Mathematics Grad. School Application Harvard

[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?

 

[twofish-quant]

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).

And where did you get that idea from? I've found that it's not true.

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.

Friendly competition is a good thing, but a lot of academia involves competition that ends up being unfriendly. The problem with academia is that it is up or out. If you make one mistake or lose one major game, you are out, and that's not good for research or life were the point is to make mistakes.

Also in most social situations, survival depends on cooperation and in some cases self-sacrifice. If we all race for the exits in a fire, the most of us are going to die, but if you set things up so that people walk out in a nice orderly way, then all of us are going to live.

Something that I find interesting is that people talk about the wonders of competition, but most of the time it's because they think that they can win the competition. If it becomes clear that you aren't going to win or that you aren't going to win all of the time, then the rules change.

Personally I'm pretty sure I wouldn't be so driven if I knew my future was somehow assured.

Would you be as driven if you knew you were doomed?

You are probably not going to make it into a big name math university, and you probably will not become a professor. If you want to keep doing math without those things, then you have to get creative. What should you do? I haven't got much of a clue. It's something that you have to work out.

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.

In fact, it wasn't. The history of Harvard is quite interesting. Also one of the things that Harvard and UChicago did in the early 20th century was to make a very strong effort to popularize art, philosophy, and literature (see the Dr. Eliot's Five Foot Shelf). In 1900, you may not have the money to go to Harvard, but you can buy the books that Dr. Eliot has listed to get you a Harvard education.

Today, it's even *easier*. All of Dr. Eliot's books are online, but the fact that Harvard isn't trying to create a 21st century equivalent says something bad about Harvard.

The problem is that if everyone is educated then it's harder to stay in power. I think it's pretty sad that Harvard isn't doing anything like the Five Foot Shelf today. MIT OCW is the closest thing, but even there the fact that you have to be "elite" keeps some interesting things from happening. Someone is going to be something revolutionary with MIT OCW, but I'm 99% sure it's not going to be MIT.

Universities are no longer educating and producing thinkers, they are producing careers. It's a trade-off one can't really avoid.

So if you want to be a thinker, then why are you giving into the system that forces you not to think? Why *can't* one avoid this?

It turns out that thinking is hard and dangerous so most people prefer not to do it, even in academia.

 

[Topologist]

@jgm40 Opportunity is something you create. For example, no-one ever told me to do mathematics when I was young. However, I did have several mathematics book in my house, mostly of my parents, on topics in mathematics such as abstract algebra, topology, analysis etc. I picked them up, started reading them, and actually enjoyed them. It got me hooked on to mathematics to such an extent that since then I knew that I want to be a mathematician. This was when I was in primary school.

That said, I needed to have those mathematics books to start doing mathematics. In fact, I still wish that my parents had introduced me to mathematics when I was even younger. It is not, by any means, that I was "old" when I started doing mathematics - I started fairly young. Nonetheless, I dwelled on this point for a long time, regretting the past, and ultimately, not moving forward as quickly as I would have liked.

But eventually I asked myself: why am I doing mathematics? I am doing mathematics because I am passionate about it. I do not want to compete with anyone. There surely would be someone younger than me who did the same advanced mathematics as me but that does not make him better than me, nor me better than him. Mathematics is not a competition like IMO suggests. It is a recreational activity, in my opinion. I finally realized that I should be proud with what I have: I might not have done well in the IMO nor would I have learned the mathematics that Daniel Kane learned to publish in Freshman year. (E.g., number theory and combinatorics.) But I did learn other branches of mathematics - not number theory or combinatorics - but mathematics like topology and algebra - and I enjoyed it.

The point is that there are so many branches of mathematics that it is impossible to compare two mathematicians, even if you know what branches of mathematics they research. And why should you need to compare? Take to heart the fact that there are plenty of mathematicians, "staring you in the face" so to speak, that have PhD's from top universities but never really became successful mathematicians.

After all, however unlikely it may be, if you solve the Riemann Hypothesis tomorrow, people will not be asking questions about your mathematical ability, no matter what your grades are, or from where you obtained your PhD. That is the beauty of mathematics. It is entirely in your hands. Plenty of mathematicians are publishing every day, even as I am writing at present. Most of them are not "spectacular" - they publish through their own love of the subject and hard work - so why can't you or I?

