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What's left to go?

  1. Jan 21, 2009 #1
    I'm currently trying to gain as broad a mathematical base as possible, and here's what I've done:
    Axiomatic ZFC Set Theory, Category Theory

    Analysis (Real, Complex, and Abstract), Algebra (Abstract, Linear), Differential Geometry, and Point-Set Topology

    What are some other areas of mathematics that would be useful in gaining a very broad level of knowledge in mathematics?
  2. jcsd
  3. Jan 21, 2009 #2
    Although it often fits in with differential geometry, I think algebraic topology is rather important to know. In addition, since it plays such a large part in pure mathematics, it wouldn't hurt to learn Lagrangian and Hamiltonian mechanics.

    Also, it's difficult to gauge your actual level of mathematics from your description. It could be the case that you need to go a bit more in depth in certain categories to have more of a mathematical base. For example, have you studied differentiable manifolds and Riemannian geometry?
  4. Jan 21, 2009 #3
    I'm well familiarized with most of the topics of all the areas I mentioned except except differential geometry, which I'm still learning. I have looked into Riemannian Geometry and I think I'll study it after Algebraic Topology. Also, would Geometric Topology be useful? Or Differential Topology? Or Mathematical Logic?
  5. Jan 21, 2009 #4


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    In terms of elementary topics, I notice that you seem to be missing both discrete mathematics (e.g. combinatorics, graph theory) and number theory.
  6. Jan 21, 2009 #5
    Just don't go lunatic, that's my only advice. :-)
  7. Jan 21, 2009 #6
    I tried studying number theory, but I lost interest rather quickly. There is something about number's I've always seemed to dislike...
  8. Jan 21, 2009 #7


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    What other have said plus


    Numerical/Applied Analysis
    Asymptotic Analysis
    Calculus of Variations
    Finite Calculus
    Difference Equations
    Ordinary Differential Equations
    Partial Differential Equations
    Integral Equations
    Integrodifferential Equations
  9. Jan 21, 2009 #8


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    Have another look. Most books and courses concentrate on dull matters. Number theory is very broad and deep. It draws on many other areas of mathematics. There are transendential, computational, algebraic, analytic, elementary, and other areas.
  10. Jan 25, 2009 #9
    If you are interested in the foundations of mathematics, it may also do well to look into Model theory (cf the text by Bruno Poizat) and topos theory.
  11. Jan 25, 2009 #10
    I second the suggestions to study geometry and combinatorics, you've overlooked some of the most interesting branches of math!
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