Mathematics Grad. School Application Harvard

[Newtime]

I think earlier you (Anonymous 111) said none of the current graduate students had published papers? In addition to this misconception, I also don't think you understand just how exceptional these students are. Now, if you're doing well in upper level grad classes as a freshman, that's pretty exceptional too, but have a look at what you're up against...

http://www.math.harvard.edu/~dankane/

Although, even at Harvard, I think Daniel Kane is not the norm; he is, and I mean this in the best possible way, an absolute freak of nature. He had a paper published in the proceedings of the AMS freshman year which means that he was likely doing the research in high school.

My point here is basically what twofish and others have been saying: from what you've told us, you've doing very, very well. Keep it up. But don't expect to get in because you aren't the only one doing this well, and you're competing for ~10 spots.

 

[Annonymous111]

I think earlier you (Anonymous 111) said none of the current graduate students had published papers? In addition to this misconception, I also don't think you understand just how exceptional these students are. Now, if you're doing well in upper level grad classes as a freshman, that's pretty exceptional too, but have a look at what you're up against...

http://www.math.harvard.edu/~dankane/

Although, even at Harvard, I think Daniel Kane is not the norm; he is, and I mean this in the best possible way, an absolute freak of nature. He had a paper published in the proceedings of the AMS freshman year which means that he was likely doing the research in high school.

My point here is basically what twofish and others have been saying: from what you've told us, you've doing very, very well. Keep it up. But don't expect to get in because you aren't the only one doing this well, and you're competing for ~10 spots.

Wow. I now see how good these students are. It wasn't that I didn't think they were that good but I never imagined that they would be this good. I've changed my mind.

Not trying to downplay this guy's achievements but his first paper was on some kind of number theory which is the sort of subject that's accessible to many people without too much math background. I don't know what kinds of math classes he took in freshman year but research is pretty amazing. But I did start Freshman year quite a bit younger than Daniel Kane so that could be a factor in grad. school admissions.

Thanks for changing my mind.

 

[negru]

I think what people were trying to say was that you shouldn't COUNT on getting into Harvard, not that you don't have a chance or that you shouldn't try and work for it. That means do everything you can, but assume for the sake of it that you won't. See what would happen in that case. Because if you're hung on just Harvard, and then don't get in, it might be rough to move on.

Though thinking just top 30 (especially if you're in a top 10 already) is a bit overdoing it, I think. I'm not sure what year you said you were in, but if you have at least one more year before applying, focus close to exclusively on research. You can probably do research for credits if that's an issue.

 

[Annonymous111]

I think what people were trying to say was that you shouldn't COUNT on getting into Harvard, not that you don't have a chance or that you shouldn't try and work for it. That means do everything you can, but assume for the sake of it that you won't. See what would happen in that case. Because if you're hung on just Harvard, and then don't get in, it might be rough to move on.

Though thinking just top 30 (especially if you're in a top 10 already) is a bit overdoing it, I think. I'm not sure what year you said you were in, but if you have at least one more year before applying, focus close to exclusively on research. You can probably do research for credits if that's an issue.

I agree. I'm not aiming just to get into Harvard. There're plenty of other great grad. schools and I've heard Harvard is kind of snobish. The only reason I was hooked onto Harvard was because I've heard that it's hard to get jobs at top institutions like the institute for advanced study or other top universities if your PhD doesn't say something close to Harvard on it and isn't from some top university in Europe. I don't really believe this but I kind of don't want to be taking chances if you know what I mean so if I do get into Harvard I'll probably go there.

The main reason I don't want to do research is because though I'm well rounded in a few areas I don't actually know one area well-enough to do research. I think at least to do research in an area you should at least know what's covered in some kind of "landmark book" in the area so to speak. So like in alg. geo. you'd want to know Hartshorne, a bit of EGA and some of Principles of Algebraic geometry or in functional analysis you'd at least want to know Rudin's real and complex and Rudin's Functional.

I've kind of been focussing a lot on trying to do the introductory and intermediate books from a wide range of areas. This means I'm familiar with plenty of mathematics from diverse areas and can probably understand most references to other branches of mathematics given in the literature on some specialized topic. It also means that I can't do research in anyone area which is a drawback. My thoughts were always that there's plenty of time to do research in grad. school why rush it? I do want to get into grad. school but I feel spending a year without getting anything done is a big risk to take when you might be able to get more ideas with a better background. So basically I don't think I have enough background. I'm still in sophomore so there's plenty of time. I'll definitely look at research somewhere in my third year. Having seen some sort of "real math" it's pretty amazing how even fairly easy "slick tricks" can get published in top journals because of their power so it's not anything beyond my scope to have a go. Maybe I'll research a couple of hours per day and see how I go.

Thanks again for all your advice everyone!

 

[Annonymous111]

I think what people were trying to say was that you shouldn't COUNT on getting into Harvard, not that you don't have a chance or that you shouldn't try and work for it. That means do everything you can, but assume for the sake of it that you won't. See what would happen in that case. Because if you're hung on just Harvard, and then don't get in, it might be rough to move on.

Though thinking just top 30 (especially if you're in a top 10 already) is a bit overdoing it, I think. I'm not sure what year you said you were in, but if you have at least one more year before applying, focus close to exclusively on research. You can probably do research for credits if that's an issue.