 

[Anonymous217]

I'd like to note that the majority of people who claim they self-study specific concepts go through such a broad detail that it would hardly be anything like a true course on that subject. For example, I could claim that I studied graduate-work Advanced Linear Algebra regarding Umbral Calculus and Affine Mappings simply by reading a page on what the definition of the two are. After all, it's technically true that you did "self-study" graduate-work Advanced Linear Algebra. And even if you did try to go through a lot of detail, self-studying is almost always a broad overview until you actually learn how to do it well in your latter years of undergraduate or in graduate.

 

[deluks917]

^No one would call that self-study. To me self-study means reading a textbook or lecture notes and doing some exercises. I'd think most people would define self-study similarly.

 

[Anonymous217]

Yes but in that manner, you can certainly claim that you self-study. Self-studying is in no way rigorous like a course where you're forced to learn the material, which is what the problem of self-studying is. It in no way shows you well you've mastered the material. I took an extreme example obviously, but it works out in normal cases as well: take for instance a linear algebra book. You could have read through all of it and understood most of the material. However, without doing the majority of the exercises, you only passively learned the material. It is much harder to reproduce a proof rather than to understand why the proof works. And if you simply just read the book, that would also be included as self-studying. This allows you to spend less time self-studying on that course since you're not doing any of the exercises. Then you could use the saved time to "self-study" another course. And generally, if someone claims they "self-studied" many different concepts such as what the OP claims, then it is usually similar to the case I explained: that it was an extremely broad overview and is in no way a true mastery of the material.

 

[Annonymous111]

Yes but in that manner, you can certainly claim that you self-study. Self-studying is in no way rigorous like a course where you're forced to learn the material, which is what the problem of self-studying is. It in no way shows you well you've mastered the material. I took an extreme example obviously, but it works out in normal cases as well: take for instance a linear algebra book. You could have read through all of it and understood most of the material. However, without doing the majority of the exercises, you only passively learned the material. It is much harder to reproduce a proof rather than to understand why the proof works. And if you simply just read the book, that would also be included as self-studying. This allows you to spend less time self-studying on that course since you're not doing any of the exercises. Then you could use the saved time to "self-study" another course. And generally, if someone claims they "self-studied" many different concepts such as what the OP claims, then it is usually similar to the case I explained: that it was an extremely broad overview and is in no way a true mastery of the material.

Yet again people on this forum jump to conclusions about me without knowing me. You've all decided that I must be telling the truth but some kind of bad math student who can't do anything beyond understand the material. Evidence?

Here's how I study the material and how I've studied the material without assistance from anyone for a long time. I pick up a book look at its prerequisites as carefully as possible and only when I have a lot more than what's assumed do I start reading.

At the beginning of each chapter I come to a definition that is the heart of the chapter. Say something like "parallelizable manifold". Then what I do is close the book and think for 1 week about what that definition. No exaggeration. 1 week. And in that process I often work out on my own most of the theory in the chapter. This develops theory-building skills.

Next once I've done this, I actually start reading the chapter. Whenever I come to a theorem or lemma, I close the book and prove it on my own if I haven't already "discovered it" in the 1 week of thinking. It doesn't matter how long it takes. I'll do it. And mostly I've sucessful. I've been doing this so long that it's becoming good practice. Sometimes I've come up with original ideas of proofs that would be presented for a week in class and some proofs that go for 5-6 pages. Often it takes me a day of continuous non-stop thinking concentrating to prove a result that has a 2-3 page proof and the bigger proofs take me a bit more. Often I come up with different proofs. Sometimes I come up with similar proofs to the one in the book and it's usually correct and this shows my undrstanding. And remember these materials are NOT basic stuff. Some of the proofs I've come up with were actually published in top journals fairly recently.

On top of that, I conjecture my own results and write them down as well as come up with new definitions to think about. As you can imagine after going through this thought process the exercises (say 10-12 of them some of which are considered "challenging or even "very challenging) take me no more than 30 minutes to solve, and say an additional 1 hour to write down.

