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[Marine Farinha]

Iam very new to set theory and have only learned the basics up to cardinal and ordinal arithmetic. I would like to learn about large cardinals and I am reading Thomas Jech's Set Theory. I have read that Kanamori's book is a good resource but I think that one is a bit advanced for me still. Are there any other books out there? Also is there a book that looks at the philosophical aspects of the subject and is less technical?Thanks:)

The question got bumped, and so let me take this opportunity to say that Kanamori's book is definitely the right source for anyone who is interested in entering this fascinating subject. Some time ago, I wrote a review of it, which you can find at:

If you find his book to be too advanced, then the solution is simply to learn a little more set theory, which you can do by studying Jech's book Set Theory, among others. Much of Kanamori's book, however, is not so technical, and he has a delightful habit of weaving the historical development into the unfolding story. It is a pleasure to read!

I would recommend picking up a good text on model theory, in particular Chang/Keislers reference on the subject. I'm in a similar position to you -- I own Kanamori's book but am as of yet unable to unlock the treasures within, however I'm very comfortable with ordinal/cardinal arithmetic as a result of Donald Monk's excellent text on MK class theory.

Other research concerns have prevented me from really digging in to Chang/Keisler as of yet so I can't say this with total certainty, but from the skimming I've done of Kanamori's text it seems like the main ingredient missing for a good, comprehensive reading (in my case) is the toolbox of model theory.

I am looking for book recommendations on the Queen's Gambit. I already own some general opening books, which cover many openings and also many variations in each opening. However, I am not advanced enough to figure out the strategic plans attached to the many variations by myself.

So what I am looking for is something that is less broad than e.g. Fundamental Chess Openings. On the other hand, if possible, I would not want to delve only into the details of a single reply, e.g., QGD. So what I am looking for is a book that somehow covers a reasonable amount of different defenses in reply to the Queen's Gambit. Of course I understand that this is a huge topic. But at least a book that covers roughly QGA, QGD and the Slav would be nice.

The best for QGD is Mathew Sadler-Queen's Gambit declined in my opinion. Every line in the Queen's Gambit declined has been explained. It is dated, so you will need to do your own research to find latest theory. Still, QGD is "stable" opening so theory rarely changes here...

For Queen's Gambit accepted, you could try a book from Starting Out series, or Semko & Sakaev-Queen's Gambit Accepted. QGA is simple, its all about not letting Black to finish development. You just need the moves to survive the opening. Maybe you will need to learn how to play against Isolated pawn. If that is the case, you can start with this post on Chess SE.

As for Slav defense, again Black aims for solidity and White tries to hinder his harmonious development. The lines are sharp but easy to understand, and you need to know theory. Maybe sometimes you will end with "hanging pawns" but there is a book about this type of middlegame. The best book I found on Semi-Slav is one from David Vigorito. As for Slav, again you can Google for one of the Starting out series.

Lars Schandorrf-Playing the Queen's Gambit 2nd edition could be what you want. It gives you most up to date theory at this point. Combine it with the above books and you should be fine. For latest opening theory you could consider getting 2 volumes from Avrukh ( Grandmaster repertoire 1 and Grandmaster repertoire 2 ).

Shandorrf and Avrukh give opening moves but don't explain basic ideas. I have listed them so you can survive opening as there were recent novelties in the Semi-Slav and Slav. For basic ideas get Starting out series and Sadler's book, they should be enough for you to start.

Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, however, the self written syllabus was not self study friendly (good syllabus overall though).

So, now I want to ask whether any of you know any good books on abstract algebra, which lift off at basic ring theory and continue to more advanced ring theory and to finite fields, Galois theory, ...

There's always the classic Abstract Algebra by Dummit and Foote. Section II of the text gives a nice treatment of ring theory, certainly providing plenty of review for what you have already covered while introducing more advanced concepts of ring theory. Section III will cover the field and Galois theory you're interested in. Some of the exercises can be difficult at times, especially for self-study, but the authors tend to give a number of examples and always provide the motivation for why they are doing what they are doing.

