Can the order of learning be changed? I have been advised by many people to learn from scratch. So I decided to learn it. But I have the following questions in my mind. 


*

*Can we skip "the Trinity" and learn something else directly? ("the Trinity" refers to analysis, topology and algebra)

*Why are these things so important and why are they taught for 3 years? 

*Are there any books that give a integrated introduction about this subject and take us to the advanced level?

*What are the most important concepts that one needs to master in them? I saw many mathematical problems that are applied, like finding integrals etc. Can they be skipped? 


Thanks a lot. 
 A: Re the question of "famous problems": at my first international conference (Syracuse, Sicily, 1964) I met Saul Ulam, a very friendly person. He made a comment:" A young person may think the most ambitious route to take is to tackle some famous problem or conjecture. However this might distract them from developing the mathematics that is most appropriate to them." I thought it very interesting that such an excellent and original mathematician should say that. 
It is often suggested that you have to be in a famous institution to do significant mathematics. However I am told the following about Solomon Lefschetz: his first position was in Kansas, then a mathematical backwater. By himself he worked out some fundamental new ideas in algebraic geometry, and eventually was invited  to Princeton. When asked : "Would you not have done so much more in a more substantial environment?" he replied: "No: just the opposite." 
There are many ways of getting on  in mathematical research, but there is a real problem in getting started, as I found. Writing and rewriting to make things clear to me turned out to be very valuable in suggesting new ideas. 
For some discussion of aims in university mathematical education, you might like to look at 
at our article: "What should be the context of an adequate specialist undergraduate education in mathematics?"
http://education.lms.ac.uk/wp-content/uploads/2012/02/Brown_and_Porter.pdf
and the references there. 
A: You could probably learn a fair amount of geometry, combinatorics, and number theory before learning abstract algebra or calculus.  A well chosen set of texts might even take you to a somewhat sophisticated level, but you'd need learned experts to choose the texts.   I don't have any arguments in favor of learning those topics first, but I haven't thought about that.
A: I have come to realize there is a marked difference between a math/scientific education and a liberal arts education which might be applicable to your question. In a liberal arts education, you can hop on the train at any station. There is nothing stopping you from looking at the great problems of Shakespeare long before you confront Chaucer. In many ways this dynamic devalues the liberal education.
Whereas in math, science or technical fields, there is a cumulative progression.This imputes a sense of rigor into the dynamic, as many topics assumes an explicit familiarity with prerequisites - and they mean it. Granted there is something very satisfying in studying, e.g., a number theory text rather than a text with a litany of theorems on commutative algebra. That is, you get to use something you know more so than accumulating theorems. 
But without a sound foundation, you're not in the game. And will at some point wish you had cultivated more capacity when you had the energy and opportunity to do so.
A: You asked:

  
*
  
*Can we skip "the Trinity" and learn something else directly? ("the Trinity" refers to analysis, topology and algebra) 
  
*Why are these things so important and why are they taught for 3 years?
  
*Are there any books that give a integrated introduction about this subject and take us to the advanced level?
  
*What are the most important concepts that one needs to master in them? I saw many mathematical problems that are applied, like finding integrals etc. Can they be skipped?
  

Almost all of this depends on what you want to do. Do you want to follow Conway and Berlekamp into the world of combinatorial game theory? Do you want to throw yourself against the millennium problems? These are very different questions. One cannot even understand the statements of the millennium problems without a good math background, and it takes much more to appreciate them (except maybe, PvNP, which somehow feels to me to be the easiest to 'grasp' even though I've no idea at all how one would go about approaching it). Because it's easier to talk about things in context, I'm going to contextualize this question to algebraic number theory.
[1] Is it possible to 'skip' learning analysis, topology, and algebra independently from more advanced topics? I suspect that it would be possible if the resources were there. Were someone to have written books/lecture notes that explain algebraic number theory, say, while explaining the necessary algebra when needed, then it would be possible. But I doubt that anyone has for three main reasons:


*

*Firstly, there is little sense in duplicating other material, and it is so easy to say "read Dummit and Foote" at the beginning of a text.

