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Nicest/most beautiful area/field in maths

  1. Nov 5, 2008 #1

    tgt

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    Which field in maths is the nicest to work in. Nicest is subjective but there are some general occurences like minimal amount of computations but still get a feeling of perfect exactness. Also highly non trivial theorems and results from simple definitions. And a general feeling of 'geez that's clever'.
     
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  3. Nov 5, 2008 #2

    HallsofIvy

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    Whichever one I was working in last!
     
  4. Nov 5, 2008 #3

    tgt

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    What might they be?
     
  5. Nov 5, 2008 #4
    Number theory fits your definition pretty well, but it requires a departure from the 'lower' mathematics mindsets
     
  6. Nov 6, 2008 #5

    tgt

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    What kind of number theory?

    Isn't it very computational?
     
  7. Nov 6, 2008 #6
    Like the study of primes, look up Terrence Tao
     
  8. Nov 6, 2008 #7
    Number Theory is crap! =-P

    If you don't have at least an uncountably infinite domain, you're limiting your potential! I find topology to be a really pretty subject. It's like painting. You start off with fuzzy blobs of paints, smear it in a continuous fashion on your canvas space, and you can end up with either something that looks just like real life or something abstract and weird you didn't know anyone could imagine!
     
  9. Nov 6, 2008 #8

    mathwonk

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    even topology apparently began with euler's bridge problem, a question about a finite graph with maybe 7 vertices.
     
  10. Nov 7, 2008 #9

    tgt

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    It might be possible to start off with something countable and derive a special consequence of it that is countable and related to number theory?
     
  11. Nov 7, 2008 #10
    Clearly the significance of number theory at the very foundations of mathematics is lost on some people. Look up Godel numbering and Godel's theorems of incompleteness.

    One of my personal favorite areas is analytic number theory, its almost a contradictory field, using analysis to deal with the discrete building blocks that are numbers, but despite that, or perhaps because of it, the results therein are beautiful.
     
  12. Nov 7, 2008 #11

    symbolipoint

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    This thread is scary. Could someone earn an undergraduate degree in Mathematics without studying Number Theory?
     
  13. Nov 7, 2008 #12
    Plenty do, don't they? At undergraduate level the main requirements (atleast for pure mathematics) seem to almost invariably be courses in algebra and analysis with appropriate electives.

    I'm only saying that number theory has its own (arguably supreme) beauty, just as the foundations of mathematics does (and the two even entwine, and relate to about every other field of mathematics in the process) regardless whether they are given an emphasis.
     
  14. Nov 7, 2008 #13
    Actually in my university (which is somewhat well regarded), you can easily get a math major without taking number theory (but you may be forced to take something like partial differential equations).
     
  15. Nov 7, 2008 #14

    turbo

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    Beauty? Graphically represented, fractals can be awesome. Once you get beyond the "pretty-picture" comprehension, the self-similarity (and departure from such) and complexity as you drill down to finer scales are mind-boggling. Some fractal patterns are well-represented in nature on small scales, and they may be represented in large-scale cosmological structure as well.
     
  16. Nov 7, 2008 #15
    Almost all universities offer something similar, there's not a great emphasis on number theory or foundations and discrete mathematics as I said before.

    Depending on that you can also avoid number theory at graduate level, sometimes even as a pure mathematician (though most have contact with it at some point).

    Fractals are a bit 'fickle' in their mathematical beauty, no matter how popular they are with the layman. Similarly things like geometry and topology which are ultimately reduced to algebra or taken as a basis for higher analysis, while beautiful on the surface have much deeper intricacies when considered at their core.
     
  17. Nov 7, 2008 #16

    Integral

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    Yes, I did. My interest was then, and remains with numerical methods and applied math.

    I encoutered someone online who claimed to have a math degree and had never taken Real Analysis. Is Real Analysis considered Number theory?
     
  18. Nov 7, 2008 #17
    Sounds a bit far-fetched, unless it was statistics or something, since real analysis is almost always a core component in undergraduate mathematics (though it may not be called that, for instance, in my university there's a Real and Complex analysis course offered for sophomores and then in third or fourth year a Complex variables course and an Analysis course that covers both real and Fourier analysis are available.
     
  19. Nov 8, 2008 #18

    HallsofIvy

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    In some colleges "Real Analysis" is called "Advanced Calculus". That a little misleading because in other (particulary Engineering schools) "Advance Calculus" mean more of what we would think of as regular calculus.
     
  20. Nov 8, 2008 #19

    mathwonk

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    some subjects are concerned with techniques or methods, others are concerned with specific topics or examples. number theory is the study of a specific example, the integers, and generalizations.

    analysis is a technique, the method of approximation via limits, derivatives, integrals, series,...

    analysis as a tool can be used to study number theory. e.g. in complex analysis, the zeroes and poles of holomorphic functions are discrete sets. so one can ask whether certain discrete sets can or cannot be the zeroes of a given holomorphic function.

    such considerations led to the solution of fermats last theorem when frey asked whether there could be an elliptic curve defined by a function with zeroes constructed from the putative solutions of fermats equation.

    riemann also constructed his zeta function from the discrete sequence of prime numbers and asked where the resulting zeroes of that function would lie?

    dirichlet used similar considerations to analyze the behavior of a complex function constructed from sequences of primes and prove there must be infinitely many primes in certain arithmetic progressions.

    i think the lesson is that not every one needs to know number theory, but a number theorist, along with everyone else, needs to know analysis. i.e. you may be missing out on a beautiful subject, but without number theory you can still do most other subjects, whereas without analysis you have trouble doing anything deep.
     
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