Don't make assumptions about people without knowing them. I may not have done research - that's simply because I haven't tried it not because I've tried it and failed. But I don't read passively. You may ask: if you can do all this why haven't you started research? THis is because I'm "investing knowledge" so to speak. Learning math in this way gives me new insights into the material. I have so much time on my hands still to learn that I'm not rushing research. I'll start specializing more in a couple of areas and when I think I'm ready I'll do research. The way I've learned has given confidence that I can do research and I'm confident enough to undertake this. It's all a matter of time. I don't want to publish low quality papers that people publish who don't know much math. I want to publish high quality work. That's why I'm delaying research. Another reason is that I want to broaden my knowledge as much as possible. And doing research would mean specilizing too early which is often a bad idea.

So some people here say stuff like "the best person from Moscow is better than you" without even knowing me? That strikes me as snobbish. I could go on telling you about what I've done but the point is not to tell you that. THe point is to teach you that making assumptions about people you don't know is a very bad idea. Can the person from Moscow come up with 2-3 page proofs in 1-2 days of concentrated thinking when the material is highly esoteric and requires math from a diversity of areas? If he's done research then remember that I'll do research too after a couple of years. If the person from Moscow is 18 or older then by a couple of years (even more than that 3 years), I'll STILL be younger than him, so it doesn't mean that I've done research later than him by any means.

 

[Newtime]

So you're a 14 year old taking upper level grad courses and have developed most of the classical theory of mathematics on your own?

Even if everything you just said was true, how is it you've studied that much volume yet you spend a full week just THINKING bout a main definition from any given chapter?

Only now am I beginning to doubt you. Something isn't adding up. But then again, maybe I'm wrong and you are indeed the next terence tao or something.

 

[deluks917]

Troll blew his cover with that last post. lol.

 

[Ryker]

So you're a 14 year old taking upper level grad courses and have developed most of the classical theory of mathematics on your own?

Well he actually said he's not 18 yet, not that he's 14

 

[Annonymous111]

So you're a 14 year old taking upper level grad courses and have developed most of the classical theory of mathematics on your own?

Even if everything you just said was true, how is it you've studied that much volume yet you spend a full week just THINKING bout a main definition from any given chapter?

Only now am I beginning to doubt you. Something isn't adding up. But then again, maybe I'm wrong and you are indeed the next terence tao or something.

Because if you think about the definition for a week and develop the theory it makes it much easier to actually read everything (you've already seen it!).

How come I've studied that much volume? Let's say a chapter is 30 pages (could be longer sometimes could be shorter but on an average). After thinking about the key theme of the chapter for 1 week it makes at least half of the chapter easy and then it takes about 3 more days to read the whole thing. So it takes about 10 days to read 30 pages, or 3 pages per day. It sometimes goes quicker and sometimes goes slower as I said.

I didn't want to sound like I'm great or anything. Just that people shouldn't come to conclusions before knowing the person. For all you know I could be a dog on Neptune connecting to the internet! There're so many people these days with a similar kind of background - they've done math young - in the US let alone across the world. In the US alone I do know 2-3 people PERSONALLY who've done that math. And to know that many people personally means there must be many more. The sole point is that people shouldn't decide something without knowing the person.

Similarly, you don't know how much math I've studied. I don't think I've done all that much anyway. But the point is that I have done something. (just an example, there's another guy on tis forum called tom1992 who was 14 when he learned topology (btw, I never said I was 14). see https://www.physicsforums.com/showthread.php?t=152365 ).

The whole reason I started this thread was to get a sense of what kind of competition I'm up against when I apply to Harvard. Obviously the thread has blown up in a couple of directions which wasn't my intentions. this is unfortunate. I don't think I'm great and I never said that but what annoyed me was the way that people said stuff about me without knowing me. I'm not a troll. I don't need to prove this but I'm not. If i were making this stuff up I'd say I was a 5 year old who's published papers! I've self-admitted myself that I haven't published papers.

(see also https://www.physicsforums.com/showthread.php?t=156013 for a guy who took advanced math courses when he was 14)

 

[Newtime]

Because if you think about the definition for a week and develop the theory it makes it much easier to actually read everything (you've already seen it!).

