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Right now I'm a senior in high school and I will be submitting a number of applications for undergraduate admissions with deadlines ranging from November $30^{\text{th}}$ to early February. However I recently had the chance to talk with a number of rising high school seniors who plan to major in physical sciences e.g NMSC finalists and other national awardees, and it seems all of them claim to have conducted "research" in there planned field of study. Whether it be shadowing a graduate student or even claims of co-authoring papers, the whole thing just seems very silly to me. I mean the idea that a high school student is capable of finding or writing anything substantial seems far-fetched in my opinion. With that said I am sure there are obviously a few special cases with students who are exceptionally "gifted", of whom have actually written something substantial. But the shear number of people I have been talking to who claim to have conducted "research" in one form of another just seems unrealistic to me.

So my problem is that i'm really worried I might not be making my applications realistically competitive with other students who are applying to many of the schools that I would like to attend. Now generally speaking I am not at all familiar with academia and much more so I am clueless when it comes to the topic of research, but I feel like I should try and write/submit something for my applications. Personally, I have been self studying mathematics for about $4$ to $5$ years now, not including general college preparatory mathematics, and through this time I have taken note of curious identities as well as interesting bits/proofs/ideas, so I want to know if there is someway I can use any of this and possibly turn some of it into a paper and then get it 'published' or something, so I can put that on my application.

So my question is how would I even start to do something like that?

Also I am sort of confident most of the stuff I have written is likely un-substantial or trivial in nature, if I did 'publish' something or 'submit' work. Would there be any way I could take it down in the future? Out of fear of possible embarrassment somewhere down along the road when I become more educated, at which time I wouldn't want a paper like that floating around. If it helps I posted several of the identities/bits bellow I have scribbled down in the past, to help anyone recommend a place where I could 'submit' or 'post' something of this nature.

$$\frac{1}{\pi}=\frac{1}{3}-8\sum_{n=1}^\infty e^{-2\pi n^2}n\coth(\pi n)-2\sum_{n=1}^\infty e^{-2\pi n^2}\text{csch}(\pi n)^2$$

$$\sum_{m=1}^\infty\frac{1}{m^7}(\frac{\sinh(\sqrt{2}\pi m)+\sin(\sqrt{2}\pi m)}{\cosh(\sqrt{2}\pi m)-\cos(\sqrt{2}\pi m)})=\frac{13\sqrt{2}\pi^7}{56700}$$

$$L(2,\chi_4)=G=\frac{11\pi^2}{120}+6\sum_{m=1}^\infty\frac{e^{2\pi m}}{m^2(e^{2\pi m}-1)^2}$$

$$\sum_{n=1}^\infty\frac{x^{2n}q^n}{y^{-2}-q^n}=\sum_{n=1}^\infty \frac{y^{2n}q^n}{x^{-2}-q^n}=\sum_{n=1}^\infty (xy)^{2n}q^{n^2}\frac{(1-xyq^n)(1+xyq^n)}{(1-x^2q^n)(1-y^2q^n)}$$

Ethan Splaver
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    This might have been worthwhile if you had submitted a paper to a journal earlier, but it usually takes several months between submission and publication. Still, you might at least be able to claim that you've submitted a paper to journal, and you can even include the paper in your application. I wrote a paper in high school that got rejected, but I still discussed the attempt in my application. – Keshav Srinivasan Nov 01 '13 at 01:22
  • This doesn't directly address the question, but If I were you I wouldn't worry too much about having to do fancy things to satisfy admissions officers. My impression of Caltech (I don't know about MIT) is that they mostly look at more traditional indications of ability in math and science such as standardized test scores and transcripts. – Trevor Wilson Nov 01 '13 at 01:28
  • @TrevorWilson I have taken all the standardized tests in mathematics indicated for each school I am applying to and there is no room for improvement on any of them. I really think I need to peruse this or something similar to account for my lack of participation in extracurricular activities with other students, and further to verify and show that I really am interested in mathematics and have been studying it for these past years. – Ethan Splaver Nov 01 '13 at 01:33
  • @Ethan It probably can't hurt to do such things, but I'm not convinced it's necessary (it wasn't for me.) It might help to document your self-study somehow. Was it supervised by a teacher? If so, a recommendation from that person might be a good idea even if it documents effort rather than results. – Trevor Wilson Nov 01 '13 at 01:38
  • @TrevorWilson Unfortunately it wasn't, I learned most of my college preparatory mathematics through sites like khan academy and E-books up until I was around $13$, and the rest of my self study was through online open access courses and undergraduate texts in mathematics I found online and borrowed, I only recently started participating on online sites like this about a year ago. – Ethan Splaver Nov 01 '13 at 01:43
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    Doing research in mathematics and in natural sciences are two different things. For example, you can "do research" in ecology by literally counting the number of bugs on a tree and things like this. Research in mathematics requires usually a very serious background not expected from a usual average high school student. So I would not worry too much about adding any "research" to your applications. – Artem Nov 01 '13 at 01:47
  • @Artem If that is the case I am worried admission officers who read my application wont be able to make that distinction. – Ethan Splaver Nov 01 '13 at 01:52
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    @Ethan maybe you can direct them to the Math Stackexchange so they can see for themselves your ability :) – Tyler Nov 01 '13 at 01:56
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    I believe you can remove what you upload from the arXiv, and if so I think that is the kind of thing you are looking for. You can make your work known publicly, update it with revisions or take it down completely. Also, I would not underestimate the ability of admissions committees to appropriately judge potential. They will know that a research publication in mathematics I more difficult than in other areas and that it is not reasonable to expect such. When a high school student has a publication, a large factor is good opportunity rather than being smarter than those who don't. – Ragib Zaman Nov 01 '13 at 03:20
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    @RagibZaman You cannot remove your texts from arXive. – Artem Nov 02 '13 at 23:53
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    @Artem I agree that nobody's going to expect a high school student to have done any research and that mathematics and science research are completely different. But the converse to your other statement is that you can "do research" in mathematics by literally writing formulas on a piece of paper, whereas research in the natural sciences often requires access to very serious equipment and laboratories. – David Richerby Mar 01 '14 at 18:29

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The only part of your question which I feel I'm qualified to answer is the last paragraph:

Yes, whatever you write now you will likely look back on as insubstantial or trivial. However, you should not be afraid of your junior accomplishments. Nobody will hold the quality of your first papers against you - you will be judged on the basis of either your most important or your most recent work. This is actually very important, because if you are aiming to seriously go into some field - whether it is pure math or something else - you will, at some point, have to write your first paper. If you are afraid of embarrassing yourself, you will never get it done.

I know someone who published a paper in high-school. Sure, it wasn't very good, but a few years later down the line, he's really made a name for himself, and is enjoying a very prestigious position at a very prestigious school. His career is well on its way, and I'm sure he would agree that it's important to start early.

And don't forget: if you're really afraid of embarrassing yourself, remember that you can excuse yourself by referring to your relative academic immaturity - as you are still in High School. Nobody will hold your work against you.

Newb
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