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Is it possible to ever be a mathematician if I struggle with Olympiad problems?

  1. Jan 13, 2016 #1
    Hi there,

    When I was a child I had a natural inclination towards numbers and learned numbers before alphabets. However, due to bad teaching, I lost interest in the subject but still managed to score good marks in it throughout school. When I came to pre university classes, I regained some of my lost passion for Maths. I am in college now and have started reading a lot of Maths and this always bothers me.

    I have noticed about by personal temperament that I learn better on my own than I do under authority. I learn better when I am reading books and studying on my own than when I am sitting in a class and listening to six hours of people talking.

    So, when I develop an interest in a subject, I learn a lot about it on my own. I had a lot of interest in Maths, not necessarily the Maths taught in college and in the second semester I realised I could indulge my need for learning on my own. Till that time, I always depended on school to learn subjects like Maths and Science and never thought I could do it on my own. With this revelation, I went down the rabbit hole. I started reading lots of Maths books. Books about Maths history (Journey Through Genius, A History of Mathematics by Victor Katz), books about recreational Maths(books by Ian Stewart, Martin Gardner, Ross Hosenberg), books about problem solving(Thinking Mathematically, Arthur Engel, Alan Schoenfeld, Sanjay Mahajan), and books about particular topics in Maths that I had no idea about like Visual Complex Analysis by Tristan Needham. There were many other books that I started reading. I should note that I never finished any of these books. I would read a little bit, and then get scared of not understanding something and then reading another book and returning to it once again (or sometimes not).

    I did this for the sheer joy of learning. I enjoyed studying Maths more than I ever did in all my school and pre-university years. In fact, for many years I did not enjoy studying Maths in school inspite of an inclination towards the subject in my early childhood. I often got intimidated by books like Arthur Engle which dealt with training for Olympiad. If I could not solve those basic problems, how could I hope to understand differential equations. However, my worry may have been misplaced because the Engineering Maths class in college which included differential equations needed a greater amount of mathematical knowledge, but not a greater amount of mathematical skill than the problem solving book.

    So, even though some of my peers think I'm good in Maths because I solve certain questions in class quickly or that I understand mathematical ideas quickly in class, or because I've heard the name of a famous mathematical problem, anecdote or mathematician a teacher brings up in class, they don't understand it's because I already spent a lot of time reading about other such ideas even if I didn't read that exact same idea. I didn't know it would help me in understanding that particular idea in class nor was that my motive, but it coincidentally did. It's just because I have spent a long time developing a vocabulary of mathematical ideas, and tricks, and the history o mathematics, not because of some divine talent.

    I am made severely aware of my lack of talent when I go through books like Arthur Engel's, especially when I realise kids work through that book to prepare for the Olympiad exam which I didn't know what was until a year ago (at 18 years). So yes, maybe I can solve that differential equation or Fourier transform quickly but mostly because they're just quick applications of formulas ... Just advancing knowledge not skill.

    It always bothers me that if I don't have enough skill to solve the simple (or perhaps elementary is a better word) problems of an Olympiad book, how could I ever apply skill I'm more complicated concepts like differential equations and Fourier transforms and such. Even though my marks may be high in the university exams because they deal with the same type of standard questions, which test breadth and knowledge not depth and skill, I feel like a fraud because I'm simply aware of my stark limitations by Olympiad questions. Is it still possible to be a computer scientist or a mathematician ?
     
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  3. Jan 13, 2016 #2

    Krylov

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    Of course it is. I'm living proof. No, I'm certainly not a "divine talent" (as you phrase it), but I know and love my trade :smile:

    I did well during mathematics university studies, in spite of absolutely detesting anything that has to do with "olympiads" of any sort or form. Since I was little, I have always refused to do mathematics (or anything else, for that matter) in competition like a trained monkey. Whether I would be good at it or not, I don't know, although I suspect the latter, since I do not excel at dealing with that kind of pressure.

