Mathematics Grad. School Application Harvard

[Annonymous111]

I'm interested in applying to a top mathematics grad. school like Harvard, Chicago, or Princeton. I'm wondering what my chances are and I'm especially interested in what would happen in the following scenario:

I've taken only math courses my entired undergrad. education and I've taken grad. level math courses since my first year of undergrad. I've also taken lots of independent study courses on advanced topics in many areas of mathematics. Like in 4-manifold theory, algebraic geometry, algebraic group theory, representation theory, noncommutative ring theory, harmonic analysis, C* algebras, several complex variables ... well you get the picture. Not only that, I've got A's in all these math courses. I've also got a near perfect score on my GRE. And the undergrad. and grad. program in this university is very strong so all the courses are advanced.

I know there are other factors to grad. school like letters of recommendation and personal statements, but would I be a shoo-in to grad. school if I had such a transcript - full of advanced math courses, only math courses, with only A's? Or would grad. schools in math frown upon me not taking other subject courses and reject me? Suppose I hadn't done very much research either. Does anyone know about how such a situation would be viewed?

This is a hypothetical situation but I'm interested in knowing how grad. schools would view it and whether it'd guarantee admission. My gut feeling is close to yes since I don't know whether anyone has such a transcript full of A's and advanced (and this is *really advanced*) math courses. But the big question is how grad. schools would view the "not done research" and the "not done other courses in other subjects". Bear in mind that "not done research" also come with "done advanced math subjects that provide nearly all the background necessary for reading papers in numerous areas of math. I'd think grad. schools would think highly if someone had the background to do research in numerous areas of math (we're talking algebra, analysis topology and geometry - the knowledge is broad) but I could be wrong. Also bear in mind that the courses are highly fast-paced and very difficult grad. courses. (BTW, I'm taking top grad. schools like Harvard here.) Thanks guys ...

 

[snipez90]

A hypothetical situation? So you're *not* actually going to do all that? What a tease.

 

[cristo]

There is never such a thing as a "guaranteed admission". What if everyone applying that year has a similar transcript but with research experience too?

 

[Annonymous111]

A hypothetical situation? So you're *not* actually going to do all that? What a tease.

No I am. But it's still early in the undergrad. education to do all that so at this stage it's hypothetical. But it won't be after a while. I'm still eager to know how it'll play a role in the admissions decisions. Could someone please tell me?

 

[Annonymous111]

There is never such a thing as a "guaranteed admission". What if everyone applying that year has a similar transcript but with research experience too?

OK and thanks for your answer. I get that there's no such thing as "guaranteed admission". But the point is that obviously people get accepted without having such a transcript. I'm interested to know what transcript of the average applicant to Harvard (say) looks like and whether admissions committees will strongly consider accepting someone if he's got such a transcript.

 

[deluks917]

From browsing this forum I'd say a "normal" strong applicant (good research, some grad classes, 900+ GRE, well known university) is enough to get in somewhere in the top-15. MIT/Harvard seem to reject even extremely strong appplicants.

 

[Newtime]

What year are you? I think you're getting ahead of yourself, which is something easy to do , and something I do often. Just take the most challenging classes you can, do as well as you can in them, and see what happens. Whether or not someone on a forum says "yeah you could for sure get in" means nothing unless they are on the admission committee at one of those schools.

One more minor thing: in your list of advanced topics, what is algebraic group theory? Typo? Otherwise it's sort of redundant/vacuous.

Also: I've known people who've come in with about as many hours as one can, but how can you possibly only take math courses for 4 years? Surely your degree has requirements which can only be satisfied by classes at your institution.

 

[Annonymous111]

What year are you? I think you're getting ahead of yourself, which is something easy to do , and something I do often. Just take the most challenging classes you can, do as well as you can in them, and see what happens. Whether or not someone on a forum says "yeah you could for sure get in" means nothing unless they are on the admission committee at one of those schools.

One more minor thing: in your list of advanced topics, what is algebraic group theory? Typo? Otherwise it's sort of redundant/vacuous.

