Reddit reviews Contemporary Abstract Algebra

"Analysis" just means proof-based calc, at least at the undergraduate level.

It sounds like the main thing you're missing is abstract algebra, i.e. the contents of this book. If you feel like you remember how to prove things, you could try to either self-study the material (potentially difficult) or enroll in a two-semester abstract algebra course through Ohio State.

If you're not comfortable with proofs, you might want to start by taking a introductory proofs course to refresh yourself, or maybe something like linear algebra or another relatively accessible proofs-based math course such as number theory, graph theory, or combinatorics.

It might also be worth learning a little bit about point-set topology, though not all Masters programs will assume this.

First and foremost discard the idea of contributing to an area of research mathematics. It's not that it's impossible for you to do so, but it's not a good goal to set. It's best for you to try and explore a field of mathematics that interests you to learn more about it. After all this is what mathematical research actually is, we have questions that we would like to know the answers to so we figure them out. It is also a much more attainable goal, whether the material is new to the mathematical community or not you will have learned something new.

Second, if you really want to try and get to the forefront of mathematical understanding, expect to put in about a year or two at minimum to get there, and that's only if you pick a new or obscure field whose frontier is not as far removed from where you are. Fields like combinatorics and graph theory also have frontiers that are easily approachable for beginners.

If you're really dead set on algebra I would put forth two different fields. The first is combinatorial group theory, which is a bit older, but a lot of people have vacated the field. The classic text on that is Combinatorial Group Theory by Magnus. I don't know much about the status of open questions in the field, but I do know that combinatorial methods crop up in solutions to open problems in group theory all the time. You might be able to get the background needed to understand and work though most of that book. You'd need at minimum a solid understanding of presentations of groups and a bit of knowledge about combinatorics.

The field that most mathematicians have moved into after working in CGT is geometric group theory. It's a relatively new field with lots of interesting accessible open questions, but requires a bit of background knowledge of metric topology. There aren't any English language classical texts that are approachable at your level, but these notes by Alessandro Sisto are a quite good introduction. (The ending of the notes tapers off with fewer and fewer details, I wouldn't read past page 64 or 65). There are also some errors you have to catch in his proofs, and statements of theorems throughout.

This is all assuming that you've read an introductory book to abstract algebra and done all of the problems, such as Contemporary Abstract Algebra and actually know all the basics solidly.

I don't know that I can help because everyone learns so differently, but I'll say a couple of things. (I'm going to warn you right now that I'm kind of tired and I didn't proofread very much.)

I think my best advice about (1) is to play with small examples. If you're asked to prove something about all real vector spaces then look at R, R\^2, R\^3, and see if you can understand what's going on in those situations. If you're asked to prove something about all differential functions, pick one and play with the statement in that case. Once you get the small examples, move onto bigger examples. What is the craziest, wildest differential function of which you can think? What vector space really pushes the boundaries of the hypotheses that you're given? It's not a fool-proof plan, but I find that working with examples is the single most helpful thing I can do when I'm working on a hard proof.

Also, about number (1), this isn't very concrete advice, but I like to tell my students to try to understand a statement before attempting a proof. Proofs are a linear line of logic between some assumptions and a conclusion. Our brains, however, aren't always compatible with the rigor and linearity of a proof; it can be hard to see everything that's going on from step seven in a proof. If you can really, *really* convince yourself of something and understand it then your intuition will guide you much better during the proof. It can be easy to fall into a trap of saying "this is on my homework so it must be true" and then diving into a proof but it's better to think critically about the statement first. Along the lines of the previous paragraph, an attempt to construct a counterexample (even if you know something is definitely true) can be really helpful.

As to part (2), all I can really say it that it's very frustrating and challenging. I'm about to graduate with my PhD and I still struggle with learning new math from a textbook or paper. You're not alone in that. It really helps to find the right book but it's also very hard. Some books are excellent because they are packed full of content and make a good reference (sort of like an encyclopedia) and other books are excellent because they have great exercises and make a good companion to a course. Rudin's analysis book is notorious for this; the exercises are fantastic which means it gets used in the classroom a lot, but the exposition is pretty brief (and, in my opinion, quite poor at times) which makes it difficult to use when self-teaching. Unfortunately, the best way to sell the most books is to write a book that is meant to be used in a class. It can be really hard to find books that are good for learning on your own. (The best example I know is Gallian's Abstract Algebra book.)

My only real advice for finding good references is to ask your teachers. I'll also say that it's usually easier to learn math in small chunks. It can be really daunting to decide "I'm going to learn all of Real Analysis" but deciding "I want to learn how the real numbers are constructed" is much more doable and it likely won't feel quite as discouraging when you get stuck. Pick a specific topic that interests you (ideally somewhat close to what you already know), and try to understand that one topic. Find videos, find websites, ask a teacher, and look into references. A complaint I often get about this technique is "I don't really know much about Topology so I don't know what interests me" but a good place to start can be "I'd like to understand what the area of Topology is really trying to study."

So, anyway, I've rambled long enough now. Good luck and try to stick with it!

Do Carmo's Differential Geometry of Curves and Surfaces

Gallian's Contemporary Abstract Algebra

There is a book that is a little more of a "lay-person"'s guide to this stuff and I think you should check it out and read through it before going onto some more technical stuff. It'll give you an idea and overview of some of the more interesting parts of the subject.

It's also a book that I think every mathematician should read, too, but I won't get on my soapbox here.

The book's called Symmetry by Hermann Weyl, who was a world-class mathematician.

For more advanced references that will work for your level I would recommend the Gallian or Beachy and Blair which are better suited for someone without an extensive math background. Many of the texts that have been suggested so far are better suited for someone who is either an honors undergrad with a strong background or a first-year grad student. The Beachy and Blair is cheap new, I would look around for a used copy if you want to go for the Gallian. There's no reason you need the most recent edition, either.

If you have questions about the material I'd actually be happy if you PM'd me, it's always good to go over this stuff again for a mathematician.

Source: I'm a PhD student in math and professional nerd, and apparently this qualifies me to tell people what books to read and not read.

I'm a big fan of Gallian's Abstract Algebra Text as an introductory text. It's a good pace if you're not very familiar with the material. Artin's text that was already suggested is also a very good text, but it moves much more quickly.

OK..

http://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/0547165099/

http://www.amazon.com/Algorithm-Design-Jon-Kleinberg/dp/0321295358/

http://www.amazon.com/Introduction-Algorithms-Thomas-H-Cormen/dp/0262033844/

http://www.amazon.com/Introductory-Combinatorics-5th-Richard-Brualdi/dp/0136020402/

http://www.amazon.com/Artificial-Intelligence-Modern-Approach-3rd/dp/0136042597/

And I guarantee you those nice amazon prices aren't that steeply discounted around the start of the semester, and certainly not in any brick-and-mortar bookstore.

Discrete mathematics with ducks!

Contemporary Abstract Algebra

An Illustrated Theory of Numbers

Visual Complex Analysis

Number-Crunching

The Magic of Math: Solving for x and Figuring Out Why
The Princeton Companion to Mathematics
The Number Devil: A Mathematical Adventure

Professor Stewart's Cabinet of Mathematical Curiosities
Gödel, Escher, Bach: An Eternal Golden Braid

One Two Three . . . Infinity: Facts and Speculations of Science

Recently made a move and these are a few books on my shelf that stand out more.

There are so many beautiful mathematical images that could be chosen though and I do wish more publishers would have more creative book covers also.

A friend of mine used this book in her undergraduate abstract algebra course. It looks fantasitic: Gallian - Contemporary abstract algebra.