I agree with the top 30. I don't want to brag or boast or anything like that but I'd be pretty surprised if I didn't get into at least one of the top 20. I may not be in the top 10 students in the world but (at least I hope!) I'm in the top 100 so getting into a top 10 grad. school should be OK. In the absolute worst case scenario that I don't get into a different grad. school to my undergrad. I can still do my PhD at the school I'm still in since the people here are willing to give me special permission. So not all is lost. I do have a back up plan that'd get me into a top 10 grad school but my school still strongly advises me to look elsewhere and they'd only accept me into their grad. school in extreme circumstances that I don't get anywhere else, but it's still a back up plane (of course, they'd accept me provided I've got good grades and done well! but I don't look like dropping grades anytime soon hopefully)

 

[negru]

Staying in your own school is another good option I think. Especially if you have people there working on interesting stuff and are well known in their field. Cause btw this is also something you need to take into account.

There's no point in going to Harvard if your advisor will be a no-name, or just bad at advising. The advantage of Harvard is that it has higher density of good profs, but that doesn't guarantee you anything. This is why even if you're good enough to get into Harvard, you might not be good enough to get the advisor you want.

Personally I'm applying only to schools with good people in my field. Based on this, UCLA would be just as good for me as MIT.

Sure, the buzz that you're a topshot will be there maybe the first year or semester. But then when the time comes to pick a research area or advisor and realize there's nothing you like it's not going to be pretty.

 

[twofish-quant]

Staying in your own school is another good option I think.

1) a lot of schools will refuse to admit their own undergraduates (MIT for one, Feymann mentions this in his autobiography and they still have this policy)
2) it's a good thing that they do this.

One thing that you should do for graduate school is to go to a school that is a different type of school than your undergraduate. If you go to a large public school an undergraduate, try to get into a small school as a graduate student, and vice versa.

The reason for this is that going to different types of schools gives you a broader appreciation for how things can be done a different way.

 

[twofish-quant]

Grad school admissions are not like undergraduate admissions. Life is not like undergrad or grad school admissions.

You say "hard work is not everything". What if I work hard enough to publish a math paper in a top math journal? Or what if I take enough math grad. classes that it covers more than what's covered in all the math grad. classes at Harvard?

Harvard has ten places. If you have one hundred people in the world do the same thing, it's not going to help you. If you have ten people in the world that do more than you, then you aren't going to get in.

If you have a limited number of spaces then there is a limit to the usefulness of hard work, because you work harder, so is everyone else, and in the end it ends up being a roll of the dice.

Graduate school is very different than undergraduate, because the supply/demand difference is higher, and the higher the ratio the more things depend on luck or things that you can't control. Also, the supply/demand difference is in fields other than in physics or math. Harvard admits about ten people per year for its Math Ph.D. program, but it admits about 1000 per year in its MBA program.

Dude I know PhD math graduates from Harvard and all these top tier grad. schools who really aren't that spectacular.

So do I, which makes me wonder why you are so intent on getting in.

One thing that I suspect is that "hyper-elite" admissions processes actually cause people to be less spectacular. The problem with being obsessed with Harvard is that in order to get into it, you have to do exactly what the Harvard math grad admissions committee wants you to do. That may not be a good thing. You may find that in order to get into Harvard, you'll have to do things that you don't think are good for your math development.

For example, if you spend any time teaching math rather than doing homework, that drastically lowers your chances of getting into a big name grad school. If you have a life outside of math, that reduces your chances. If you get interested in a field of math that just isn't hot, that reduces your chances.

So I'm pretty sure I'm going to get into at least one of the top 10 schools. I don't necessarily think that I'll get into Harvard but I'd be pretty surprised if I didn't get into at least one top school like Princeton, Stanford, MIT, Chicago etc.

The biggest math grad schools typically admit about ten or so people per year. The smaller ones, one or two. The places are few enough so that you could find yourself not getting into any of the big name schools, and sometimes for some random reasons.

You ask me What if I don't get into any math grad. school in the top 30?! Your argument is that since the chances of getting into Harvard are nearly zero I should have a back up plan. But the same argument tells me that the chances of NOT getting into a top 30 grad. school is also very nearly zero. That's the absolute worst case scenario that it's not worth preparing for that.

What I'm telling you is that not getting into a big name grad school is a *LIKELY* scenario that you should be preparing for, at least psychologically.

If that really happens then not getting into grad. school would be the least of my worries! It's like saying that the US will be bombed tomorow. Ain't going to happen ...

Yes it will. You may get lucky and get into Harvard grad school, but then you have post-doc hell where you run into the same then, then faculty hell where you run into the same dynamic. If you roll the dice long enough, the odds will turn against you and you will get wiped out. The important thing is that you set yourself off, so that you can keep going even when the dice go against you, and mathematically, they will at some point.

You clearly aren't interesting in absorbing what I'm telling you now, which is fine, since I'm just hoping that enough of it sticks with you so that when you do get a stack of rejection letters, you don't do anything crazy, but just feel miserable for a few days and then go forward with your math studies.

One general problem with math and physics Ph.D.'s is that people go into graduate school with wildly unrealistic views on what their situation is, and part of the problem is that the system tends to reinforce those views, so I figure it's a good thing to throw a cold dose of reality early. The good news is that reality ain't that bad.

 

[twofish-quant]

I don't know what kinds of math classes he took in freshman year but research is pretty amazing.

He probably didn't take any classes at all. A lot of high level mathematicians don't *need* to take any classes. They figure stuff out on their own.

Also, you'll find that a lot of math geniuses are extremely odd people. They can come up with hugely interesting things when they do number theory, but they have difficulty with daily life things like ordering food at a restaurant.

 

[Annonymous111]

Grad school admissions are not like undergraduate admissions. Life is not like undergrad or grad school admissions.



Harvard has ten places. If you have one hundred people in the world do the same thing, it's not going to help you. If you have ten people in the world that do more than you, then you aren't going to get in.