    So please, go on and learn more in the way that suits you best: by studying topics of your choice in the comfort of your own desk, possibly using your acquired skills in the classroom afterwards. In my view, olympiads are not the gold standard of mathematical skill or talent.
     
  4. Jan 13, 2016 #3
    I am too told to enter the Olympiads now. But I am asking not from the competitive angle of Olympiads. I am asking about the kind of problems that they ask. I have no serious ambition of writing that exam. Olympiad problems were a way of explaining the kind of problems outside my grasp. The problems are very simple and require no calculus. Yet, I find them so I inapproachable and their solutions strike me like they came from nowhere. So, it always bothers me if I can ever accomplish anything with complicated mathematical concepts like transformations, differential equations, abstract algebra, etc. if I can't even do these elementary questions.
     
  5. Jan 13, 2016 #4

    Krylov

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    Are you interested in solving these olympiad problems? Do these problems appeal to you, do they make you curious?

    If yes, then I'm sure that you can train for them and become pretty good at solving them rather quickly. Specially in Middle and Eastern Europe there is a long tradition of "drilling" students for this purpose. However, if you merely feel that you should be able to solve them, then I would just leave them alone and instead concentrate on something you find more interesting.
     
  6. Jan 13, 2016 #5
    It's more the latter. I feel like I should be able to solve them because they're so teasingly simple, yet so hard.
     
  7. Jan 13, 2016 #6

    Krylov

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    In that case, just solve the ones you like, perhaps use them as "mental chewing gum". However, I would recommend giving up on feeling obliged in any way to solve them, or to be able to solve them. As long as you enjoy their mental challenge, good, but otherwise you don't have to prove anything to anyone.
     
  8. Jan 13, 2016 #7
    True. What are some tips you'd like to give on how mathematics should be studied ? Both for developing mathematical knowledge (learning more concepts and mathematical ideas) and mathematical skill (solving problems). Also, what were some influential books on your life and career ?
     
  9. Jan 13, 2016 #8

    Krylov

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    I'm not sure whether I'm really in the position to answer these (nice) questions, because I'm just at the start of my "professional life" and because the answers are probably very much dependent on the individual, but I can make some brief attempts. Maybe before answering it is a good idea when I ask about your current program of choice in university? (I don't think you told that yet.) Is there a certain direction in mathematics that you tend towards?
     
  10. Jan 13, 2016 #9
    Actually, I'm from India and I'm enrolled in a Computer Science Engineering course. I like studying Maths on my own. To be frank, the areas of Maths I enjoy most are well written expository works dealing with its history, and which present ideas in innovative ways like Flatterland did.

    Some of the things I like about Maths are mathematical induction, Fibonacci numbers, Abstract algebra and problem solving.
     
  11. Jan 13, 2016 #10

    Krylov

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    Probably the kind of mathematics that connects well to your interests and to CS and will help you a lot there is discrete mathematics, such as combinatorics and graph theory. There exists a nice guide with tips on how to study mathematics in general, irrespective of the specific field. Perhaps you find it useful?

    I could give you some names of authors and titles of books that I found inspiring, but they are mostly on specific topics that do not really have your primary interest.

    However, I can say that it has always helped me to focus on specific problems (either purely mathematical or of an applied nature) while finding my was through any field. It is easy to get lost in the large amount of available information. By making mistakes I learned to work from the particular to the general instead of the other way around and this has served me well.
     
  12. Jan 13, 2016 #11
    Stick with engineering. That's my recommendation. You'll be able to use math there.
     
  13. Jan 13, 2016 #12

    Krylov

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    How does this relate to the original question and what was discussed so far?
     
  14. Jan 13, 2016 #13
    Krylov, are all the books about mathematics that you have read text books about specific subjects ?

    I like a lot of work which talk about the history of mathematics like the works of William Dunham. Victor Katz is another good mathematical historian. I also like books on recreational maths like Works of Ian Stewart and Martin Gardner. Books dealing with particular gems of mathematics, involving certain pieces of ingenious thought are also liked by me like Ross Hossenberg. A text book I liked because of its elegant style is Visual Complex Analysis by Tristan Needham. I have a problem finishing the books though.

