Grad School Thread

[Reid]

So I'm sure several people here have attended, so I thought we could share some experiences and/or advice. So anything you want to share about graduate school, please share!

I've applied for this coming fall semester for mathematics, a few people I know have received rejections from a few schools but I haven't so far, I suppose a good sign. So, I'm very probably going to get accepted into a program so my train is headed down those tracks. For those who have gone, what am I in for? I know the basic "I'm going to have absolutely no life or fun for four plus years" thing, which yeah, I've at least taken a couple graduate courses so I kind of feel what I'm getting myself into, still curious though to hear about things I may not know in regards to this plan.

[Reverend Jones]

Congratulations so far.

I am not a grad student, but I would suggest you study for your preliminary exams early on, a little bit each day.

You might consider obtaining this book, which was expressly written for the purpose of helping graduate students learn the material expected of you in your analysis exam.

This book is a course on real analysis (measure and integration theory plus additional topics) designed for beginning graduate students. Its focus is on helping the student pass a preliminary or qualifying examination for the Ph.D. degree.

I would also keep an eye out for Iranian terrorists.

[Baconfish]

Masters or Doctorate?

[ECHOMAN]

I was thinking about grad school because of things like this which means, in the future, it may take 4+ years of undergrad plus 1+ years of grad to get a PE for civil engineering. With the cost of education so high, this falls on the back of students and workers, not companies and certainly not the schools. I just don't get the safety and public welfare argument because grad school requirement wouldn't cull the supply of lousy engineers. Thank you professional society that I pay dues to...

[Zloc_Vergo]

Lots of typical advice re: bureaucracy is doubly important in graduate school. Get things in writing. Get yes/no answers (academics are bad at these). Don't assume anything for the positive if you were not explicitly given a yes. You getting paid will depend on these.

Make sure you have chemistry with your adviser, talk to other students to figure out how s/he runs the group. Math is probably less collaborative than physics, but hopefully they can still shed light on how things operate.

Also it's not so much of "not having fun" as constantly feeling inadequate. A good adviser will probably help with the latter, a bad adviser will just remind you that you're inadequate at your (weekly/monthly/annual, depending on how absent s/he is) meeting.

[Eversor]

I did a masters degree in philosophy. When I arrived at grad school, I was in the strange position of feeling like I was better educated than most of my peers, though significantly less well prepared. I was more widely and more deeply read on most thinkers in the philosophical canon, and didn't have many of the half-baked ideas that undergraduates take away from survey courses. But I had been taught the material I'd learned in my undergraduate degree in a way that forced me to relearn a lot of it once I got to grad school. The entire approach was different, and the expectations were qualitatively different (not simply "higher"). It required some adjustment, but I learned at a much more rapid pace than at any time during my undergraduate degree. Overall, my undergraduate degree was formative, but my graduate degree was even more so.

In most of the departments I've gone through, (and I've gone through several, each at a different university), there was a strong pressure to conform and adopt whatever prevalent orthodoxies are perceived as being defining of the identity of the department. I wouldn't be surprised if that happens less outside the humanities, where I'd hope professors and students hold fewer commitments to polarized schools of thought (they can even be cult-like at times). But I'll admit I have no idea how it works in the natural sciences/mathematics (and I'm very curious if it does happens there).

Another thing to anticipate is how much professionalization is part of being a grad student. It will be stressed more in a doctoral program than a masters program. But still, over the course of your degree, you'll be expected to develop some of the hard and soft skills, and acquire many of the credentials, that a professional academic is expected to have.

[Nikumubeki]

I went to a school once.

[Brian]

I'm so judging that book by its cover.

[Jon`C]

I'm so judging that book by its cover.

It's not always a bad idea.

[Reverend Jones]

The cover is **** because it is self published.

[Reid]

Congratulations so far.

I am not a grad student, but I would suggest you study for your preliminary exams early on, a little bit each day.

You might consider obtaining this book, which was expressly written for the purpose of helping graduate students learn the material expected of you in your analysis exam.

...

I would also keep an eye out for Iranian terrorists.

Ay, thanks for the suggestion, I've lined up a job for myself working at a summer camp this summer, I'll have to get a book like this to study in my spare time.

Masters or Doctorate?

Doctorate

I was thinking about grad school because of things like this which means, in the future, it may take 4+ years of undergrad plus 1+ years of grad to get a PE for civil engineering. With the cost of education so high, this falls on the back of students and workers, not companies and certainly not the schools. I just don't get the safety and public welfare argument because grad school requirement wouldn't cull the supply of lousy engineers. Thank you professional society that I pay dues to...

Yeah, the growing consensus seems to be that a Masters is the new Bachelors.