"Contemporary Abstract Algebra 7/e provides a solid introduction to the traditional topics in abstract algebra while conveying to students that it is a contemporary subject used daily by working mathematicians, computer scientists, physicists, and chemists. The text includes numerous figures, tables, photographs, charts, biographies, computer exercises, and suggested readings giving the subject a current feel which makes the content interesting and relevant for students."

It's a unique book that covers the basics of group theory, ring theory, and even a tiny bit of Galois Theory, but it does it almost entirely through problems. Every chapter begins with a short section defining some terms and giving a few basic proofs, and then it leads the reader through the rest of the exposition in a series of problems, some difficult, some not. The end result is that if you actually do all the problems, you've written the book yourself. It's impossible not to be comfortable with basic abstract algebra if you take this book seriously.

One book that I did not see mentioned, but which really deserves some accolades is the recent book Visual Group Theory by Nathan Carter. There are some excellent accompanying videos by Prof. Macauley on his youtube channel. These go really well together.

The biggest trouble I ran into with group theory and abstract algebra was the dizzying set of definitions that most books present at the beginning. You get a bunch of definitions with little or no motivation and with little description of the underlying geometry of how the binary operations work. The nice thing about the Carter book and the videos is that it spends a lot of time working though group diagrams and showing the "symmetry" of a group. It is easy to get caught up in the formalism, but without a good intuitive understanding of how different groups work--and how simple groups differ--it is easy to get frustrated--especially in self study. Also note that the Carter book has exercise solutions at the end.

There is no easy or right answer. I know brilliant professors who cannot easily decide what textbook to use for an advanced math course, and for good reason. Every book has its own strengths and weaknesses. I suggest you go to your math library (assuming one is available during this pandemic) and examine several books. A book you like might be hated by someone else, it is highly individual. You likely will need at least two or three books so you can go back and forth. Even a good book can be bad in a particular section and vice versa. Use a common textbook that has gone through at last two or three editions as a guide as to what topics to cover and then be prepared to use alternate books to actually learn the topic.

Use the Internet. Don't be afraid to read lecture notes or check Wikipedia. Also, Professor Keith Conrad (Univ. of Conn.) has dozens of expository papers on algebra on his web site, some are easy, some are difficult, and some are advanced or specialized. I have found that lectures by professors at lessor known universities to often be better than those by professors at famous brand name universities. That being said, I have found lectures by Unv. of Berkeley professors to be quite good, and lectures by MIT professors to also be good, but the latter are often very fast paced and better for review than to self learn from as they are so intense.

I suggest you get an easy book, an intermediate book and eventually a hard book. Herstein's: Topics in Algebra is harder than Birkhoff and MacLane's book, but Birkhoff and MacLane's book is good for learning the fundamentals. As an undergraduate I used Herstein, but I think it is too difficult to self study from.

It is critical to learn the definitions and other fundamentals cold and then go on to a more advanced treatment. (One really smart professor basically told me: memorize definitions, but do not memorize proofs, just understand them.) Herstein loves to give problems and results that are hard using elementary methods, but easy using more advanced methods. In my opinion this a bad way to learn, as not everybody is clever at solving hard problems or following highly technical arguments, and I think it is more useful to put one's energy into learning the concepts and theory that makes it possible to eventually easily understand what is really going on, rather than rely on clever technical tricks or manipulations to get a result with no real deep understanding as to what is really going on. Neither Herstein nor Birkhoff and MacLane cover everything a graduate course would cover. Herstein, in my opinion, makes the subject seem more difficult than it is.

I also use: Algebra: A Graduate Course by I. Martin Isaacs, it has its strengths and weaknesses. It is is elegant and the proofs are carefully done, but it may be too abstract and condensed to self study from.

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