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I immediately apologize for my English, Google translator is my assistant. I couldn't find the information in my own language. My question is addressed to people who understand mathematics. I hope for your help.

As you know, any branch of mathematics cannot exist by itself. Accordingly, as a student, having knowledge at the level of limits/integrals, having opened a textbook on category theory, homological algebra or topological groups (all these as examples), I will not be able to understand anything there. At the same time, there is no explanation anywhere regarding which specific sections of mathematics and at what level should be mastered before starting to study, again, for example, manifolds and cell complexes. I also understand that people who study mathematics as a specialty somehow come to understand what and in what order they should study. But I am not a student of mathematical directions, and I plan to study mathematics for myself. Not as an applied discipline, but specifically mathematics for mathematics. Which particular branch of mathematics would I like to study? I don't know. I would like to take a look at the cutting edge of this science, and what questions are being raised there. It is obvious that this region is very wide, and the issues there are sometimes diametrically opposed. I just don't want to spend time studying everything, because mathematics is not easy. To study random processes when I actually find topology interesting? What for?

In general, my question. What is the relationship between the beginning and the edge of mathematics? Or that there is a root and branches (if mathematics is a tree)? Is there some kind of math roadmap? Or a web? I would like to see its most complex, advanced areas, and what you should know to start diving into one of them. I've come across similar things before, but they don't cover everything. Once again, please note that I am not interested in applied mathematics. No physics or finance, just pure math, only hard.

I really hope for your help!

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    $\begingroup$ You might like the Princeton Companion to Mathematics: press.princeton.edu/books/hardcover/9780691118802/… $\endgroup$
    – Ben McKay
    Commented Feb 15, 2022 at 10:36
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    $\begingroup$ I’m very sympathetic to this question, but having answered (the beginning of) it for myself the only advice I can offer is “spend several years studying the math you think is most relevant to your interests, and adjust as necessary to get where you want to go”. You will be wrong about what you need along the way, but it’s a necessary part of the process and fun if you enjoy doing math. $\endgroup$
    – Alec Rhea
    Commented Feb 15, 2022 at 11:35

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The "road maps" are entertaining, but they are wholly dependent on the point of view of whoever drew them. Do not attach any importance to them.

I would even go as far as to say that you should not even attach importance to the syllabuses of degree courses in universities.

When I was a student in the 1980s, nobody questioned that in pure maths you should start by learning group theory and real analysis. Applied maths (ie applied to physics and engineering) was basically about differential equations and (the mathematical aspects of) computer science amounted to combinatorics.

I became a categorist in theoretical computer science. At that time, only people with pure maths degrees could understand what we were doing.

Now it is all much richer and more chaotic (in a good sense of the word). Physics, computer science and philosophy graduates can understand and contribute to theoretical computer science as well as maths grads can. Ideas from category theory are found in all those subjects and cross-pollinate one another.

Study whatever (numerate) university subject suits you and take your mathematical ideas wherever they lead you.

PS Your English is fine.

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Good question. 100 years ago it was much easier to answer it. For example, the "map" of mathematics drawn by Janiszewski in 1915 in the book "Poradnik dla samoukow" looked as follows:enter image description here

Now everything became much more complex, complicated and diverse. Though there are analogous maps of modern mathematics, too: enter image description here

The last picture (by Dominic Walliman, 2017) was taken from this youtube presentation which contains explanations to this map (drawn during the presentation).

So, if you want to study some part of math, just look at that map and decide what you are interested in and what prerequisites are necessary for that. In any case one should start with the upper left corner of this map in order to understand the language of math.

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    $\begingroup$ I haven't watched the presentation, but the "modern" map you link to seems frankly very idiosincratic. Among the astonishing omissions there's algebraic geometry (!) and commutative algebra (!!). In general it seems strongly biased towards the more "applied" side of things and the math that's used there (like analysis) $\endgroup$
    – Denis Nardin
    Commented Feb 15, 2022 at 10:57
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    $\begingroup$ I'm not convinced that the 2017 diagram is a map. It seems to be more a visual representation of an ontology than of the true relationships / distance between the topics represented. If you try to use it to figure out prerequisites then you'll conclude that mathematical chemistry is required for biomathematics, whereas the former is closely tied to combinatorics and the latter could be dynamics or computer science depending on what it's understood to mean. $\endgroup$
    – Peter Taylor
    Commented Feb 15, 2022 at 11:37
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    $\begingroup$ @DenisNardin Of course you are right. There are many many omissions in this map. For example, there is no Banach space theory there and no good portion of Functional Analysis. Anyway it is just attempt to cover uncoverable. As for me more-or-less successful. Everyone is of course invited to create its own Map of Moderm Mathematics. Maybe after many attempts some acceptable optimum will appear... $\endgroup$
    – Taras Banakh
    Commented Feb 15, 2022 at 11:41
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    $\begingroup$ As someone who works at the thriving interface of group theory and topology, I am amazed to discover that the two subjects are separated by something called "order theory", of which I have never previously heard. The presence of "fluid flow" on the pure side is also very strange. Obviously creating such a map is difficult, but this attempt would give a newcomer a completely erroneous view of the structure of pure maths. $\endgroup$
    – HJRW
    Commented Feb 15, 2022 at 14:53
  • $\begingroup$ @HJRW I am also working in Topological Algebra (but also in Order Theory) and understand your concern. A (partial) justification of this Map of Mathematics is that such a Map is expected to be 2-dimensional whereas the real graph of relations between math disciplines is rather nonplanar. So, it is just an approximation. $\endgroup$
    – Taras Banakh
    Commented Feb 15, 2022 at 15:21

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