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I am a high school senior and I am interested in doing a math research. I hope someone can recommend areas or topics of research that are challenging, rewarding, and yet do not exceed my capability. (I acknowledge this is quite hard)

My math background:

a. I have done competition math (Elementary number theory and combinatorics, Euclidean Geometry, and Algebraic manipulation) and I'm fairly comfortable with proofs.

b. I had my first courses in Multivariable Calculus, Differential Equation, and Linear Algebra (Familiar with fundamental concepts, basic techniques and motivations)

c. I have learned a portion of Abstract Algebra on my own and in summer programs including topics like Lagrange theorem, Vector spaces, Polynomial Rings, and Morphisms.

d. I don't have a good background in statistics and probability

e. I have been exposed to Knot theory and Chaos theory

f. I do have basic programming skills in python and Mathematica, and I can work with LaTeX.

I really appreciate your help!

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  • $\begingroup$ Have you done anything to do with topology? Dynamical systems might also be an option if you like differential equations, calculus, and programming. $\endgroup$ – recursive recursion Aug 19 '14 at 20:50
  • $\begingroup$ I second the above recommendations. Also, rigorous analysis if you haven't seen much (my favorite, I must admit). Special relativity? You seem to be more of a pure guy though... Geometry (as in algebraic geometry, etc.) leading on from topology if you've seen some/are going to look at some. $\endgroup$ – ShakesBeer Aug 19 '14 at 20:56
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    $\begingroup$ Personally, I would recommend solidifying your current knowledge before getting into research. You'll probably learn a lot more by reading and solving problems. Perhaps pick a topic, and learn all you can about it. $\endgroup$ – Bruno Joyal Aug 19 '14 at 20:58
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    $\begingroup$ I think you should search for one recent journal article that sounds really interesting to you, and seriously read it. This could take weeks, but you will learn a lot, and you will also learn something about the cutting edge of research in that area. It's very hard to find research topics of the sort you describe, but apart from making personal connections with mathematicians, reading new papers is a great way. Perhaps check out www.arXiv.org, and don't be intimidated if it takes you a long time to get through a paper. $\endgroup$ – Eric Tressler Aug 19 '14 at 21:13
  • $\begingroup$ Please avoid the tags undergraduate-research and research. We are trying to remove them. $\endgroup$ – 6005 Jul 24 '16 at 19:52
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Ok, here is my sincere suggestion. First: +1 for your question. Since you have done multi variable calculus and differential equations, how about studying the Laplace Transform? This is (I think) new to you but yet, with your back ground, accessible. It is a cool topic (my opinion) with applications within the framework of calculus you have studied. You can solve (systems of) differential equations with it, as well as certain types of convergent improper integrals. I believe this will be a doable challenge for you!

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  • $\begingroup$ I have learned the basic computational aspect of Laplace transform like the translation theorems, convolutions, etc. However, now I only know how to calculate and play with the formulas. What would you suggest for me to have a deeper understanding on this subject?Those integral transforms seem really cool! In addition, in what way can I find things to research that can really did out my originality? I don't wish to do a exploratory project, but rather make up something new. I know it's bold for me to say this, yet that's my end goal. $\endgroup$ – Bohan Lu Aug 19 '14 at 23:43
  • $\begingroup$ @BohanLu Here is one: math.stackexchange.com/questions/376945/… But when I searched the internet, I could not find much of examples. It seems like calculating convergent improper integrals with Laplace is not a common thing. Mostly Laplace is used for Diff Eq related problems. When I did this stuff, I got very intrigued by the fact that a whole class of improper integrals could be evaluated this way, I made my own study out of it. And so I developed categories of integrals that work that way. Anybody else has a suggestion here??? $\endgroup$ – imranfat Aug 20 '14 at 15:19

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