4

I don't think there's a lot of sense in learning things that will be in your high school classes. You might want to take in some elementary set theory. Once you understand that, you can start plunging into any of the modern formal math subjects -- axiomatic linear algebra, group theory, graph theory, and so on. You don't have to dive too deeply into any ...


3

At https://www.maa.org/math-competitions/putnam-competition it says, "The Putnam Competition covers a range of material in undergraduate mathematics, including elementary concepts from group theory, set theory, graph theory, lattice theory, and number theory," so all you have to do is trundle off to the library/bookstore to find books that cover those topics....


3

I have taught College Algebra a number of times, and can recommend two pretty good books. The first is Dave Cohen's book, he who lectured at ucla for many years. The second is by Gustafson and Frisk. All the topics you listed would be important, btw.


3

I followed a course on set theory in my third year of studying mathematics. The formal logic language, though useful and very important for lying the foundation of mathematics is also hard to intuitively get, and I would recommend it as a place to start learning mathematics. Instead I would start with calculus and some basic set theory, like the intro ...


3

Here is a list of freely available books I know, but since I've personally learnt proofs before most of them were written, I am no good judge of their quality as proof tutorials: Eric Lehman, F Thomson Leighton, Albert R Meyer, Mathematics for Computer Science. Part I is specifically about proofs, and parts II and III should provide good practice. It's ...


3

You can follow this one "Contemporary Abstract Algebra" by Joseph A. Gallian It's an excellent book for a strong base for algebra group theory, ring theory, field theory, vector spaces, eigen values eigen vectors, canonical forms. The great thing is that this book talks to you. It is interactive. Chapter are small in size and there are a lot of examples ...


3

David Hilbert. Foundations Of Geometry. Donald Coxeter. Introduction To Geometry.


2

I would consider Algebra: Chapter 0 by Aluffi, although it doesn’t satisfy all your requirements. The voice of the text is light and conversational, and Aluffi does give some allusions to analogies in other interesting categories at times (e.g. category of smooth manifolds). The book’s language is highly categorical — Chapter I is entirely dedicated to ...


2

I really like Discrete and Computational Geometry by Satyan L. Devadoss and Joseph O'Rourke—very clear and well illustrated. A nice touch is including "unsolved problems," which motivates students and shows this is a very living discipline.


2

I recommend Cover and Thomas, https://www.goodreads.com/book/show/433439.Elements_of_Information_Theory as well as Gallager https://www.amazon.com/Information-Theory-Reliable-Robert-Gallager/dp/0471290483


2

Malliavin's stochastic analysis is rather difficult, it's mostly approachable when you've already gotten familiar with the themes of Malliavin Calculus (it assumes familiarity across several topics, in and outside probability). I personally don't like Oksendal's writings because he's a financially focused researcher, but if that's what you're aiming for then ...


2

I'm certainly no expert, but I have particularly enjoyed An Introduction to Ordinary Differential Equations by Earl Coddington. I used it along side Elementary Differential Equations and Boundary Value Problems by Boyce and Diprima and it is definitely a nice complement to that.


2

Well, Ramanujan's notebooks contain quite a few mistakes. However, he didn't generally provide proof along with his theorems, so it's really just a list of theorems. If I'm not mistaken though, he later went through them with Hardy and added some proofs. Even today, people are still working through his notebooks and proving his theorems. A lot of the proofs ...


1

I would suggest Michael Taylor's first book on PDEs ("Basic Theory"). In particular, the third chapter seems to contain everything that you're looking for.


1

The typical school math up to college is Algebra I → Geometry → Algebra II → Precalculus → AP Calculus AB/BC, or Calc I/II in college → Multivariable calculus → Linear Algebra. I do not know which of these you have learned already, or not, but these are sort of the "general" math, sort of like general chemistry. Considering you are in ninth grade, I do not ...


1

Good Thinking, by I.J. Good is a great book for you. The author analyzes the conceptual underpinnings of probability theory and brings a philosophical perspective to the theory. There is also a focus on applications. A little about the author, who worked with Alan Turing at Bletchley Park: https://en.wikipedia.org/wiki/I._J._Good


1

The books Differential Equations : Theory, Technique, and Practice by G.F simmons and Steven Krantz An introduction to Ordinary Differential Equations by E.A. Coddington fulfills your requirements!


1

I would recommend Tom Apostol's Calculus. It starts with an introduction to basic set theory and then develops the properties of the real numbers. Then it moves into one-variable real analysis at a relatively relaxed pace but is nonetheless completely rigorous. I would suggest you study the first few chapters till you become comfortable with the basic ...


1

Riemann surfaces is a very standard topics in math, then you can find a lot of books talking about Riemann surfaces under different point of views. I can suggest you: -Riemann Surfaces - S.Donaldson, -Riemann Surfaces - Farkas and Kra, -Algebraic curves and Riemann surfaces - R.Miranda -Lectures on Riemann Surfaces - Otto Forster Donaldson's book is ...


1

I recommend Patrick Billingsley’s beautiful book “Ergodic theory and information”. Rigorous but very ‘readerfriendly’. The treatment of the concept of entropy is measure theoretical, not topological. It is published in 1965, so you should probably consult a university library. Btw, for both treatments of entropy and their interrelationships, you should ...


1

Maybe take a look at this one:https://ia600905.us.archive.org/25/items/ComputationalGeometryAlgorithmsAndApplications2e/Computational_Geometry_-_Algorithms_and_Applications_2e.pdf


1

Not a direct experience, but a friend of mine was taking a course on probability and the text-book used was: "Probability and Measure Theory" by Robert B. Ash and Catherine A. Doléans-Dade https://www.amazon.com/Probability-Measure-Theory-Robert-Ash/dp/0120652021/


1

The book Combinatorial Optimization: Algorithms and Complexity by Papadimitriou and Steiglitz covers much of what you require. It is a little bit dated perhaps (written in 1982) but well-written and pretty comprehensive.


1

These algorithms are based on linear algebra, multivariable calculus and probability theory (when the algorithms are stochastic). Specifically, rather than attempting to learn generic results from these branches, I would recommend starting with reviews of different optimisation algorithms, such as 1 or the more in-depth 2, and refer back to Khan-academy/...


1

There are hundreds of 1800s algebra texts that have been digitized and are freely available on the internet (in the U.S. at least), and many of those would be useful for what you want. The following are some of the better known “advanced school level” algebra texts: William Steadman Aldis, A Text Book of Algebra George Chrystal, Algebra. An Elementary Text-...


1

If you’re just looking to re-up your commutative algebra, you may want to consider these notes by Gathmann. If you’re looking for a general algebra book, I’d recommend Algebra: Chapter 0 by Aluffi. It gives a categorical perspective on algebra, which may be useful if you’re looking to move from algebra to algebraic geometry. It’s also just a great book.


1

In addition to Joseph Gallian's, Contemporary Abstract Algebra, I would also like to recommend John B. Fraleigh's, A First Course in Abstract Algebra for the more advanced student. Although it is indeed, an undergraduate text, I found it to be a nice bridge between undergraduate and graduate abstract algebra, as I often consulted it for good, basic, ...


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