If you have a limited number of spaces then there is a limit to the usefulness of hard work, because you work harder, so is everyone else, and in the end it ends up being a roll of the dice.

Graduate school is very different than undergraduate, because the supply/demand difference is higher, and the higher the ratio the more things depend on luck or things that you can't control. Also, the supply/demand difference is in fields other than in physics or math. Harvard admits about ten people per year for its Math Ph.D. program, but it admits about 1000 per year in its MBA program.



So do I, which makes me wonder why you are so intent on getting in.

One thing that I suspect is that "hyper-elite" admissions processes actually cause people to be less spectacular. The problem with being obsessed with Harvard is that in order to get into it, you have to do exactly what the Harvard math grad admissions committee wants you to do. That may not be a good thing. You may find that in order to get into Harvard, you'll have to do things that you don't think are good for your math development.

For example, if you spend any time teaching math rather than doing homework, that drastically lowers your chances of getting into a big name grad school. If you have a life outside of math, that reduces your chances. If you get interested in a field of math that just isn't hot, that reduces your chances.



The biggest math grad schools typically admit about ten or so people per year. The smaller ones, one or two. The places are few enough so that you could find yourself not getting into any of the big name schools, and sometimes for some random reasons.



What I'm telling you is that not getting into a big name grad school is a *LIKELY* scenario that you should be preparing for, at least psychologically.



Yes it will. You may get lucky and get into Harvard grad school, but then you have post-doc hell where you run into the same then, then faculty hell where you run into the same dynamic. If you roll the dice long enough, the odds will turn against you and you will get wiped out. The important thing is that you set yourself off, so that you can keep going even when the dice go against you, and mathematically, they will at some point.

You clearly aren't interesting in absorbing what I'm telling you now, which is fine, since I'm just hoping that enough of it sticks with you so that when you do get a stack of rejection letters, you don't do anything crazy, but just feel miserable for a few days and then go forward with your math studies.

One general problem with math and physics Ph.D.'s is that people go into graduate school with wildly unrealistic views on what their situation is, and part of the problem is that the system tends to reinforce those views, so I figure it's a good thing to throw a cold dose of reality early. The good news is that reality ain't that bad.

I appreciate your advice twofishquant. What I'm trying to say is that if the situation were so bad that I'd get a stack of rejection letters from every one of the top math departments then that's pretty much diametrically opposite to the situation of getting into Harvard. Both really aren't too likely. Besides, I will consider applying to the UK and to top math departments in Europe if such a situation occurs. I'll pick up languages like French and German as necessary. So there're simply too many options for me that it really isn't worth thinking that I won't get in.

It's not that I don't expect rejection letters. I know how it feels like getting rejection letters. But there are so many odds against me not getting anywhere that it isn't worth assuming that that'll happen. I will assume that I won't get into Harvard that's a bit harder. Daniel Kane changed my mind but it hasn't really changed my chances. as I said I was quite q bit younger than Daniel Kane when I entered undergrad. and still am younger than the time when he published his first paper. So I still have time to "beat him to his first publication" so to speak. Grad. schools must be looking at the age of their applicants right? I don't believe it should play a part in selection but they should consider the dichotomy between age and math experience. Give an example: if a 10 year old applied with the same knowledge of an "average undergraduate" (as in taken all the classic undergraduate math classes) and a 24 year old applied with more knowledge the 10 year old might get the upper hand simply because he hasn't had all that experience to learn math and if he's accumulated that much math already he probably won't burn out in grad. school. (this is hypothetical btw I'm not 10!)

The reason I do want to get into top. grad. school is that it'll make it easier to get a job. Is that right? I don't want to go to a grad. school just because of it's name but I don't even know what I want to research so there's still time to have a look. But the point is this: it seems in every one of the top 20 math departments, every single faculty or at least most are PhD's from Harvard, Princeton or some other top tier. school. I certainly don't want to be a PhD from something like Ohio State University either.

 

[Fragment]

What's better, publishing a not-so-good paper at age 19, or publishing quality work at age 24? It seems far unlikely that the application committee admitted him on grounds of early work, as opposed to quality work that will help him during his grad years.

Furthermore, after reading all of this thread, the point is that you are NOT in the top 100 best in the world(probably). Truly, there is no way to grade this. Even Daniel Kane is probably not in the top 15 (for his age). There are a lot of clever students out there, being among the top 10 to apply to Harvard is highly unlikely. It is better to use this as your premise than to expect people to fling open their doors when they hear about your coming (As twofish is trying to explain to you).


Of course you might get into Harvard, and, I speak for everyone here, the aim is not to discourage you, but more to give you better footing when things don't go the way you want. If you do get in an Ivy League school, then congratulations, but if you don't, then don't stop doing mathematics out of defeat.

Good luck to you in your endeavors.

 

[deRham]

So I still have time to "beat him to his first publication" so to speak. Grad. schools must be looking at the age of their applicants right? I don't believe it should play a part in selection but they should consider the dichotomy between age and math experience. Give an example: if a 10 year old applied with the same knowledge of an "average undergraduate" (as in taken all the classic undergraduate math classes) and a 24 year old applied with more knowledge the 10 year old might get the upper hand simply because he hasn't had all that experience to learn math and if he's accumulated that much math already he probably won't burn out in grad. school. (this is hypothetical btw I'm not 10!)

Look, I know where you're coming from. I agree that some posters downplay the extent to which your efforts will help. But take it from someone who is certain - people who are "not that great" are probably very smart and had glowing letters of rec. Professors trust each other as colleagues for judgement as to which candidates should be taken seriously.