Also: I've known people who've come in with about as many hours as one can, but how can you possibly only take math courses for 4 years? Surely your degree has requirements which can only be satisfied by classes at your institution.

Do you know what an algebraic group is? You can look it up on Wikipedia or on the internet I think. It's basically an algebraic variety that has the structure of a group. Elliptic curves and abelian varieties are examples. Algebraic group theory is the study of algebraic groups.

Yup I'm in my second year and I've already taken some of the classes I listed. So I've been taking lots of advanced grad. courses in my first and second year. My question I guess (which I think was misunderstood) is that: does the typical very strong applicant to a top grad. school have this kind of transcript?

 

[Annonymous111]

Could someone please answer my question? Thanks.

 

[Newtime]

Do you know what an algebraic group is? You can look it up on Wikipedia or on the internet I think. It's basically an algebraic variety that has the structure of a group. Elliptic curves and abelian varieties are examples. Algebraic group theory is the study of algebraic groups.

Yup I'm in my second year and I've already taken some of the classes I listed. So I've been taking lots of advanced grad. courses in my first and second year. My question I guess (which I think was misunderstood) is that: does the typical very strong applicant to a top grad. school have this kind of transcript?

Regarding your main question: yes, absolutely that's what they look for. I'm still an undergrad, but everyone I've talked to (peers, grad students, professors, etc.) says this. Take the most challenging (and don't forget interesting!) courses you can, and do well in them. If you're making good grades in courses like this beginning from your freshman year, then you're doing all the right things I would say. Of course you'll want to consider doing other things like research, etc. but I'm sure you're doing that too.

Unrelated: I had no idea algebraic groups existed. I assumed that since you already have a variety, which has more structure than a group, you wouldn't be interested in what other structures you can attach to it. Very interesting. http://en.wikipedia.org/wiki/Algebraic_group

 

[Anonymous217]

Grad classes since freshman year?

 

[deRham]

To the original poster, you do have a very good shot at schools, but I would not get hopes too high up. I don't think it is uncommon to have your kind of transcript at the highest caliber schools, probably not at Harvard.

 

[TMFKAN64]

*No one* is a shoo-in at the top schools.

I would about your lack of research experience as well... students who get into the top schools tend to have great grades *and* some experience... so having just great grades makes you a below average candidate.

The good news is there is still time for you to do something about that!

 

[deRham]

I don't know about research. It doesn't have to be on there for top schools and I don't think it will help you much unless you do something really significant, which is unlikely for math theory.

 

[flemmyd]

I'm wondering what my chances are and I'm especially interested in what would happen in the following scenario:

wait, so you haven't actually done any of this stuff? where along this timeline are you? what have you actually done and what is hypothetical?

and if you haven't started university yet, what makes you think you can take (and succeed) in graduate level classes?

 

[yenchin]

wait, so you haven't actually done any of this stuff? where along this timeline are you? what have you actually done and what is hypothetical?

and if you haven't started university yet, what makes you think you can take (and succeed) in graduate level classes?

I have nothing constructive to say... just to point out that he's in second year, see Reply 8 above.

 

[General_Sax]

You should just be proud of the fact that you're doing such advanced courses at such an early stage in your education!

Good things come to those who wait =]

 

[Anonymous217]

What university is this that has an excellent undergraduate and graduate math program where it's possible (and manageable) for you to be taking only math courses for all four years and to be taking graduate math courses freshman year? Just curious.

 

[Annonymous111]

What university is this that has an excellent undergraduate and graduate math program where it's possible (and manageable) for you to be taking only math courses for all four years and to be taking graduate math courses freshman year? Just curious.

Obviously, the only reason I can do grad. classes in math is if I've already done math before going to university which is what I've done. So I finished the undergrad. program for math (on my own) before going to uni. For reasons of "annonymity" as I'm called Annonymous111, the only information I can give you is that the university is in the top 10 (probably much better than 10th) math departments in the world and is located somewhere in the US.