[Jon`C]

Yeah, the growing consensus seems to be that a Masters is the new Bachelors.

That is sad for about 100 reasons.

[Reverend Jones]

There are good and bad reasons to get a masters.

Some people who ****ed up in undergrad to do it to prove to the PhD programs that they are capable of getting good grades, and shell out the money without a stipend.

Many more, particularly foreign students in C.S., just go through the motions of a masters in exchange for their hefty out-of-sate tuition, and the universities are happy to take their money in exchange for the rubber stamp.

Those cases aside, it IS genuinely sad if the pace and quality of high school education in the U.S. is so retarded that people don't figure out what they really want to now try to do in until their senior year of college. But then again, college in the U.S. is often one big country club with so many social activities and extracurriculars, that they hardly have time between the booze and the polishing of their resume and Facebook with this and that.

[Jon`C]

There are good and bad reasons to get a masters.

Some people who ****ed up in undergrad to do it to prove to the PhD programs that they are capable of getting good grades, and shell out the money without a stipend.

Many more, particularly foreign students in C.S., just go through the motions of a masters in exchange for their hefty out-of-sate tuition, and the universities are happy to take their money in exchange for the rubber stamp.

Those cases aside, it IS genuinely sad if the pace and quality of high school education in the U.S. is so retarded that people don't figure out what they really want to now try to do in until their senior year of college. But then again, college in the U.S. is often one big country club with so many social activities and extracurriculars, that they hardly have time between the booze and the polishing of their resume and Facebook with this and that.

Proliferation of advanced degrees is fully explainable through game theory. It doesn't require student indecision.

[Reverend Jones]

I am sure you are right, and my comment was just a cynical half tongue-in-cheek remark about my observations from university.

I would imagine that by game theoretic explanations, you are probably talking something about the job market, in which employers can shift the cost of training their new employees onto the students and the government, financed through debt.

[Reid]

Lots of typical advice re: bureaucracy is doubly important in graduate school. Get things in writing. Get yes/no answers (academics are bad at these). Don't assume anything for the positive if you were not explicitly given a yes. You getting paid will depend on these.

So like, be annoying enough to get answers, get solid answers with evidence? Thanks. I've been using a similar strategy to get anything done in undergrad but I guess it gets worse. How will payment be affected by my handling bureaucracy? I was under the assumption that I should be getting funding from TAing and fellowships?

Make sure you have chemistry with your adviser, talk to other students to figure out how s/he runs the group. Math is probably less collaborative than physics, but hopefully they can still shed light on how things operate.

Also it's not so much of "not having fun" as constantly feeling inadequate. A good adviser will probably help with the latter, a bad adviser will just remind you that you're inadequate at your (weekly/monthly/annual, depending on how absent s/he is) meeting.

The inadequacy feelings come often enough as an undergraduate, they get worse in graduate school? I'll be okay. I hope.

I did a masters degree in philosophy. When I arrived at grad school, I was in the strange position of feeling like I was better educated than most of my peers, though significantly less well prepared. I was more widely and more deeply read on most thinkers in the philosophical canon, and didn't have many of the half-baked ideas that undergraduates take away from survey courses. But I had been taught the material I'd learned in my undergraduate degree in a way that forced me to relearn a lot of it once I got to grad school. The entire approach was different, and the expectations were qualitatively different (not simply "higher"). It required some adjustment, but I learned at a much more rapid pace than at any time during my undergraduate degree. Overall, my undergraduate degree was formative, but my graduate degree was even more so.

In most of the departments I've gone through, (and I've gone through several, each at a different university), there was a strong pressure to conform and adopt whatever prevalent orthodoxies are perceived as being defining of the identity of the department. I wouldn't be surprised if that happens less outside the humanities, where I'd hope professors and students hold fewer commitments to polarized schools of thought (they can even be cult-like at times). But I'll admit I have no idea how it works in the natural sciences/mathematics (and I'm very curious if it does happens there).

Another thing to anticipate is how much professionalization is part of being a grad student. It will be stressed more in a doctoral program than a masters program. But still, over the course of your degree, you'll be expected to develop some of the hard and soft skills, and acquire many of the credentials, that a professional academic is expected to have.

I don't think there are really "orthodoxies" in math the same way, because there aren't alternative interpretations of texts, but certain departments do have their strong and weak points in what fields they actively pursue. UCR is pretty dead for any sort of topology, the one topologist I know basically does group theory, but if you want representation theory then this school is a fantastic climate. So on and so on, even between schools who are strong in the same fields may have different ideas about what's important in that field. At least this is what I've gathered from the conversations I've had with math faculty.

