Areas of Mathematics

October 22, 2013 Leave a comment Go to comments

For one of my upcoming talks I am trying to include an exhaustive mindmap showing the different areas of Mathematics, and somehow, how they relate to each other. Most of the information I am using has been processed from years of exposure in the field, and a bit of help from Wikipedia.

But I am not entirely happy with what I see: my lack of training in the area of Combinatorics results in a rather dry treatment of that part of the mindmap, for example. I am afraid that the same could be told about other parts of the diagram. Any help from the reader to clarify and polish this information will be very much appreciated.

And as a bonus, I included a \LaTeX script to generate the diagram with the aid of the tikz libraries.

\tikzstyle{level 2 concept}+=[sibling angle=40]
\begin{tikzpicture}[scale=0.49, transform shape]
  \path[mindmap,concept color=black,text=white]
    node[concept] {Pure Mathematics} [clockwise from=45]
      child[concept color=DeepSkyBlue4]{
        node[concept] {Analysis} [clockwise from=180]
          child { 
            node[concept] {Multivariate \& Vector Calculus}
              [clockwise from=120]
              child {node[concept] {ODEs}}}
              child { node[concept] {Functional Analysis}}
              child { node[concept] {Measure Theory}}
              child { node[concept] {Calculus of Variations}}
              child { node[concept] {Harmonic Analysis}}
              child { node[concept] {Complex Analysis}}
              child { node[concept] {Stochastic Analysis}}
              child { node[concept] {Geometric Analysis}
                [clockwise from=-40]
                child {node[concept] {PDEs}}}}
          child[concept color=black!50!green, grow=-40]{ 
            node[concept] {Combinatorics} [clockwise from=10]
              child {node[concept] {Enumerative}}
              child {node[concept] {Extremal}}
              child {node[concept] {Graph Theory}}}
          child[concept color=black!25!red, grow=-90]{ 
            node[concept] {Geometry} [clockwise from=-30]
              child {node[concept] {Convex Geometry}}
              child {node[concept] {Differential Geometry}}
              child {node[concept] {Manifolds}}
              child {node[concept,color=black!50!green!50!red,text=white] {Discrete Geometry}}
              child {
                node[concept] {Topology} [clockwise from=-150]
                  child {node [concept,color=black!25!red!50!brown,text=white]
                    {Algebraic Topology}}}}
          child[concept color=brown,grow=140]{ 
            node[concept] {Algebra} [counterclockwise from=70]
              child {node[concept] {Elementary}}
              child {node[concept] {Number Theory}}
              child {node[concept] {Abstract} [clockwise from=180]
                child {node[concept,color=red!25!brown,text=white] {Algebraic Geometry}}}
              child {node[concept] {Linear}}}
    node[extra concept,concept color=black] at (200:5) {Applied Mathematics} 
      child[grow=145,concept color=black!50!yellow] {
        node[concept] {Probability} [clockwise from=180]
          child {node[concept] {Stochastic Processes}}}
      child[grow=175,concept color=black!50!yellow] {node[concept] {Statistics}}
      child[grow=205,concept color=black!50!yellow] {node[concept] {Numerical Analysis}}
      child[grow=235,concept color=black!50!yellow] {node[concept] {Symbolic Computation}};
\end{tikzpicture}
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  1. October 23, 2013 at 8:55 am

    Hi, I have real doubts about capturing present day mathematics in a tree. I would think but that a graph would be needed. The following might be too specialized but I will mention them.

    Umbra calculus l: I didn’t see any mention of Umbral calculus. I have been trying to understand Roman and Rota’s method of rationalizing it by contrasting it with the Pascal matrices. The correspondence is fairly strong with some cross-breeding. Incidentally both can be considered a case of merging discrete linear combinatorics with parameters/polynomials from the real line.
    Differential Equations: I also didn’t see any mention of partial and ordinary differential equations.
    Algebraic Geometry: Putting Algebraic Geometry under abstract seems problematic since part of it is quite concrete.
    Foundational Mathematics: I also don’t see any Foundational Mathematics. People think that they have found a new foundation that is an alternative to set theory. “Univalent Foundations of Mathematics ” from the Wikipedia web page “homotopy type theory”. Don’t bother to ask m anything about it 🙂 Nonetheless foundational mathematics is not dead.
    Optimization: Linear, convex, geometric “programming”

    • October 23, 2013 at 6:22 pm

      I agree! I have an option to create “connections” between the nodes of this tree, and I will go for that at some point.

      I did include ODEs and PDEs (see the second-level leaves in Analysis), but did not mention many other sub-areas of Analysis… I simply didn’t want to overwhelm my listeners with all those “Clifford Analysis”, “p-adic Analysis”, “hyperreal numbers”, “Tropical Analysis”, etc. I would have put “Umbral Calculus” under the same roof as “finite differences,” but I am sure someone out there would be outraged at this oversimplification.

      I decided to wave my hands on the “Foundational Mathematics” issue. If questioned during the talk, I shall point to the big black dot in the center, and say: “Logic, Set Theory and all that stuff is in here!” 🙂

  2. October 24, 2013 at 10:31 am

    Yes, I do have tunnel vision on the things that I consider interesting. You definitely have to draw the line on how specialized/detailed a presentation or organization will be.
    I think there is a vast mathematical reorganization lurking in the future; although looking at history it will probably be a long time coming.
    A case in point is Polya’s combinatoria theorem where, in the class I was taught, group theory and combinatorics are mixed. Now that I think about it; that looks like the work I have been doing with Pascal’s matrix and Umbral calculus. That’s just an idle thought but it illustrates how interconnected mathematics has become without admitting and unifying it.
    Professionals know the connections but to those of us on the outside it seems we have to discover them.
    I will try out your graphing method; but, as usual, don’t expect that immediately 🙂 It does look effective. There is duality in mathematical graphing where subsets that seem to be internal to an abstract layer have to be made external branches because there is never enough room inside for the sub-sub divisions 🙂 But it makes just as much, or more, sense. Upon reflection the subdivisions might _have_ to be brought out, to show where other abstract layers overlap certain subsets. For instance Poyla’s theorem is a subdivision of combinatorics but that particular proof drags in group theory; and it’s probably not the only cross application.

    Good luck on your presentation!!

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