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I have 5 months to go to 17 years and I want to finish calculus 1 and in the other 5 months maybe finish lang linear algebra. I don't think I have talent or creativity (I'm so scared) but I want to commit myself to the maximum so that with hard work I can compete with guys maybe born with more creativity than me (is it possible?). Before I go to university I want to finish what you do in the first three years, and my question is how can I make the most of my time? Is it worth spending all day on a problem? (when I can read the solution), how can I improve my creativity in math? (I'm afraid I don't have it), I have so much difficulty with the problems of the mathematics Olympics but I am also working on those in order to have a medal at the national level (if only I had committed myself a few years earlier. ..). How should I approach the proofs (they also scare me), in my calculus book there were some problems that asked to prove simple things like 0b = 0 but I couldn't do it, is it serious? Could you give me a method to follow in order to become an excellent mathematician and solve important open problems? Thanks

math_lover
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    Consider math programs like https://www.promys.org/ or http://rossprogram.org/. –  Nov 01 '20 at 00:14
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    Study mathematics that you are interested in intensely, and don't worry about becoming great. – Derek Luna Nov 01 '20 at 00:17
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    @DerekLuna I'm afraid of failing – math_lover Nov 01 '20 at 00:18
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    better to do math to live than live to do math – user2661923 Nov 01 '20 at 00:39
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    Stop being in such a hurry. After 36+ years as a math professor I can tell you that trying to cram too much too fast is no solution. Take your time, savor learning actual mathematics, rather than just mass quantities to "exempt" standard courses. – Ted Shifrin Nov 01 '20 at 00:59
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    It is quite normal at your age to have big dreams. It's not when you will be at 70 that you will have ambitions like that. BUT, beware of the trend to consider to be a great mathematician; I have known somebody who was rather good at maths, but was desparate at 45 because (in his own terms) he "hadn't found a theorem". for this reason he was depressed, considering his whole life as a failure. This is the danger if you think to pure maths; in applied mathematics, it doesn't happen in this way; you always make at least little improvements to "systems" you work on.... – Jean Marie Nov 01 '20 at 01:29
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    ...Consider that the opportunities you will meet in the future will make you hopefully make a 45° turn into maths for physics, maths for biology, or maths for physics, for meteorology etc. – Jean Marie Nov 01 '20 at 01:29
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    If there were such a thing as "a method to follow in order to become an excellent mathematician and solve important open problems" then we would all have adopted it, and there wouldn't be any important open problems left. My lack of success at solving any of the Millennium Problems (https://www.claymath.org/millennium-problems) should stand as proof that the method you ask for does not exist. – Gerry Myerson Nov 01 '20 at 02:00

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Don't put pressure on yourself to be a great mathematician. It's no way to start out. It's also, I think, the wrong way to approach math.

Math is a massively collaborative discipline with many, many overlapping and intersecting subfields that run the gamut from applied to theoretical. Mathematical discoveries are generally not the product of a lone genius making a sudden breakthrough, but the fruit of a community of mathematicians making incremental progress on a problem.

The point being: you should think less about how you'll become a great mathematician and more about how you'll eventually fit into the mathematical community.

If you really want to be ahead of your peers, see if you can find a topic or a result that interests you. Dive into the literature and textbooks related to that topic or result and see what you can learn. Don't understand what you're reading? Try to figure out what you need to learn in order to learn that thing. Maybe even pick another topic. Have a university in mind? What are the professors at the university researching? Can you understand their work? What would you need to know to understand their work? Is it interesting? These sorts of investigations are how you get involved with the mathematical community and start to become a productive mathematician.

If you were destined to be "great" (whatever that means), you will be, but only after you've figured out how to work with other mathematicians and build on the results of others. If you can get a jump on these sorts of things, I promise you'll be well ahead of your peers, even if you don't know as much math as them at the start of your degree.

I hope this isn't too scattershot. These are just some of the things I wish someone would have told me before I started my undergraduate math degree. Great mathematicians didn't just start great (some of them may have even started out quite mediocre), they also knew how to keep getting better. A big part of that continual growth is knowing how to learn from your peers and how to research on your own.

Charles Hudgins
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