You compete with an international pool. International students tend to be insanely accomplished. You are not anywhere close to the top person from Moscow, and yes I'm sure of this without knowing you.

By all means, if you do what you set out to do, you will very likely make at least one top 10 school (note - I do not endorse any specific 10, but Harvard, MIT, etc are up there). Harvard, the odds are still scary as hell against you. People may have connections you don't. Some (even if not many!) will be better than you.

He probably didn't take any classes at all. A lot of high level mathematicians don't *need* to take any classes. They figure stuff out on their own.

A professor I know went to Harvard for undergrad, took Math 55 and some graduate course in a lit dept. He got his PhD at age 20 or 21 from Princeton. Do not discount that real mathematics is beyond classes by far. You and I need classes because we're not geniuses.

Your saying "I want to take all these advanced classes" sounds like someone going "I want to take 10 AP classes in high school so that college work will be a breeze later" ... they're different playing fields. Research mathematicians of the highest caliber will write papers that require thousands of pages of background to follow and that you couldn't come up in your wildest dreams over a lifetime, every few years.

Again, they are the minority, but know that while you're probably going to be a top caliber math students, the pool for top academics is smaller, and top grad schools are home to many top academics.

 

[negru]

I'm not sold on the clear benefit of changing schools. If you're fine with the weather, location and general atmosphere, why change, unless there's an obvious academic reason? It's not like in grad school you'll have time to do anything other than study anyway. Your preoccupation will mainly be research.

I think you're already sacrificing enough by going to grad school, I don't think the minor benefit of a different atmosphere is worth considering (of course, depending how minor it really is). If you're bored with the people you've been working with, there are probably others in the department you can switch to.

There's certainly no point in going to a "lesser" school (considering profs, etc, not rank) than your own, if you can stay. Already knowing the profs, the surroundings, probably other grad students staying, etc can be very useful. I mean, you'll actually know who the good advisors are. Just visiting a school or being there for only a year (or less if you don't want to waste time) before choosing an advisor can easily cost you your potential career. Much more so than going to school #4 instead of #2.

 

[Annonymous111]

Look, I know where you're coming from. I agree that some posters downplay the extent to which your efforts will help. But take it from someone who is certain - people who are "not that great" are probably very smart and had glowing letters of rec. Professors trust each other as colleagues for judgement as to which candidates should be taken seriously.

You compete with an international pool. International students tend to be insanely accomplished. You are not anywhere close to the top person from Moscow, and yes I'm sure of this without knowing you.

By all means, if you do what you set out to do, you will very likely make at least one top 10 school (note - I do not endorse any specific 10, but Harvard, MIT, etc are up there). Harvard, the odds are still scary as hell against you. People may have connections you don't. Some (even if not many!) will be better than you.



A professor I know went to Harvard for undergrad, took Math 55 and some graduate course in a lit dept. He got his PhD at age 20 or 21 from Princeton. Do not discount that real mathematics is beyond classes by far. You and I need classes because we're not geniuses.

Your saying "I want to take all these advanced classes" sounds like someone going "I want to take 10 AP classes in high school so that college work will be a breeze later" ... they're different playing fields. Research mathematicians of the highest caliber will write papers that require thousands of pages of background to follow and that you couldn't come up in your wildest dreams over a lifetime, every few years.

Again, they are the minority, but know that while you're probably going to be a top caliber math students, the pool for top academics is smaller, and top grad schools are home to many top academics.

Can you elaborate a bit on the the person from Moscow?

You see I'm not just taking "classes". These classes go way beyond just learning stuff. It's kind of like the Moore Method classes with some research if you've heard of the Moore Method before.

Maybe some people've published research in undergrad. But many don't. Take Terence Tao for example. He never published anything in undergrad. but still got in Princeton.

 

[negru]

Yeah but Tao went to IMO at age 10. But that was probably a joke

 

[twofish-quant]

I appreciate your advice twofishquant. What I'm trying to say is that if the situation were so bad that I'd get a stack of rejection letters from every one of the top math departments then that's pretty much diametrically opposite to the situation of getting into Harvard. Both really aren't too likely.

And I'm telling that this is not the situation. Over the last few decades (i.e. between 1940 and today), you've seen a huge increase in the number of undergraduates who are able and willing to go into math grad school, however, the number of spots in the big name schools has not increased. What has increased are the number of spots in schools outside of the big names.

So there're simply too many options for me that it really isn't worth thinking that I won't get in.

What I'm curious about is why you are considering only "big name" graduate schools. My guess is that you've been "brainwashed" by the undergraduate admissions process to think in a certain way about school admissions, and what I'm trying to get across to you is that do do reasonable math work, you have to unbrainwash yourself.

Yes, if you don't go into a big name grad school, you are less likely to get a faculty position, but just as you should plan not to go to Harvard, you should also structure your academic career with the assumption that you will *not* be able to get research faculty position.

Now right now math education is a hot field, and people that have Ph.D.'s in that area aren't having too many problems getting faculty jobs, but that's a different animal.

It's not that I don't expect rejection letters. I know how it feels like getting rejection letters.

After the first dozen, you don't care any more.

But there are so many odds against me not getting anywhere that it isn't worth assuming that that'll happen.

You increase your chances by not prematurely removing options. If you absolutely insist on getting into a big name grad school, you are removing all of the options that go with not going to a big name grad school. Also at some point, you have to create your own odds rather than accepting what is going around you.

Grad. schools must be looking at the age of their applicants right?

Not really. Also the people that do end up being math uber-geniuses often have rather unimpressive coursework because they just taught them the math on their own.

