A single topic of mathematics , you can learn most from?

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I've heard people around me say , they have learned more mathematics then they have in grad school just by reading this topic.

Is this true , I know mathematics has a huge diverse background and specializing in many topics , well I don't know if it's even possible.

The implications this has in autistic rigidity and success is tremendous , your basically recommending me a future career in pure or applied mathematics. We typically succeed in highly specialized professions and we can focus for hours on end on a single topic.

So I'm asking the question , is thier a topic of mathematics that is very highly specialized , where it encompasses most fields of mathematics , that you learn from?

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Some barriers:

Would it be too advanced considering that thier is a sort of ladder analogy in mathematics , would you be able to learn it from a top-down sort of way? Would the person have to combine most or all fields? Is it like a fundamental theorem of mathematics? How specialized would it be? How many theories does it encompasses? How abstract is it to learn from? Would it be the most difficult?

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Best Example:

Could you clarify what is your current background and what is it that you trying to accomplish? Category theory does sound as what you are looking for (I don't think I know anything *more abstract* but then again I'm more into the applied side), however, there are quite a few good reasons against jumping into abstractions without having stepped a foot into the field first. I would also add that almost everything one might do as a scientist would be super specialized. I would probably suggest starting with Abstract Algebra.

The term "abstract nonsense" is not always meant in a negative way.
Abstraction doesn't necessarily take you away from concrete problems. It may, on the contrary, provide a more elegant way of solving them.
It will probably move from being 'abstract nonsense' to something akin unifying theories in topology and algebraic geometry later this century. It may eventually even have something to say about physics.

Thier have been several attempts by great mathematicians in finding a unifying theory in mathematics they failed , instead thier have been attempts at uniting theories , examples include the langlands program , fundamental theorem of galois theory , taniyama-shimura conjecture and other's like the monstrous moonshine.

Could you tell me why are you asking an impeding question on the attribute of one , sure it's too know if one would understand such but back to the point , I would just like the experience of others , assess why one would ask such , a pure mathematician wouldn't. Moreover it would be someone as an undergraduate. Mathematicians are pattern seekers , older one's are better at it , a younger one could also do the same but it would be a little more erroneous.

Any other suggestions there , chronologically Mar00? Thanks..

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My mindset is based on:

Three important ideas for every kind of math include assumptions, transformations and representations.

Assumptions include any initial data and constraints that apply to your problem of interest. It could be pure or applied mathematics (or even statistics). Usually pure mathematics have constraints that are more relaxed (i.e. broader) and this can make it harder.

The transformations are simply converting one thing to another. It is another way of way of "decomposing" things. You might do this with a simple algebraic trick, you might do it via some integral transform, or you might do it in a way that doesn't preserve the object itself (like doing an integral transform and keeping only the dominant terms). Learning transforms and their uses is absolutely critical, but what is more critical is understanding why different transforms are used.

The representationis just the explicit definition of the object you are dealing with. In order to analyze or even think about solving a problem, you need to explicitly define both the problem and the "stuff" that the problem refers to. You would be surprised that it's rather recent for mathematicians to define something like continuity in a fairly rigorous way.

By the way , Ruveyn sounds like a mathematician , that was a good example!

Other's include dynamical systems , euclid's elements with hartshorne's companion book - Geometry: euclid and beyond.

Anymore?

All I am trying to say is that in my opinion and experience Category Theory is immensely difficult and those grad students probably ment something else. I simply asked what is your background, this is not an impending quesiton (and one which a pure mathematician would). I think this is highly relevant - I am just trying to be practical here because I do agree with what you are saying on this *philosophical* note. I got the impression from your post that you are in a high-school, apologies if I am wrong. Yet if you are then there are plenty of topics which would give the same effect you are seeking for(if I understand it, that is). Pretty much all of the pure stuff. Philosophical mindset is all good but is only the first step. Hope I make sense. What I mean is that modern maths is so intertwined that there are many many examples of how one theory generalizes another. It's basically what maths is. I doubt this is a theoretical problem because you know its answer if it was.

To Mar00:

I am a newly 1st year undergrad at Waterloo! Close! Going for a general B.Math Degree. I did very well in Math (88 Average) in highschool though. Apology accepted.

From telling from my post , I do seem like a lost highschool student , and I do feel like one , I need your wisdom but don't tell me a can't do it lol.

Oh , so you can't really offer me a chronological course of study then from Abstract Algebra -> ? -> ? -> ?

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Questions:

Well could you give me an idea about how intertwined it is?
How many theories does it encompasses?
What areas of modern math?
What's the most abstract level of math , is it the foundations , I thought the foundations were easy?
Were would you recommend me to start from and end from?
Did you feel the same way when you were a kid?

Congratz!! I got my BSc in maths and it was the most rewarding experience. I really like this link so I am posting whenever I have a chance http://hbpms.blogspot.com/ this is somewhat chronological. Also see http://en.wikipedia.org/wiki/Lists_of_m ... ics_topics (but I am sure there must be something more visual/better than this such as http://www.math.niu.edu/~rusin/known-math/ ).

As ruveyn noted linear algebra and group theory are perhaps starting points for abstract algebra. Some standart textbooks can be found (which I am sure one can dig out of the web) at www.maths.cam.ac.uk/undergrad/course/schedules.pdf . While for linear algebra some stuff is prerequisite one could get into groups right away - you could try Herstein Topics in Algebra to see what it looks like. What would follow next, well these are vast and deep topics, other objects are introduced and various relations between them.

When I was a fresher I remember feeling somewhat similarly, I really wanted to get my hands on the most abstract and fundamental suff, that's just how I am wired so I think your enthusiasm is amazing And sure you can and will do it but its good to have a very clear perspective and keep it real at times I know I wish I did. Its someting like measuring the heigth of the mountain to climb. Then again it is sometimes better not to see how much is left to go.. Sorry, I am a bit cynical these days.

I think maybe someone more pure than myself could be more of a use and particular; I eventually was more inclined to physics. I will think how I would anwer these questions now I am a bit sleepy (sorry if that causes any incoherence). Category Theory is probably it as far as I can tell and Foundations no matter of what is usually mostly advanced and requires a solid knowledge of the matter itself. The other thing might be philosophy of mathematics but it might appear readable only because it is written (mostly) in words

All in all I would advice to enjoy your studies and do a thorough job at it. Once you are in a midway you will probably have a very good sense of where you are at and what is what, there is no need to think about it too much beforehand (not saying that at all, but I think empasis is on present). In the process reading this and that to improve on your own is what probably counts the most. It might sound trivial but that's my wisdom part.

End of thread then , Thank you very much Mar00

For what it's worth, I did an undergrad degree in statistical mechanics.
Maybe my professors sucked, but in all the n courses I took, I had the constant feeling it was impossible to understand anything from the i-th course, for 1 < i < n, without understanding all of the other (n - 1) courses; yet none of the (n - 1) courses were sufficient to understand any of the other courses.

So it's not a ladder; I just stumbled around, half understanding everything until a day of enlightenment when, I supposed, I'd collected enough pieces of the jigsaw to figure out what the picture's supposed to look like. Then bang! I understood everything.

That threshold moment of enlightenment came during a course in propositional logic. It could be that, cos propositional logic elucidates quite nicely all about logical systems, which is what all the abstract areas are generally about: algebra, galois theory... axiom, theorem, corollary, lemma, ad infinitum.