 

[Annonymous111]

*No one* is a shoo-in at the top schools.

I would about your lack of research experience as well... students who get into the top schools tend to have great grades *and* some experience... so having just great grades makes you a below average candidate.

The good news is there is still time for you to do something about that!

I don't understand. So suppose someone has published multiple papers to a top mathematics journal and has great grades, and he gets rejected? There's a lot of hype about Harvard being a great university and all but surely they have some sort of maximum standard and surely there can't be too many people publishing papers to top math journals applying.

Anyway as I said I haven't published anything. But I'm interested in what people mean by "research". Does this mean original research that is published in a journal? Or does this just mean library research or survey articles that survey some highly advanced topics in a given sub area of math but that don't really constitute original research? Does the latter count as research experience?

Surely Harvard doesn't expect people to publish original research before going for a PhD? And math is something that it's really difficult to get some original research done without having a specialized enough background.

So what kind of research is expected of me when I apply? Could someone please clarify? I've also read plenty of forums of top grad school applicants and none of them seem to have taken more than so-many grad. classes. Besides, surely there can't be too many people who've had enough math background to be able to take advanced grad. classes in their freshmen year?

Suppose also the grad. classes contain material that's more advanced than and subsumes every single one of the grad. classes offered at Harvard and you've got A's in them. Does *that* make you a shoo-in?

Thanks for all the answers so far. I appreciate them!

 

[Annonymous111]

Could someone please answer my question? Thanks guys ...

 

[negru]

Take into account that Harvard takes what, 10 students per year? So basically you WILL need to be somewhere in the top ~20 students in the US. Are you sure those classes are enough to earn you such a spot? I've been pretty much told that I'll need a few math grad classes since at least junior year to have a real chance, and I'm going for physics grad school!
You really need to do research. Take some of the grad classes but start doing research (all summers etc). If you've started since freshman year, and take only math, there aren't enough courses anyway. And there's no reason to take ALL courses. Take the basic ones, find out which field you like best, take some advanced/specialized ones in that field. Then start doing research with the profs in those courses. Ideally this will happen somewhere around second sophomore semester, so you can get in two summers of research on the topic you like.

No one is expecting you to do research as an undergrad, but you need to be on the same level with the other applicants. Find grad students at Harvard and look for their resumes to see what you're up against.

 

[Phyisab****]

Does this post amount to anything more than bragging about your hypothetical accomplishments? If you are so talented, there is no reason not to publish some quality research before you graduate. Certainly you have professors on their knees begging you on a daily basis to work on their projects?

 

[deRham]

"So what kind of research is expected of me when I apply? Could someone please clarify? I've also read plenty of forums of top grad school applicants and none of them seem to have taken more than so-many grad. classes. Besides, surely there can't be too many people who've had enough math background to be able to take advanced grad. classes in their freshmen year?

Suppose also the grad. classes contain material that's more advanced than and subsumes every single one of the grad. classes offered at Harvard and you've got A's in them. Does *that* make you a shoo-in?"

Listen to what I say carefully (I hope that doesn't sound too "knowing"), because I have a realistic answer for you.

Yes, with your approach, you will impress many top grad schools. No, that doesn't guarantee you a spot. It is perfectly possible to get in the high 90s percentile, have taken a ton of insane courses in a top school for undergrad, and get rejected.

Think about it, Harvard (as someone said) may have 10-ish people in the entering class. It almost is to the point where when you take which graduate course doesn't matter. Yeah, of course your schedule will make anyone think you're a hardcore student, but 10 students is *tiny*, and they will look for people who bring research talent in some field and really know what they are doing. It is NOT, unlike what you seem to believe, important when you take your graduate courses. Real research work is insane even if you've taken every course in your dept, because the courses are the foundation, and then it takes years and years of struggling around poorly documented information in mathematics to start to follow what's going on.

Spend your time figuring out what you want to study, and being serious. I promise when you take which course isn't going to matter beyond the fact that at the end, you should seem a promising researcher.