The reason I do want to get into top. grad. school is that it'll make it easier to get a job. Is that right?

Never get a Ph.D. in math and physics for the purpose of getting a research professorship. Your odds of getting one to first approximation is zero. Your main reason for going to grad school in math or physics should be because you want to go to grad school.

But the point is this: it seems in every one of the top 20 math departments, every single faculty or at least most are PhD's from Harvard, Princeton or some other top tier. school. I certainly don't want to be a PhD from something like Ohio State University either.

The problem is that decision is not up to you, and the degree to which you can influence that decision is rather small.

If you want to get into the top 1000, then you can do that with hard work. If you want to get into the top 10, then anything that goes wrong will wipe you out and a lot of stuff may be things that you can't control.

I'm not that good at math, and I make up for not being good by working hard. However, it turns out that I have to work ten times as hard to learn a tenth as much as some of the uber-geniuses that I know. Sometimes it doesn't matter. It will take me five times as long to get some concept in number theory as a lot of people that I know, but I'll get it in the end. However, for things like Ph.D. graduate math programs at top level schools, it does.

Curiously, I think the fact that I'm not that good at math, makes be a much better physicist and computer programmer.

 

[twofish-quant]

Furthermore, after reading all of this thread, the point is that you are NOT in the top 100 best in the world(probably).

Also you have to ask the question the top 100 at what?

At world class levels of competition, things become extremely specialized. Someone that is a champion baseball would be expected to be reasonably good at tennis but that would be unlikely to be a good tennis player.

There's also the "chess boxing" strategy. You can be merely good at chess, merely good at boxing, but be the world champion at chess boxing. It's not hard to carve out a territory in which you are the number one in the world at.

Something that I've seen college students that are good at physics and math run into trouble is that the elementary and secondary school system is a rather artificial environment, in which you get graded and evaluated, and you get good grades and make teachers happy, you make it to the next level. People that go through that environment assume that this is how the world works, and it's not.

Of course you might get into Harvard, and, I speak for everyone here, the aim is not to discourage you, but more to give you better footing when things don't go the way you want. If you do get in an Ivy League school, then congratulations, but if you don't, then don't stop doing mathematics out of defeat.

The problem is that even if you do get into Harvard, there is then post-doc competition, and then tenure-track faculty competition and then tenured faculty competition. At some point you will fail.

 

[jgm340]

Can you elaborate a bit on the the person from Moscow?

You see I'm not just taking "classes". These classes go way beyond just learning stuff. It's kind of like the Moore Method classes with some research if you've heard of the Moore Method before.

Maybe some people've published research in undergrad. But many don't. Take Terence Tao for example. He never published anything in undergrad. but still got in Princeton.

I think this cuts to the root of the matter.

It confers a great advantage on one to start studying advanced mathematics at an early age. It takes two things for this to happen:

1) A young child who is more or less intelligent, but who has a strange interest in mathematics in the same way a child might have a strange interest in monster trucks
2) Adults who recognize this and continually provide a path of development

In Russia, I gather it is far more common for adults to push children early in a particular direction, and so there are many talented young mathematicians there.

The advantage to this is that one learns the fundamentals of mathematics when one is young and most flexible mentally. After this, learning mathematics becomes easier. It is like learning to read: one will struggle for a long time learning letters and words, but a few years later, hundreds of pages seems like nothing.

In the Unites States, kids just coast along in their development without being introduced to real mathematics until college or late high school. It seems such a waste, retrospectively, to have never been challenged in school. I only realize now that with the slightest bit of effort, I could have been so much farther in my education than I am.

I take it that the pool of applicants to graduate school in mathematics is incredibly diverse in terms of well-versedness. It makes me question: to what extent are applicants judged by how well-versed they are in mathematics in general? Certainly all applicants are expected to know basic analysis and algebra, but does prodigious knowledge far above a sparse few graduate courses really make one a better applicant? Or rather, does the fact that one has started later in the game than the child prodigies mean that one will never be as great a mathematician as them? Does it make it less likely that I will be as great a mathematician as Gauss was back in the day, or as Terence Tao is today?

I would like to think that young mathematicians are admitted into graduate school solely on the basis of their promise (so long as their backgrounds are solid enough to handle graduate work). In other words, someone with almost supernatural knowledge of mathematics may be denied entry to a top graduate school if it is deemed that their knowledge was the result of years and years of studying, not of especial talent in mathematics. I would also like to think that someone like myself, who has been lazy and unmotivated mathematically most of his teens, could be considered to be one of the top applicants to one of the top graduate schools by his senior year in college if he clearly had talent in mathematics yet reaped by decades of studying.

 

[Topologist]

Can you elaborate a bit on the the person from Moscow?

You see I'm not just taking "classes". These classes go way beyond just learning stuff. It's kind of like the Moore Method classes with some research if you've heard of the Moore Method before.

Maybe some people've published research in undergrad. But many don't. Take Terence Tao for example. He never published anything in undergrad. but still got in Princeton.

I do not think that you quite understand the meaning of the comments here. You are clearly a top mathematics student but it is important to realize that there are people equally as good as you and perhaps even some with better looking CV's than you. One cannot judge whether one mathematics student is better than another if neither has publications since it often turns out that undergraduate plays little or no part in the future career of a mathematician. However, you should be aware that there are people who are taking courses as advanced as those that you are taking.

By way of comparison, I plan to take many graduate classes in my undergraduate beginning from my first year. I have not done so yet (I am still in my first year), but I plan to do so. I might not take classes in the same direction as you but I still will take some graduate classes. And I am actually planning to do research since in one or two areas I do have close to enough background to do so if I invest some more time.