How come I've studied that much volume? Let's say a chapter is 30 pages (could be longer sometimes could be shorter but on an average). After thinking about the key theme of the chapter for 1 week it makes at least half of the chapter easy and then it takes about 3 more days to read the whole thing. So it takes about 10 days to read 30 pages, or 3 pages per day. It sometimes goes quicker and sometimes goes slower as I said.

I didn't want to sound like I'm great or anything. Just that people shouldn't come to conclusions before knowing the person. For all you know I could be a dog on Neptune connecting to the internet! There're so many people these days with a similar kind of background - they've done math young - in the US let alone across the world. In the US alone I do know 2-3 people PERSONALLY who've done that math. And to know that many people personally means there must be many more. The sole point is that people shouldn't decide something without knowing the person.

Similarly, you don't know how much math I've studied. I don't think I've done all that much anyway. But the point is that I have done something. (just an example, there's another guy on tis forum called tom1992 who was 14 when he learned topology (btw, I never said I was 14). see https://www.physicsforums.com/showthread.php?t=152365 ).

The whole reason I started this thread was to get a sense of what kind of competition I'm up against when I apply to Harvard. Obviously the thread has blown up in a couple of directions which wasn't my intentions. this is unfortunate. I don't think I'm great and I never said that but what annoyed me was the way that people said stuff about me without knowing me. I'm not a troll. I don't need to prove this but I'm not. If i were making this stuff up I'd say I was a 5 year old who's published papers! I've self-admitted myself that I haven't published papers.

I don't mean to criticize and I'm sorry my post came across that way, although reading it back now it seems it couldn't be taken any other way. All I'm saying is what I've been saying: if everything you are claiming is true then you are an exceptional student. Not many have the discipline you have nor the background knowledge at your age (I assumed 14 because you said in a few years you would be less than 18). It just seems (very) odd that such an advanced student would be this oblivious to how advanced he is. Surely you have classmates?

 

[Annonymous111]

I don't mean to criticize and I'm sorry my post came across that way, although reading it back now it seems it couldn't be taken any other way. All I'm saying is what I've been saying: if everything you are claiming is true then you are an exceptional student. Not many have the discipline you have nor the background knowledge at your age (I assumed 14 because you said in a few years you would be less than 18). It just seems (very) odd that such an advanced student would be this oblivious to how advanced he is. Surely you have classmates?

No apology needed. You weren't criticizing at all (you were the only one who didn't criticize). Yes I do have classmates. But the thing is this: no-one in class knows me at all and I don't know them. So I practically don't know anyone (so in particular people don't really know how old I am). But that's no problem really for me since I'm used to that from high school. And I kind of don't know how to make friends really except on the internet ;)(btw, I know I'm advanced but not very. As I said, there're people who know the kind of math I know, some who know more. I can understand why and how people think I am advanced. But I really don't see what I've done great. I simply had the opportunity to learn math earlier than others and I took use of it. I'm sure at least 50% of people my age could've done it if they had that opportunity).

 

[G037H3]

Why don't you go solve a few problems in the math subforum?

 

[axeae]

Troll blew his cover with that last post. lol.

I called it quite a while ago

 

[Math Is Hard]

You've exhausted the answers that this forum can give you. We're not a Magic 8 Ball.

Here's what Harvard has to say:

http://www.gsas.harvard.edu/programs_of_study/mathematics.php

The graduate Mathematics Program at Harvard is designed for students who hope to become research mathematicians and show definite promise in this direction.

So, I think the important things to show to Harvard are:
1) Why you hope to become a research mathematician
2) What you have done that shows promise in obtaining that goal

When I hear about students getting infatuated with attending a certain graduate school based solely on prestige of the institution, it seems very shallow and naive. When I looked at graduate programs, all I really cared about was what areas people at different universities are working on, if there was a researcher there who was expert in something I wanted to work on, and if the program fit with my long term career goals. If you don't consider those things, then you might be in for 6-7 years researching something you don't give a rat's patoot about.

As you gain maturity as a student, you'll find a number of schools and programs that meet your goals and expectations. It's wise to apply to as many as you can.