"Does this post amount to anything more than bragging about your hypothetical accomplishments? If you are so talented, there is no reason not to publish some quality research before you graduate."

I don't even think it's *THAT TALENTED* - people overstate the importance of taking this many or that many courses all the time. I think people misunderstand how lost you will be anyway when you enter and try to research a hard topic - because all you can do is take the few courses that give you the foundation and then you have to rely on your advisor.

To the original poster, if you want to be *GUARANTEED* admission at Harvard, you should not even be asking these questions, you probably should have been doing high quality PhD level work at some absurdly young age (it's not unheard of, but even at a school like Harvard, it's exceptionally rare, to the point where I doubt anyone could deny you admission).

I would wager you will get into *a* top school if your letters are fairly good and you do all you set out to, but don't underestimate how hard it is to actually do it (I'm not saying you are, but you might be), and don't believe that this is the most important thing.

The problem is real research mathematics is insane to succeed at. A professor I know went to Harvard as an undegrad and was a serious, great student, but his true accomplishments shone when he went to grad school. He outdid many people who had traditionally impressive padded resumes.

 

[zpconn]

If you do all that and also do research in the summers (which does not necessarily mean getting published results but just attempting to do so at REUs, for instance), then you will have about as high a chance as you could ever have of getting into Harvard, and if you apply to all the top grad schools I'd be shocked if you got rejected from every single one of them.

Most grad students at Harvard didn't do any really serious research as undergraduates. They all tried, but most didn't get any really noteworthy results. To be a *shoo-in*, maybe it's necessary to have significant results. But to have a good chance, it's not necessary; you just need to have given it a serious attempt.

That said, I can safely say the chances of not having to take any non-math courses for your entire undergraduate degree are about 0%. How exactly are you planning on pulling that off? One of my friends, who's a freshman at Harvard doing Ph.D. level research, is only taking two math classes this semester, and I've heard of some other extremely talented undergraduates at Harvard who do the same thing.

I've been pretty much told that I'll need a few math grad classes since at least junior year to have a real chance, and I'm going for physics grad school!

Which classes out of curiosity?

 

[deRham]

They all tried, but most didn't get any really noteworthy results. To be a *shoo-in*, maybe it's necessary to have significant results. But to have a good chance, it's not necessary; you just need to have given it a serious attempt.

This, I agree with. I just wanted to emphasize that it's perfectly possible to take a million hard courses, get A's in nearly all of them, and not get into Harvard, and it happens all the time. But yes, non-geniuses do get in, I know that for a fact. They're probably geniuses to the standard math major, but not to the talented individual who has a serious shot at top schools.

 

[negru]

Which classes out of curiosity?

Well basically topology, differential geometry, algebraic geometry and complex analysis. For high energy these are pretty useful. And it seems like common sense. Your application will be taken much more seriously when you say you want to do string theory, AND actually know what a Kahler manifold is.

Of course it also depends how the admission committee works. If they give your app to profs in the field, you'll need to show you know details. I mean you don't HAVE to, but that's what will raise your chances.


Plus here we're talking about what it might take to have a considerable chance (eg higher than the default ~5%), for one particular school, not at least one out 5 or 10.

 

[Anonymous217]

Obviously, the only reason I can do grad. classes in math is if I've already done math before going to university which is what I've done. So I finished the undergrad. program for math (on my own) before going to uni. For reasons of "annonymity" as I'm called Annonymous111, the only information I can give you is that the university is in the top 10 (probably much better than 10th) math departments in the world and is located somewhere in the US.

I sincerely doubt the validity of your preexisting academic credentials, but okay.

 

[negru]

I sincerely doubt the validity of your preexisting academic credentials, but okay.

Why does everyone doubt what he's saying? Skipping undergrad courses since fresh/soph isn't that uncommon (in my experience in physics at least). Based on competition for top schools, I'd say it's pretty close to required if you want a decent shot.

 

[Anonymous217]

Skipping lower division courses is not uncommon. However, being able to skip upper division and immediately go to graduate courses freshman year is.