*Secondly, if one's goal is to learn algebraic number theory so that one might be able to perform research, then there is the problem of not knowing will be necessary to continue, or to make the next 'breakthrough' in a sense. The material in intro analysis/complex variables/algebra/topology classes is comparatively basic and widely applicable (within math), and so in all likelihood one will have to learn them eventually. And learning them early facilitates all future classes.

*Thirdly, in keeping with the algebraic number theory example, what might be some of the earlier things learned? I opened up my copy of Ireland and Rosen's Intro to Number Theory and skimmed the first 20 pages or so. It's about unique factorization domains, principal ideal domains, units in groups, polynomial rings, and lots and lots of ideals. And this is in the review-ish part - 200 pages later, in the actual "Algebraic Number Theory" section, it's all about field extensions, isomorphisms, separability degree, etc. These are all assumed to be known, so it's not as though those 200 pages were reteaching the reader about the basic concepts from the first year of algebra. So to get through the material in this (very good, highly recommended, yet still introductory) book, one needs much of the material from a year of algebra. One has to describe groups, rings, ideals, and fields to get through the first 10 pages. So to 'skip algebra' before learning number theory is a misnomer, as one would have to learn it anyway to begin. It would simply take an author much longer to get to anything, forcing much longer and belaboured books.


Now a harder bit - could one learn the basics of algebraic number theory without analysis or topology? Maybe for a bit. But one will likely want to topologize things, use some topological groups and rings, talk of the convergence/divergence of functions over number fields, etc. Maybe you'll want to integrate something over adelics using a  Haar measure, maybe associated with the Langlands Program or something. This requires intense amounts of algebra, analysis, and topology to understand. Intense enough that I think of creationists' 'irreducible complexity' when I think of it, although this is a hyperbole.
[2] Why are these things so important and why are they taught for 3 years? I don't know about how long they're taught - at my undergrad, one more or less takes 3 semesters of calculus, 2 semesters or real analysis, 1 semester of complex analysis, 2 semesters of algebra, 1 semester of topology, among other things (we also required combinatorics, linear algebra, and differential equations, which I would say are also widely applicable). One could fit the algebra, topology, and analysis into 1 year if one really, really wanted to do so. 
But as to why they're important? It just turns out that way, I suppose. But I'm surrounded by mathematicians (current and blossoming), and we all use topology, analysis, and linear algebra with great regularity. I've always sort of wondered how much (abstract-ish) algebra an analyst might use - I don't know much about that. But knowing a little (like a semester or two each) might be enough to give sufficient background to recognize when some bit of unknown material is afoot, find a source to learn more about it and, if you have a really good understanding of the basics, maybe even the ability to understand the source more easily. It can be a pain to see something (say, an integral on a manifold or something) only to go to the wrong source (in this example, a calculus textbook because you saw an integral), just like it's a pain to go to the right book (one of Lee's Manifolds books or something) only to be unable to understand it, forced to backtrack.
[3] I don't know of any books integrating advanced topics and basic material. I sort of wonder about them - knowing the basics of algebra,analysis, and topology beforehand meant that when a new theorem/lemma/corollary/exercise came up, I had to search for an answer from a wide domain. I wonder if such a book would read like a calculus book, in the sense that material is presented 1 idea at a time, things are proved from it using 1 idea at a time, and the exercises at the end would be based on this idea ("now, boys and girls, we're going to solve 20 integrals using partial fractions"). But I digress.
[4] It is hard and mysterious to say what is or is not important. The basic classes of topology, analysis, and algebra are pretty pared down already. If one knew, say, Munkres Topology, Herstein or Dummit&Foote Algebra, Rudin and/or Folland Real Analysis, and Conway or Ahlfors Complex Analysis, then one would probably have a good start.
And there is something to be said for being able to do something. Computation can facilitate learning. It can elucidate and make something concrete. It might provide heuristics, etc. I wouldn't ever advise anyone to skip all sort of 'applied bits' or computation, just like I wouldn't tell someone to only do computation.
A: you use these list http://www.math.niu.edu/~rusin/known-math/index/index.html of mathematics subjects, and you should follow the order in learning mathematics it's important in order to build a strong knowledge.
