which math classes shall I take and which direction shall I choose?

Discussion in 'Scientific Statistics Math' started by comtech, Nov 19, 2005.

  1. comtech

    comtech Guest

    Hi all,

    I am an engineering student who recently come back to school from
    industry part-time. Previously I did a lot programming. I found in
    math, probability, stochastic processes and matrix theories are
    particular interesting.

    Since I am back from industry, I want to do some research work that is
    more theoratical. I want to find academic research job in the future.
    However since I was from engineering background, I cannot go pure math
    since that's not my advantageous points.

    I am thinking of doing applied math and applying math to other
    applications and fields.

    I found statistics is an interesting field of applied math because it
    has both theory and applications. Optimization might also be good. Data
    mining and finance might also be good. I haven't decided whether I
    should become a statistician or probabilist yet. I equally like them
    both. Which direction shall I choose in order to do good research in
    the future? In which area of research will my past experience in CS and
    programming help? Any other applied math directions that allows some
    great research to be done?

    Working towards statistics theory and applications, optimization, data
    mining and finance, I want to strengthen my math levels. I am planning
    to take some undergraduate and graduate level pure math classes. There
    are many to choose; for example, is Group Theory useful for me?
    Representation theory? Category Theory? Topology? Differential
    Geometry? Algebreic Topology? String theory? Manifold? Number Theory?
    etc. ?

    I hope I can find a map that guides me through all the math classes and
    their hierachies. I want you know that I really love math... although I
    have been spending many years on programming previously... please help
    a willing heart! Thank you very much!
     
    comtech, Nov 19, 2005
    #1
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  2. My .02 follows:
    Suggestions: Probability modeling, stochastic processes, optimization,
    computational statistics, algorithms, computer science, network theory,
    artificial intelligence, bayesian methods, operations research.
    Web-related data mining? Genetics and proteomics research? Knowledge
    representation? Modeling?

    to take some undergraduate and graduate level pure math classes. There
    are many to choose; for example, is Group Theory useful for me?
    Representation theory? Category Theory? Topology? Differential
    Geometry? Algebreic Topology? String theory? Manifold? Number Theory?
    etc. ?
    To be honest, all those seem way too theoretical if you're talking
    about job-relevant learning.
     
    John Uebersax, Nov 19, 2005
    #2
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  3. P.S. If you want to try something really "avant garde" you might check
    out quantum computing.
     
    John Uebersax, Nov 19, 2005
    #3
  4. comtech

    Herman Rubin Guest

    You will find a strong foundations of analysis course
    essential, and an abstract algebra course useful.
    The group theory in an abstract algebra course should
    be sufficient.

    In the courses listed, probably reading in abstract
    point-set topology would be helpful; finding such a
    course may be difficult at this time. One cannot
    understand probability, including the computational
    aspects, without measure theory, and analytic function
    theory (functions of a complex variable) is computationally
    very useful.

    In statistics also, get the concepts and "abstract" ideas.
    The rest is much easier with these. Learning how to
    calculate first can make understanding difficult, and in
    any case, it takes longer. If you understand something,
    you can figure out how to apply it, and you will find
    most of the topics you have listed as such applications.
     
    Herman Rubin, Nov 20, 2005
    #4
  5. Take this with a mountain (as opposed to a grain) of salt.
     
    Stephen J. Herschkorn, Nov 20, 2005
    #5
  6. comtech

    Herman Rubin Guest

    One does not have to prove the measure-theoretic theorems,
    but statistics is based on the Radon-Nikodym Theorem, also
    known as the likelihood ratio. As for the computational
    usefulness of these, I have so used them.

    One problem posed to me about 40 years ago, of interest to
    the person posing it for applied reasons, was to find to
    good relative accuracy the probability of extreme tails
    (probability around 10^-8) of the sum of many absolute
    normal random variables. It was relatively easy.
     
    Herman Rubin, Nov 20, 2005
    #6
  7. comtech

    comtech Guest

    Hi Herman,

    Thanks for your invaluable advice. I know you are an expert in
    statistics. Could you please name a few current "avant garde" research
    directions in statistics? I am trying to read some overview articles in
    statistics to have a general grasp of the current trend in
    statistics...

    Also it looks to me that probabilist and statistician are two type of
    different professions... I am still have difficulty in figuring out
    which direction I should go...

    As for the math classes, I was trying to draw a roadmap for myself in
    the next 1-2 years... Also it is interesting to learn what will be my
    advantage in statistics comparing with people with math backgrounds? I
    guess even I finish all the math classes mentioned, since I am old, and
    I am parttime, I won't be able to compete with math-degree-holders in
    terms maths... But I have programming, signal processing, EECS back
    ground, will that make me a unique researcher and survive in the
    statistics/probablist community?

    Thanks a lot!
     
    comtech, Nov 20, 2005
    #7
  8. I second all of the remarks by Herman Rubin in the message above,
    especially this one.

    Robert Dodier
     
    Robert Dodier, Nov 20, 2005
    #8
  9. comtech

    comtech Guest

    For example, what kind of concepts and "abstract" ideas in statistics?
     
    comtech, Nov 20, 2005
    #9
  10. comtech

    comtech Guest

    I think if I want to be a professor I have to stand a little higher and
    be more theoretical... such that my work can last a little longer ...
     
    comtech, Nov 20, 2005
    #10
  11. comtech

    quasi Guest

    Then, as you aim towards the applied areas, resist the tendency of
    some students (and some teachers) to settle for the "how" without the
    "why". Of course, this costs you a lot in terms of time so it won't
    always be practical -- you'll have to make some compromises.

    But from things you've discussed in the past, it's clear that you are
    missing some of the basic tools of sets, logic, proofs, and these
    tools are absolutely essential if you want to achieve your stated goal
    of becoming "more theoretical".

    You probably should try to make up this gap via self-study, but don't
    avoid it, otherwise there's not much chance of mastering the
    theoretical parts of any branch of math.

    quasi
     
    quasi, Nov 20, 2005
    #11
  12. comtech

    comtech Guest

    Hi Quasi,

    Thanks for your help along the time. Yeah, i am studying those basics
    now... however I have to also plan for the next 1-2 years... that's why
    I want to draw a map of driving directions... this map is going to be
    something I will stick with in the next 2 years...

    Aside from the courses I've listed, do you think Algebraic Geometry is
    useful too?

    I am just trying to figure what should be in my weapon list if I want
    to be a good researcher in the applied math fields that I've mentioned:
    data mining, theoratical statistcs, probabilities, stochastic
    optimization, game theory, math finance, etc.

    If you get a chance, please answer also my other questions, thanks a
    lot!
     
    comtech, Nov 21, 2005
    #12
  13. comtech

    quasi Guest

    Of all the many good replies you've received, reread the one above. In
    my opinon, it's right on target.

    quasi
     
    quasi, Nov 21, 2005
    #13
  14. comtech

    quasi Guest

    No.
     
    quasi, Nov 21, 2005
    #14
  15. comtech

    quasi Guest

    However to determine the size of the required mountain, you might
    might need measure theory.
     
    quasi, Nov 21, 2005
    #15
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