 

[negru]

Depends what you mean by uncommon. If more than 20 students in the US can do it, it will be pretty common when applying to Harvard, that's the problem.

 

[Newtime]

Depends what you mean by uncommon. If more than 20 students in the US can do it, it will be pretty common when applying to Harvard, that's the problem.

It just sounds fishy. He's claiming to only be taking UPPER division graduate courses, and has been doing so since freshman year. Also, he's claiming he wants to (and can) finish his undergrad years with taking nothing other than math courses. What university allows this? And if he was this advanced, surely he wouldn't be wondering about his chances of getting into top grad schools on internet forums. It seems either he is making this up, subtly bragging, or completely oblivious to how exceptionally talented he is.

 

[negru]

I'm not exactly sure what upper division math courses are. Algebraic geometry, number theory? More advanced? If "intro" level grad courses are algebra, real analysis, manifolds, it wouldn't be impossible to get over them in one-two years before/around freshman year.

There are many universities with completely no requirements other than completing a major. In particular, I will be graduating with only one course outside of math/physics.

Also, unless you have a good academic advisor, starting/thinking about research/what it really takes for grad school doesn't come automatically. A few years ago I also thought that the only thing I can do is take as many courses as possible. I didn't even imagine the possibility of doing research before completing most of the graduate courses. Thankfully I started reading forums in time.

 

[Anonymous217]

Depends what you mean by uncommon. If more than 20 students in the US can do it, it will be pretty common when applying to Harvard, that's the problem.

It is doubtful more than 20 students in the US can take graduate courses freshman year and supersede all undergraduate requirements in a top math university (note the top math university condition). I know this is definitely true for UC Berkeley and this seems very unlikely in privates, especially in those that focus on a liberal arts education like Princeton.

 

[Phyisab****]

It seems either he is making this up, subtly bragging, or completely oblivious to how exceptionally talented he is.

This.

 

[negru]

With proper advising and some previous training (eg olympiads) I think most of the top students at each university can do it. Maybe not advanced grad courses, but the intro ones definitely. And with an obvious cost in spare time. I think it's primarily the lack of a competitive spirit within schools which holds back most students, not their own potential. If all the advisors tell you to try some music classes, do some reading, take your time, you obviously won't feel any need to go hardcore.

 

[Anonymous217]

With proper advising and some previous training (eg olympiads) I think most of the top students at each university can do it. Maybe not advanced grad courses, but the intro ones definitely. And with an obvious cost in spare time. I think it's primarily the lack of a competitive spirit within schools which holds back most students, not their own potential. If all the advisors tell you to try some music classes, do some reading, take your time, you obviously won't feel any need to go hardcore.

It's not whether they can succeed in graduate courses early. It's whether they can meet all the requirements to be solely taking graduate courses early.

 

[Annonymous111]

It just sounds fishy. He's claiming to only be taking UPPER division graduate courses, and has been doing so since freshman year. Also, he's claiming he wants to (and can) finish his undergrad years with taking nothing other than math courses. What university allows this? And if he was this advanced, surely he wouldn't be wondering about his chances of getting into top grad schools on internet forums. It seems either he is making this up, subtly bragging, or completely oblivious to how exceptionally talented he is.

The reason I'm wondering about my chances of getting into top grad. schools is that they're top grad. schools. I've also heard of people who get rejected from these grad. schools after publishing papers to high quality math jounrals bc they haven't got good grades. Grad. admissions are a mystery to me as are undergrad. admissions. I was rejected from most top undergrad. schools. I know there're people more talented than me but I didn't see what I'd done wrong to get rejected. I'm trying to turn the tables when I apply for grad. schools.

I never said I was talented and nor do I think I'm talented. I wholeheartedly agree with DeRham. I haven't done serious research. I've only got good grades in courses with fancy names. Getting into grad. school is one thing and being a successful mathematician is another. Just getting into Harvard doesn't guarantee that you'll become a successful mathematician simply because grades, test scores and rec. letters don't necessarily give a good indication of research potential.