I am an international student. I am not trying to compare you and I but the point is that there are people who are really passionate about mathematics. I do not believe, and never have believed, that "talent" makes a mathematician more than "hard work". Perhaps that is something to take to heart. Most of your competition will be people with similar talent to yourself. The point is that 10 students is not a lot and about 250 students apply to Harvard each year (or so I have heard). You will probably be among the best, but please remember that there are people who really are passionate about mathematics and will do whatever it takes to become successful in the field.

I think that you will get into a top 10 graduate school. But remember that part of the admissions process is centered around the area of mathematics in which you wish to specialize. That is, if you wish to become a finite group theorist, and you apply to Harvard stating this intention, you might be rejected simply because there may not be anyone in Harvard's mathematics department who shares a similar research interest.

You are right in saying that some of the CV's of the applicants to top graduate schools are not that spectacular; I have seen some of these CV's, and often they do not contain as many graduate classes as yours does based on your description of the graduate classes that you have taken. On the other hand, it is amazing how many of these applicants have participated in the IMO and got medals. I do not think that achievements in the IMO play a part in the admissions process (but I could be wrong). However, it does show that they have the tenacity to succeed. (I was never successful at the IMO myself but I still could do decent mathematics.)

However, and this "however" is with great emphasis: there are people who have done amazing things in undergraduate, especially in research, that played a major factor in the graduate admissions. Also remember that letters of recommendation can sometimes be the deciding factor between admission and rejection. I suspect that you will get into some of the top 10 graduate schools in mathematics. No-one here wants to discourage you. But the point is that you should not think that you will get into all of them, nor should you expect to get into them. There is plenty of competition around the world to get in, and it is often close to impossible so distinguish between two candidates in some instances, so much so that luck can play a factor.

Try your best to get into mathematics graduate school at at a top mathematics department. But do not stop working hard now because you think there is no competition. The point of these comments is implicitly to motivate you to work even harder than you are working now. You can always work harder, and you should do that at this point in time rather than focussing on graduate school which is still some time away.

 

[Topologist]

You might find the following page interesting:

http://www.math.harvard.edu/pamphlets/gradsch.html

I quote:

"Many schools look at your transcript to see evidence of substantial exposure to serious mathematics (e.g. some graduate level courses)."

In this regard, you will probably have an advantage. However, there are other factors noted on the page.

 

[Topologist]

Not really. Also the people that do end up being math uber-geniuses often have rather unimpressive coursework because they just taught them the math on their own.

I am merely speculating here, of course, but judging from the names of the courses that he said he has taken, I suspect that they would be based on self-study. Even at Harvard, it is rare for faculty to offer such courses. Perhaps he is doing some kind of reading based course, although I could be wrong.

 

[twofish-quant]

I'm not sold on the clear benefit of changing schools. If you're fine with the weather, location and general atmosphere, why change, unless there's an obvious academic reason?

Because it's bad to be too comfortable.

It's not like in grad school you'll have time to do anything other than study anyway. Your preoccupation will mainly be research.

But it's interesting to see how different schools do research and teaching in very different ways. Also most of what you learn, you learn outside of the classroom. Part of what you learn in graduate schools is the process, politics, and culture of science and math, and different schools do it in different ways.

Also, your preoccupation *may not* be research. There are schools in which the ability to teach is highly valuable. There are schools in which no one cares about how well you teach. Seeing different academic cultures is quite useful. Seeing people try to change cultures is also interesting.

I think you're already sacrificing enough by going to grad school, I don't think the minor benefit of a different atmosphere is worth considering (of course, depending how minor it really is).

It's a really major benefit. Grad schools are all about learning a culture, and different schools have very different cultures. Learning to adapt to another culture is useful because it gives you more flexibility to deal with different things that happen, and to ask more useful questions. Who decided that grad school is 100% research with teaching being irrelevant? Is it a good thing that we made this decision?

There's certainly no point in going to a "lesser" school (considering profs, etc, not rank) than your own, if you can stay.

One thing that you'll find is that "lesser" schools aren't that bad, and have things that are better than "greater" schools. One thing that you do learn is that rankings are semi-bogus. I went to MIT as an undergraduate. It turns out that there are things that MIT is great at. There are also things that MIT is totally incompetent at. You get to see these sorts of things when you look at the Institute from the outside.

Also, if you put me into a top school, I'll get bored. It turns out that I'm happier being in a #50 school that is trying to get to #35.

Just visiting a school or being there for only a year (or less if you don't want to waste time) before choosing an advisor can easily cost you your potential career.

That's one good thing about considering that chances of getting an academic career is zero. Once you've said to yourself, well I'm just not going to get a professorship, it can be tremendously liberating since you don't worry about making certain people happy.

I think it's horrible that people are so career focused, and I think it's a bad thing that people just don't ask themselves some pretty basic questions about the world. One reason I got into physics was that I like to ask deep questions about the world, and if I get into a situation where I just have to accept a social system because "things are just that way" that sort of defeats the purpose.

Asking questions can get you in trouble. One question that I'm also thinking about is "so who made you boss anyway?"

Look, if you are born in the United States, you are going to be wealthier than 99% of the people that have ever lived. People in the US can study philosophy and art and history and live a life of the mind that people in the past just couldn't because someone had to plow the fields. We get the machines to do that now. Yet rather than *use* this sort of wealth, people just end up in the same sort of rat race that people were in in the past.

So why the hell are people so career focused. It's really frustrating to me to see this.

In elementary school and secondary school, you get ahead by following the rules. Do what you are told, you get prizes and recognition. Do something else, you get sent to the principals office. This stops working in college. You'll find that sometimes you have to do something even though you get laughed and humiliated at.