    Which area of mathematics do you specialise in ?
     
  15. Jan 13, 2016 #14

     
  16. Jan 13, 2016 #15

    Krylov

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    I'm interested in your reasoning behind the recommendation.
     
  17. Jan 13, 2016 #16

    Krylov

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    Yes, and research monographs. I realize that during the last decade or so, that is the mathematics literature (in book form) that I have been reading.
    That is beautiful :smile: Then, to come back to your original question, please concentrate on reading those books instead of feeling obliged to read books on olympiad type problems. This is my view, in any case.
    I have this, too. Once I was under the impression that I have to read every mathematics book cover-to-cover to benefit from it. While this may certainly be true for some basic textbooks, I found that other books can also be read in a nonlinear fashion, picking the part that interests you most at that point and returning to other parts whenever the need arises.
    I enjoy differential and integral equations and their analytical and numerical treatment, using techniques from linear algebra and functional analysis. At the moment my favorite area of application is classical mechanics as it appears in engineering problems.
     
  18. Jan 13, 2016 #17
    What are some of the books you have enjoyed ? Also, as an undergraduate, I'd like to start reading research papers. Do you know where I can start so that I can understand the papers ? I'd like to start with some preferably easy things so I understand what's going on.

    The problem with reading those kinds of books is that ultimately I feel like I'm only getting shallow knowledge and not the 'real' knowledge of the academic text books, per se. I too struggled with that notion that a book has to be read cover to cover to get benefit from it and then I realised that if it was like that, I hadn't read a single book ! And, yes, I think it's better to just read the parts you like and return to it.

    Do you lean more towards the analytical or numerical side, and towards the pure or the applied side ? Problems about Physics scare me a bit. I was never any good at Physics.
     
  19. Jan 15, 2016 #18

    Krylov

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    One book that I loved as an undergraduate was Nonlinear Dynamics & Chaos by Strogatz. Amusingly, now I don't like it anymore: I find it sloppy, it is not a mathematics book according to my evolved standards. Nevertheless, I would certainly recommend it to a beginning student to get a taste of what studying differential equations can be like, and how it applies to everything around us.

    Since then, books I return to over and over again include
    • Introduction to Functional Analysis by Taylor & Lay,
    • The Theory of Matrices by Lancaster and Tismenetsky,
    • One-Parameter Semigroups for Linear Evolution Equations by Engel and Nagel,
    • Theoretical Numerical Analysis by Atkinson and Han,
    • Analysis for Applied Mathematics by Cheney,
    • the books by E. Zeidler on (applied) linear and nonlinear functional analysis,
    • the various books by M. Krasnosel'skii on the same broad subject
    • ...
    You wrote earlier:
    I think there are some journals that focus specifically at undergraduate students. The articles are serious, but written in an accessible way. I know of such a journal in Dutch, but I'm not sure about what is available in English. Maybe @mathwonk could provide you with some advice on this? (In particular, I know almost nothing about abstract algebra.)

    I think to deepen your understanding, making problems helps best. Be very critical about the main text as well, fill in gaps as you proceed, try to give (parts of) proofs yourself. The guide that I mentioned earlier has a number of good tips about this.

    In fact, numerical analysis can be very analytical when your aim is to prove that a certain approximation does what it is expected to do. I think that mathematics is best seen as a unity, not divided into "pure" and "applied". For me, it is most satisfying to rigorously prove something about (or inspired by) a concrete problem, preferably from mechanics and preferably with a numerical component. I feel very pleased when I can start in an abstract Banach space and end up with well-crafted MATLAB or Fortran code :smile:.

    I'm not that good at physics either. I like problems from physics and engineering, but lack most of the intuition of a physicist or an engineer. That is why I prefer to rely on mathematics. Your field, computer science, has many ties to mathematics. You are fortunate to have an interest in both.
     
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