I feel that I should point out though that I'm not making this up. I'm not saying that I'm talented nor am I bragging, but I'm not someone who would lie about these things.

It's not that hard these days to get enough math background before going to undergrad. Besides hundreds of students learn calc. at an insanely young age. Some of them continue with math enough to have a strong background before even going to undergrad. I'm not saying that I do. But it isn't something that's uncommon.

BTW, the only reason I think I have research potential is because while I've never actually done solid research, they're indicators that suggest I might be good at it - for example ideas come to me if I think long and hard enough and some of them original. But I've never actually published anything. I'm not saying I'm great or anything - just that I do have some potential in research as do many other people - and this has nothing much to do with my grades.

 

[Annonymous111]

With proper advising and some previous training (eg olympiads) I think most of the top students at each university can do it. Maybe not advanced grad courses, but the intro ones definitely. And with an obvious cost in spare time. I think it's primarily the lack of a competitive spirit within schools which holds back most students, not their own potential. If all the advisors tell you to try some music classes, do some reading, take your time, you obviously won't feel any need to go hardcore.

I have to disagree here. They're plenty of students who can take intro grad. classes in Freshman year I don't doubt that. But olympiads don't actually have very much intersection with undergrad. math so acing olympiads doesn't necessarily say that you know undergrad. math. In fact it's quite possible to ace olympiads without even knowing calculus. As a former participant in suh compettions myself you're not even allowed to use calculus in olympiad problems.

 

[Annonymous111]

It's not whether they can succeed in graduate courses early. It's whether they can meet all the requirements to be solely taking graduate courses early.

There are several factors that saw me avoiding the general requiremets. I can't state them here but most were to do with my prior record in math. They gave me special exemption from the general requirements but "strongly encouraged" me to satisfy the general requirements. So I chose the former.

 

[Annonymous111]

I'm not exactly sure what upper division math courses are. Algebraic geometry, number theory? More advanced? If "intro" level grad courses are algebra, real analysis, manifolds, it wouldn't be impossible to get over them in one-two years before/around freshman year.

There are many universities with completely no requirements other than completing a major. In particular, I will be graduating with only one course outside of math/physics.

Also, unless you have a good academic advisor, starting/thinking about research/what it really takes for grad school doesn't come automatically. A few years ago I also thought that the only thing I can do is take as many courses as possible. I didn't even imagine the possibility of doing research before completing most of the graduate courses. Thankfully I started reading forums in time.

I agree. What are "grad. classes" really? I mean there're universities where grad. classes would be first or second year undergrad. classes at a better math university. To say you're taking grad. classes probably doesn't mean anything. E.g. if manifolds are what's covered in a grad. class then that's pretty basic. The point is that to do manifolds you don't need to have much more than a solid background in multivariable calculus and linear algebra (and some basic topology could help). Maybe a course in "morse theory" could be called a grad. class. I'd say a grad class is something that requires more material than what's covered in undergrad in at least one area.

A good indicator of what grad. classes really are is to compare with Harvard's selection of math grad. classes. http://www.math.harvard.edu/courses/index.html . The "primarily for graduates" is the real meaning of "grad class". But it's evident that Harvard is not equally strong in all areas from that list.

 

[Annonymous111]

A quick question. Is GPA measured in terms of your average % mark over all your courses or just your average grade? Does it matter whether you get 90% or 95% as long as you get a A+ or does your GPA take into account the difference. Thanks guys ...

 

[Annonymous111]

I'd also like to know how much it matters what area you have the best background when you apply to grad. school. I mean take an area X with no one in the Harvard math faculty working on anything related to X. SUppose you've published high quality research in X but want to get a PhD in X. Will Harvard (or any grad. school for that matter) reject you simply because your "X" doesn't match the interests of their faculty?

 

[twofish-quant]

There's a lot of hype about Harvard being a great university and all but surely they have some sort of maximum standard and surely there can't be too many people publishing papers to top math journals applying.

There aren't, but Harvard only graduates about a dozen math Ph.D.'s each year.