 

[jgm340]

I sympathize a lot with Annonymous111, because I feel like I'm in a very similar situation.

I also feel a bit jealous, because he has had opportunities I have not (or perhaps he had just seen and taken them while I bummed about in high school).

I also feel more powerful than most mathematics undergraduates at my university, because I started out in a better position than they did. I knew I was going to be a mathematics major from day 1, and I could take introductory analysis right off the bat. I feel myself advancing over a large number of my peers in a way that is due solely to a slight head start.

Also I know that a few of my peers at other universities and around the world are advancing past me, due to similar such advantages over me as I have had over others.

I do not ever worry that I am not intelligent enough. I simply do not believe that anyone human being is born so much more intelligent than any other. I believe 90% of people on the planet were capable, at birth, of becoming the one of the leading mathematicians in the world. But I DO worry about the advantages that others have over me that allow them to progress farther than I will be able to. I worry about it because I see, from my own perspective, that my own advantages will allow me to progress farther than a lot of the mathematics majors I know.

I worry about that I am not "working hard enough". I never worked hard enough in high school. I wasted time away. I rarely read in high school. I am uncultured. There are two parts of working hard: (i) the amount of time spent, and (ii) the effectiveness of that work. As I become more and more proficient, I find that it becomes easier to spend a large chunk of time studying math, and also that it becomes easier to study math. Those who have a head start have not only a head start material-wise, but also in their proficiency in studying. This is an advantage which overshadows all other advantages one could have, and I am afraid that I do not have it to the extent that I ought to.

I feel more motivated than a lot of other kids I know. I feel more interested in math, and feel like if I really put my mind to getting through a good amount of reading material, that I will start advancing at an even faster rate.

But then again, here I am, posting late night on a forum instead of reading.

I hope that if I can fully kick my sedentary habits, I will be one of the top candidates for the top graduate schools. I also hope that Annonymous111 makes it to that point to, but I also secretly hope that he doesn't, so that I have less competition.

 

[twofish-quant]

It makes me question: to what extent are applicants judged by how well-versed they are in mathematics in general? Certainly all applicants are expected to know basic analysis and algebra, but does prodigious knowledge far above a sparse few graduate courses really make one a better applicant?

Let me ask you another question.

Why does it matter to you?

The problem with being obsessed with getting into big name grad schools is that you become obsessed with what their admissions committee finds interesting and useful rather than what you find interesting and useful.

Or rather, does the fact that one has started later in the game than the child prodigies mean that one will never be as great a mathematician as them?

Part of the problem here is defining "great mathematician"

I would also like to think that someone like myself, who has been lazy and unmotivated mathematically most of his teens, could be considered to be one of the top applicants to one of the top graduate schools by his senior year in college if he clearly had talent in mathematics yet reaped by decades of studying.

Again. Why does this matter to you?

OK Harvard thinks that you are bogus, and you don't get in. Screw Harvard and go somewhere else.

 

[twofish-quant]

I do not ever worry that I am not intelligent enough.

The question is intelligent enough for what? I do think that there is something of a genetic component to how quickly you can pick up math, and some people may have their brains just wired in a way that let's them learn math more quickly.

This might matter, or it might not. For getting into a math grad school, I don't think it matters much, but for getting into the top math schools, the fact that it takes me a bit longer to teach myself differential geometry just eliminates me.

But I DO worry about the advantages that others have over me that allow them to progress farther than I will be able to. I worry about it because I see, from my own perspective, that my own advantages will allow me to progress farther than a lot of the mathematics majors I know.

Well, you may be screwed. But realizing that you could end up at the bottom of the heap for reasons that are out of your control makes you a better human being, I think.

Something that every math and physics student has to deal with at some point is being at the middle or at the bottom of the heap.

I hope that if I can fully kick my sedentary habits, I will be one of the top candidates for the top graduate schools.

And you may find that you aren't going to get in no matter how hard you work. That's why you should do things for the sake of doing them. If you organize your time better and learn more math, then you've learned more math.

One problem with undergraduates is that there are human limits to how hard you can work. At some point you have to relax, and people that try to push themselves run the risk of burnout. The big risk of smart undergraduates is burn out. So you work less hard and you get into your tenth choice of grad school, you are still in the game. If you work too hard and seriously damage your health, then you could wipe yourself out.

I also hope that Annonymous111 makes it to that point to, but I also secretly hope that he doesn't, so that I have less competition.

Seriously, that's one of the big problems with cutthroat competition. Once it becomes obvious that you get ahead by pushing other people down, the environment becomes seriously unpleasant. Friendly competition is a good thing, but once you have too few places, the competition becomes extremely unfriendly.

I get a thrill out of helping other people, so if I find myself in a situation where I end up benefiting only by messing up other people, I get myself out of that situation, which is one reason I didn't end up in academia.

 

[jgm340]

Let me ask you another question.

Why does it matter to you?

The problem with being obsessed with getting into big name grad schools is that you become obsessed with what their admissions committee finds interesting and useful rather than what you find interesting and useful.



Part of the problem here is defining "great mathematician"



Again. Why does this matter to you?

OK Harvard thinks that you are bogus, and you don't get in. Screw Harvard and go somewhere else.

It matters tremendously to me because one must find some standard against which to judge oneself. It helps tremendously to look at what other people are doing and think, "Hmmm... perhaps I should be doing what this person is doing!", or "Gee, I know not to go down that route!".

The mathematicians who get into the top universities are great mathematicians. It is natural, then for me to consider them as idols, as people whom I ought to consider modeling myself after.