[Reid]

That is sad for about 100 reasons.

Well, yeah, I wouldn't mind trying to work some programming job for a while, but I don't think I could find anything fun or interesting to do. A higher maths degree sounds far more appealing and rewarding than any job I could get straight out of college, even if I'll be poor, but I've been poor most of my adult life, I can deal with it.

Proliferation of advanced degrees is fully explainable through game theory. It doesn't require student indecision.

Basically a Masters degree is the lowest math degree that makes getting an interesting job feasible. In CA it would be conceivable for me to finish my teaching credentials and get a decent paying job teaching high schoolers, but seriously who wants to teach calculus to 17 year olds in Bakersfield? Even though the odds are extremely likely that I won't get any tenured research position, still much prefer to spend my life here than there.

[Reid]

There are good and bad reasons to get a masters.

Some people who ****ed up in undergrad to do it to prove to the PhD programs that they are capable of getting good grades, and shell out the money without a stipend.

Many more, particularly foreign students in C.S., just go through the motions of a masters in exchange for their hefty out-of-sate tuition, and the universities are happy to take their money in exchange for the rubber stamp.

Those cases aside, it IS genuinely sad if the pace and quality of high school education in the U.S. is so retarded that people don't figure out what they really want to now try to do in until their senior year of college. But then again, college in the U.S. is often one big country club with so many social activities and extracurriculars, that they hardly have time between the booze and the polishing of their resume and Facebook with this and that.

I've applied only to PhD programs, because the funding is better and I have the GPA/scores for it.

[Reverend Jones]

Das ist gut!

[Jon`C]

I am sure you are right, and my comment was just a cynical half tongue-in-cheek remark about my observations from university.

I can't generalize, but I worked the program selection problem for my alma mater so I can at least speak about the situation there. Most incoming students are sensitive to economic opportunities, and have fairly specific professional goals in mind when they start their programs. Poor outcomes usually happen for two reasons: either the market signal is out of whack due to intentional interference, as is the case with medical, dental, and other professional schools; or students do not thrive in their selected program (which can happen for many good reasons, not just because they're too busy partying) and the university administration convinces them to finish 'something', where 'something' is usually less employable and more expensive than just dropping out.

I would imagine that by game theoretic explanations, you are probably talking something about the job market, in which employers can shift the cost of training their new employees onto the students and the government, financed through debt.

It costs too much for domain experts to review every application. Companies need a cheap resume filter, so they use education. Candidates want to pass those resume filters, so they get better and better credentials. There a race to the bottom (top?) independent of any real need for that training.

[Jon`C]

Well, yeah, I wouldn't mind trying to work some programming job for a while, but I don't think I could find anything fun or interesting to do. A higher maths degree sounds far more appealing and rewarding than any job I could get straight out of college, even if I'll be poor, but I've been poor most of my adult life, I can deal with it.

There's Wall Street quant or quant dev, or most Silicon Valley companies, which do hire math undergrads as long as you aren't totally hopeless around code, at least some of which are working on stuff that's not totally boring. For further schooling there's also accounting, actuarial sciences, or economics. Like you said, though, none of this is as interesting as a math PhD. It's what I would have done if I hadn't gotten tired of being so poor.

[Reid]

There's Wall Street quant or quant dev, or most Silicon Valley companies, which do hire math undergrads as long as you aren't totally hopeless around code, at least some of which are working on stuff that's not totally boring. For further schooling there's also accounting, actuarial sciences, or economics. Like you said, though, none of this is as interesting as a math PhD. It's what I would have done if I hadn't gotten tired of being so poor.

What qualifies as "totally hopeless"? I've implemented some tricky algorithms to make use in some of my math research, and like, I can write a functional fizzbuzz. What competency is expected? I could always use some spare time in these last months of undergrad to learn more.

[Jon`C]

What qualifies as "totally hopeless"? I've implemented some tricky algorithms to make use in some of my math research, and like, I can write a functional fizzbuzz. What competency is expected? I could always use some spare time in these last months of undergrad to learn more.

As far as junior roles go, "not totally hopeless" means "can convince someone that you can learn to do the job within 8 months". There isn't a subjective measure for this.

On the plus side, tech interviews are mostly answering bull**** discrete math questions on a whiteboard, so you should be quite good at them.

[Zloc_Vergo]

So like, be annoying enough to get answers, get solid answers with evidence? Thanks. I've been using a similar strategy to get anything done in undergrad but I guess it gets worse. How will payment be affected by my handling bureaucracy? I was under the assumption that I should be getting funding from TAing and fellowships?

The inadequacy feelings come often enough as an undergraduate, they get worse in graduate school? I'll be okay. I hope.

