up producing the textbooks. An interesting outlier was the Polish school, which included analysis, logic / set theory, and even cryptology in separate sub-schools. They couldn't become as established as others because they were royally screwed by all sides in WW2. Anyway... There are more (French, German etc.). Try getting familiar with them as it can help a lot. For example, you'll see that Russian sources use tensor algebras as the basis for almost every subject!

Probability and Statistics

Combinatorics, probability, statistics, game theory, applied stats

Combinatorics: Reinhard Diestel's Graph Theory book is what you want. It's excellent.

Probability: ???

Statistics: ???

Game theory: Game theory is not taught in math departments, usually. Instead it's taught in Economics departments or Computer Science departments. The two online courses on Coursera provide an excellent, gentle introduction.

Applied stats: ???

Mathematical Analysis

Real analysis, numerical analysis, complex analysis, optimization theory

Real analysis: Real Analysis Modern Techniques and Applications by Gerald B. Folland is commonly used in measure-theoretic graduate real analysis courses. I had to use Bartle's book as it was required in my school but I didn't like it. Chapters were quite short / quick and exercises hard.

This is distinct from the undergraduate "advanced calculus" which is unfortunately often also called real analysis, for which the best is Intro to Analysis by William R. Wade. (I do not recommend Walter Rudin. Don't fall for the hype on the Internet. It's due to historical reverence. It's OK but not very good, rather dry and missing some detailed explanations, goes a bit fast / dense.)

Numerical analysis: Burden's book is the well-respected book in this field.

Complex analysis: Silverman's books are very often used because Dover publishes them very affordably in paperback for students. First one is excellent, I used the second one which is a bit too concise and not as good. Arguably a better, more modern book is this. (Again I do not recommend Walter Rudin. It's OK at best. Usually not used in higher undergrad / grad courses anymore.)

Optimization theory: ???

Abstract Algebra

Abstract algebra, category theory, algebraic geometry and topology

Abstract algebra: Abstract Algebra by Dummit & Foote is the obvious "algebra bible". It's very good, however it's encyclopedic and absolutely massive, covering everything from intro undergrad group theory all the way to higher graduate level stuff, so it should be used alongside a course and topics / exercises should be picked by someone who knows what they're doing. It can be used at any level (undergrad or grad) to cover any algebra topic. I used this book in both undergrad and graduate, for a total of 5 semesters!

Typically one semester course covers group theory, a second semester covers rings and module theory, a third semester covers field and Galois theory, and further semesters can cover various topics like bits and pieces of algebraic number theory, algebraic geometry, etc.

Category theory: Category Theory for the Working Mathematician by Saunders Mac Lane is the book that grad students / researchers use to learn the "necessary evil" of category theory if their research area requires it (homotopy theory for example). It's written from a mathematician's point of view, not programmer's or category theorist's.

Algebraic geometry: This is a very complex and interconnected subject. Some of this will be covered by Dummit & Foote, and some of it intersects with Differential Geometry and Topology courses / books. Hartshorne's book is the most popular, it's even possible to find solutions online. Arguably a better, more modern book is this.

Topology: This does not belong under algebra! For "Algebraic Topology" the best book is Algebraic Topology by Allen Hatcher. Excellent book!

[owenekblad]

I would veer away from suggesting many of the above textbooks for self-study. For example, it often takes the better part of a PhD in algebraic geometry before one can even begin to feel comfortable with Hartshorne's tome; understanding the first chapter alone is a feat for a master's student, and this is for learners who have access to advisors, more advanced graduate students, and peers. In general, most graduate-level mathematics textbooks are quite difficult to self-study from: it is often implicit that the reader be 'mathematically mature'---meaning they will be familiar with the major constructions of modern mathematics and will be unintimidated by fancy-looking symbols (which, to the uninitiated, are horrifying, and to the more experienced, are uninteresting)---which is an ill-defined skill not easily acquired.

For this reason, I would advise against listing any graduate-level textbooks for more advanced topics. There are amply many well-written undergraduate texts that deal with, for example, algebraic geometry, commutative algebra, number theory, knot theory, point-set topology, and other special topics, which don't instantly scare the reader away from the topic.

TLDR; I would advise practicing some discretion before suggesting some of these graduate-level textbooks for learning about some special topics. The undergraduate is not expected to be doing research-level mathematics and, accordingly, doesn't need to learn the most hardcore version of these special subjects occurring in these `yellow Springer books'. There is more than enough accessible mathematics an undergraduate can leisurely peruse before selling their souls to the rigor gods.