My father once told me something, "Sometimes you think someone is really great at something. If you really idolize them for this, and you start hanging around them, you'll quickly pick up on how they do what they are so good at. You'll learn so quickly, in fact, that you'll be able to beat them at their own game! They had to figure everything out for themselves, but you have the advantage of being able to learn only the right ways to do things."

I don't really have any young people I idolize, which I honestly think is a major problem! Perhaps if I were at another university, I would be surrounded by kids I idolize, but I am not here.

At the very least, I need some idea of how to live my life in a way that makes me into someone I would like to idolize. A huge part of that is not being sedentary, and always pushing myself into learning new things. Another part of that is being a kind human being. Another part of that is eventually passing my knowledge down to other people, who are passionate and who will surpass me in knowledge.

I am honestly not obsessed at this point with big name schools (I got over that a while ago). I am obsessed with being the best mathematician I can be, and with becoming a human being I can idolize.

 

[twofish-quant]

I am merely speculating here, of course, but judging from the names of the courses that he said he has taken, I suspect that they would be based on self-study. Even at Harvard, it is rare for faculty to offer such courses. Perhaps he is doing some kind of reading based course, although I could be wrong.

What you will find with math prodigies is that they are often smarter than their teachers. Teaching someone that is clearly smarter and more talented than you are is something that can be quite thrilling and is quite a bit harder than it sounds.

It's also one big difference between undergraduate education and graduate education. After you go through an undergraduate course, the teacher still knows more about the subject than you are. However, for Ph.D. programs, the whole point is train students to be smarter and more talented that you are.

 

[jgm340]

Also, the reason I speak of things in such competitive terms, such as "advantage" "surpass", etc. is because these are inherent in a self-judgement. A self-judgement will always be based around what other people are doing, on some level.

And self-judgement is necessary for a human being to progress.

 

[negru]

Look, if you are born in the United States, you are going to be wealthier than 99% of the people that have ever lived. People in the US can study philosophy and art and history and live a life of the mind that people in the past just couldn't because someone had to plow the fields. We get the machines to do that now. Yet rather than *use* this sort of wealth, people just end up in the same sort of rat race that people were in in the past.

So why the hell are people so career focused. It's really frustrating to me to see this.

Well you see that's the thing with careers in academia. You want to get a career and become a professor because it's the best way of learning and gaining knowledge (not to mention contributing). There's also the social aspect. Sure you could argue that the best way would be to just become rich and pay Witten to come to your house everyday and teach you. But it's clearly not the same thing. Plus we're not talking about what would be the best way, we're talking about what works. And the reality is that people work better when dealing with competition. And they work even better when basic survival depends on it. Personally I'm pretty sure I wouldn't be so driven if I knew my future was somehow assured.

Probably some would, but everyone functions differently and for different reasons.


Otherwise I agree with you. But the problem lies with how universities transformed over the years. A century or so ago, places like Harvard were precisely for people who could afford to study art, philosophy, literature, etc, because usually their parents had earned enough. Nowadays however, with financial aid and everything, not everyone who goes to Harvard can afford to not be productive. Universities are no longer educating and producing thinkers, they are producing careers. It's a trade-off one can't really avoid.

 

[twofish-quant]

It matters tremendously to me because one must find some standard against which to judge oneself.

Why must one do that?

Also, Yao Ming is a better basketball player than me. Since I don't care much about basketball, that doesn't bother me.

It helps tremendously to look at what other people are doing and think, "Hmmm... perhaps I should be doing what this person is doing!", or "Gee, I know not to go down that route!".

Which is one good thing about seeing people up close. I've seen some Nobel Prize winners that are completely brilliant, but they are also total jerks with awful personal lives. After seeing Professor J up close and personal, I've decided that it's not worth getting a Nobel Prize if you have to turn into him.

The mathematicians who get into the top universities are great mathematicians. It is natural, then for me to consider them as idols, as people whom I ought to consider modeling myself after.

They are people. Smart brilliant people, but just brilliant. Also after knowing some people that are totally brilliant at math, but really bad at something else, it bothers me less to not be totally brilliant at math. Also some of the people that I do admire, aren't that great at math.

Also the problem with modeling yourself after someone is that sometimes you can't.

My father once told me something, "Sometimes you think someone is really great at something. If you really idolize them for this, and you start hanging around them, you'll quickly pick up on how they do what they are so good at. You'll learn so quickly, in fact, that you'll be able to beat them at their own game! They had to figure everything out for themselves, but you have the advantage of being able to learn only the right ways to do things."

And if you are close to someone, you figure out that they are just human and have flaws and things that you don't like. One reason that Professor J is great at physics is that he is one of the most single-minded and competitive people that you can meet, but this is why everyone that has met him, hates him.

At the very least, I need some idea of how to live my life in a way that makes me into someone I would like to idolize. A huge part of that is not being sedentary, and always pushing myself into learning new things. Another part of that is being a kind human being. Another part of that is eventually passing my knowledge down to other people, who are passionate and who will surpass me in knowledge.

You'll find that those goals are contradictory. If you really want to pass down knowledge, you are going to be more effective as a high school teacher or teaching lower division undergraduate work. If you want to do that, then spend some afternoons volunteering as a tutor. This *will* make it less likely that you will get into Harvard math graduate school, but you have to make some decisions about what is really important to you.

I am honestly not obsessed at this point with big name schools (I got over that a while ago). I am obsessed with being the best mathematician I can be, and with becoming a human being I can idolize.

But you have to define what is "best". Is a Fields Medal winner better than a high school algebra tutor that teaches in poor inner city schools?