Mathematics Grad. School Application Harvard

[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.