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  No. 863
>>864 >>866 >>918
Hello everyone. I have seen a lot of threads(book threads, IMO advice...) that seem to exist solely so people give advice and give comments on your study material.
So I had an idea for a thread where we: state our goals, name the books(video series and other material) we plan to cover in a certain order, give an approximate deadline and other mathchan users can give suggestions, meaningful comments, advice and pointers to others!
I'll start:
My goal is to develop enough knowledge before I go to university where I hope to take advanced classes and spend more time on advanced topics while simultaneously getting myself ready for the national math competition(not my main focus, just for fun)
《Elementary math 1 i 2 notes》
-notes from my local university where they want to "bridge the gap between high-school and university math" (introductory logic, set theory, relations, functions, number theory, Euclidian geometry, vector spaces and analytic geometry)

《A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre /// 101 problems in algebra》
-covers similair material as the already meantioned notes but in more detail

--------------I am here----------------

《Analysis I notes》
-notes from the local university, this time in calculus/analysis (what European universities call analysis is almost always just rigorous calculus or very elementary analysis since we cover calculus in HS)

《Linear Algebra Shilov》

《Šime Ungar, Analiza 3》
-same deal as before, rigorous calculus/ begginer analysis book for R^n

《101 problems in various topics, Andreescu books, generatingfunctionology, Problem solving strategies by Engels》

《Elements of Set Theory by Enderton》

《Introduction to Logic: and to the Methodology of Deductive Sciences by Alfred Tarski》

《An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery》
-mainly for competitive math but also as an introductory text

《The USSR olympiad problem book》

《Algebra by Artin》

《The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by Steele》
-half for competitive maths, half since I heard analysis problems demand a lot of inequality knowledge

《Amann and Escher Analysis series》
-all three books

I hope to cover everything stated above in a 1.5-2 years. I covered the things above the "I'm here line" in three weeks while doing most of the exercises.
>>
  No. 864 >>901
>>863
Amann Escher do both analysis and calculus, so putting it at the end of your list isn't really ideal imo. Shilov has no prerequisites either. I'd recommend reading both earlier/now instead. The first chapter of Amann Escher is also a nice introduction to proof-based math in general.
For algebra, I read Gaitsgory's Math122 and Math123 notes (both are on AMS) combined with Lang's Algebra (didn't complete it though). Gaitsgory's notes on Math122 can also be used as a somewhat abstract course in linear algebra. I think he actually based his Math55a course on them, of which you can also find notes online. I also read the first few chapters of Gorodentsev but can't say much about it as I never finished it.
For Topology, I really like Brown's book "Topology and Groupoids". It has few prerequisites (if any) and is very interesting as it emphasizes applications in differential topology, algebraic topology and algebraic geometry (for instance, it introduces the Zariski topology early on in an exercise) whereas e.g. Munkres is far too focused on point-set topology, which you'll find in analysis books like Amann Escher anyway.
If you want a book that treats the differential geometry stuff in amann escher 3 in more detail, I recommend Sternberg's lectures on differential geometry. Kobayashi-Nomizu is the go-to book if you want to specialize in it (note that it'd most likely be a very difficult first read).
If you're interested in geometry I also highly recommend learning physics.
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  No. 866 >>901
>>863
i intend, wholely and fully, to have a good time
do tell if any of those problems are fun, i would be fine to check them out
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  No. 901 >>920
>>864
As you said, I am probably going to cover Amann and Escher quite a bit sooner. I think people scared me into thinking that analysis is more demanding than it probably is.
I also thought that set theory and logic are more basic in some sense than they are. But it seems to me like you only need basic logic (basic propositions, quantificators and so on) and just the very basics of set theory (sets, operations on sets, partitions and such) to cover math to a decently advanced level.

Thank you for the other recommendations as well.

>>866
101 problems in algebra I thought was okay. The difficulty of the problems differs quite a bit since I solved some in minutes and others left me confused even after checking the solutions.
Some of the explanations I didn't like at all. They don't explain the thought process and sometimes just begin the solution like this:

Since (critical part of the solution that greatly simplifies the problem) holds. We know that x is equivalent to y. Therefore z and therefore the statement is proved.
>>
  No. 918
>>863
A gentle intro will not be very beneficial and will waste time. The reason analysis is hard is because the ideas are presented in a way that is isolated and general. The same will go for algebra and topology. The books on those subjects should be read, but not be the center of your study. They are there for reference when needed.
I studied analysis and ignored everything else and it failed me. I then changed my approach, got a book on differential geometry (lee) and began to read that, referencing books on analysis, algebra, geometry and topology when it was necessary. Math is a slow process and the time you put in will significantly outweigh the knowledge you gain.
Also a little side note, get into complex analysis asap. Both (real and complex) deal with the same ideas, just complex analysis is more tangible due to its geometric nature. I recommend Ahlfors book.
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  No. 920
>>901
Logic and model theory are a bit weird in the sense that they're mostly disconnected from the rest of the math world despite seeming fundamental. Of course I am neither a logician or set theorist so my view may be biased but it is interesting how both camps often know very little about the other side. And then there's stuff like algebraic geometry in its formulation by grothendieck that seemingly required inventing/appealing grothendieck universes just to define some things rigorously. Turns out it wasn't necessary (see the stacks project, there's extensive discussion around size issues).

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