Surely Harvard doesn't expect people to publish original research before going for a PhD? And math is something that it's really difficult to get some original research done without having a specialized enough background.

They can and do. You are competing here with the best in the world, and there are people that are just absolutely incredible at math.

 

[axeae]

A quick question. Is GPA measured in terms of your average % mark over all your courses or just your average grade? Does it matter whether you get 90% or 95% as long as you get a A+ or does your GPA take into account the difference. Thanks guys ...

you took graduate courses as a freshmen, but you don't know how GPAs are calculated? kind of getting a troll vibe here

 

[Annonymous111]

you took graduate courses as a freshmen, but you don't know how GPAs are calculated? kind of getting a troll vibe here

Sigh ... I'm not a troll. If I wanted to be a troll I'd do other more subtle trollish things. I've realized from this thread that "don't post something so unbelievable that people will start thinking you're a troll" just be the "normal guy who gets B's and A's and has taken the odd math grad. class." Besides I've not insulted anyone here anyway.

 

[Newtime]

Sigh ... I'm not a troll. If I wanted to be a troll I'd do other more subtle trollish things. I've realized from this thread that "don't post something so unbelievable that people will start thinking you're a troll" just be the "normal guy who gets B's and A's and has taken the odd math grad. class." Besides I've not insulted anyone here anyway.

Look back at what I said. Personally, I don't think you're making this up. But I DO think you're either bragging but trying to seem like you aren't, or you really have no idea how ridiculously far ahead you are. Your question was answered in the first few posts but you asked it again twice, and then you asked about GPA and what not...so I could see where the troll accusations are coming from.

 

[qspeechc]

For what it counts, I think Annonymous111's story is credible. What really do you need to start doing graduate courses in math? Real and complex analysis (including measure theory), topology, algebra, differential geometry. Consider how many untalented people do calculus in high school and it is not beyond belief that a very hard-working and bright student can complete the above topics in high school.

Anyway, that's not what I want to write about. Have you thought that Harvard may not be the school for you? It is better to find a mathematician who does research in an area you are interested in then simply go to a school because of its name. Also, a school might not be as prestigious as Harvard, and yet have many top researchers in the field you are interested in, they may indeed be stronger than Harvard in that field. There are so many great mathematicians that they are not all at Harvard or other fancy-pants universities. You should know what field you are interested in, then go to someone in the department who works in a closely allied field-- say if you're interested in algebraic number theory then speak then an algebraist, if there is no one working in number theory itself-- and ask him what good grad schools or what good researchers he recommends. He would probably know if you have a chance of getting into that school from your academic record so far, or what you need to do if not. Even if he doesn't know what school is good for you, he probably has colleague (perhaps at another university) that does. By all means apply to Harvard, but don't kill yourself if you don't get in. As others have said, to get into Harvard you have to be one of the top math students in the world.

 

[jtbell]

A quick question. Is GPA measured in terms of your average % mark over all your courses or just your average grade? Does it matter whether you get 90% or 95% as long as you get a A+ or does your GPA take into account the difference. Thanks guys ...

Look up the rules at your school, or ask someone in the registrar's office, to be sure.

Where I teach, course instructors report only the letter grades (A, A-, B+, etc.). These are converted to numeric values for calculating GPAs. Something like A = 4.0, A- = 3.7, B+ = 3.3, B = 3.0, etc.

Some colleges and universities may report grades on a 0-100 scale, but I don't know any examples myself. It's not like in high school where this is fairly common.

 

[deRham]

But I DO think you're either bragging but trying to seem like you aren't, or you really have no idea how ridiculously far ahead you are.

My humble input? I don't think it's bragging, after all it's legitimately Harvard math we're talking about, and it's insane to get into, and wondering what exactly people did to get is something that is worth asking.

Sure, there are other researchers elsewhere, but there are a ton of very great ones at Harvard concentrated in certain fields. For instance, if you want to do number theory, it can be a dream to study there, not just because of the professors but because of your peers!