If you can navigate undergraduate bureaucracy efficiently, you'll be fine.

Re: getting paid, I'll give you my anecdote. A potential adviser told me he had funding for me, with no qualifications, which I took to mean "I have funding for you, come work with me and it'll be great!" and I joined his group. What it actually meant was he had funding for me to finish my first year (that my department was willing to pay for as well), but following that first year, he had no money left for even that first summer (God help the experimentalists in the group who had no money left to buy equipment with). He didn't tell me until ~1 month before the summer started that there was no money for me for the summer. Luckily my department offers full first year funding so I got to stay on that summer, and I lucked into a TA position from a prof I was acquainted with for the fall. I do not have any guaranteed funding at this point, so I have to TA indefinitely or find another group to switch to, and I still don't know what I'm going to be doing to pay the rent this summer while being geographically limited for personal reasons. It's a lot of stress, but this was my fault for not asking very specifically what "funding" meant to my adviser.

My situation is very much a "worst case," but it's still a possibility, and given the new administration NSF money is probably going to be even harder to come by. TAing will work, and since you're in Math there will always be ****ty freshman/sophomore classes to TA over the summers, but it's not ideal for getting research done.

The impostor syndrome absolutely got worse for me in grad school, but it's only as debilitating as you allow it to be. As with many things, it's about maintaining perspective and realizing that the student down the hall with four theory papers this year is not who you need to be. Students develop into researchers at different rates and through different routes.

[ECHOMAN]

I'm sure it varies a lot, but TAing actually provides some decent cash?

[Zloc_Vergo]

TAs get paid the same rate as RAs at my institution (and other schools I know about), so it's preferable to have an adviser with funds to pay your tuition/stipend as opposed to needing to TA. It effectively lifts a part time job off of your shoulders that gives you no extra compensation and allows you to just do research.

E- to clarify, TAing will pay the bills and get your tuition waived. The issue is the time it takes away from doing the work you're in graduate school to do, and you aren't compensated any extra for it. And, depending on the school, TA positions may not be guaranteed (I know several graduate students struggling to find a TA position for the spring semester which starts in a week). This stress and time sink is avoidable if you have an adviser with enough grant money to cover your expenses instead.

[Vin]

I did a semester of community college. Now I'm too old.

I work at a gas station.

[Alco]

It costs too much for domain experts to review every application. Companies need a cheap resume filter, so they use education. Candidates want to pass those resume filters, so they get better and better credentials. There a race to the bottom (top?) independent of any real need for that training.

This is very true and many hiring managers could probably care less if you even went to college (especially if you already have a work history in the field). This is why a lot of people lie on their resumes about their education. The student debt isn't worth it and I'm not convinced the quality of education received is either.

[Reid]

As far as junior roles go, "not totally hopeless" means "can convince someone that you can learn to do the job within 8 months". There isn't a subjective measure for this.

On the plus side, tech interviews are mostly answering bull**** discrete math questions on a whiteboard, so you should be quite good at them.

Sounds alright, now I just need to learn how to write a resume properly.

If you can navigate undergraduate bureaucracy efficiently, you'll be fine.

Re: getting paid, I'll give you my anecdote. A potential adviser told me he had funding for me, with no qualifications, which I took to mean "I have funding for you, come work with me and it'll be great!" and I joined his group. What it actually meant was he had funding for me to finish my first year (that my department was willing to pay for as well), but following that first year, he had no money left for even that first summer (God help the experimentalists in the group who had no money left to buy equipment with). He didn't tell me until ~1 month before the summer started that there was no money for me for the summer. Luckily my department offers full first year funding so I got to stay on that summer, and I lucked into a TA position from a prof I was acquainted with for the fall. I do not have any guaranteed funding at this point, so I have to TA indefinitely or find another group to switch to, and I still don't know what I'm going to be doing to pay the rent this summer while being geographically limited for personal reasons. It's a lot of stress, but this was my fault for not asking very specifically what "funding" meant to my adviser.

My situation is very much a "worst case," but it's still a possibility, and given the new administration NSF money is probably going to be even harder to come by. TAing will work, and since you're in Math there will always be ****ty freshman/sophomore classes to TA over the summers, but it's not ideal for getting research done.

Fortunately for math people, lots of people use calculus so getting a decent TAship is basically guaranteed by you being there. Meaning, I shouldn't have as much difficulty funding.

So, what the **** is wrong with that advisor? Did they not think, at all? Did you tell them they are a moron? I know you didn't, but Jesus. There's room there to be a little more than pissed.

The impostor syndrome absolutely got worse for me in grad school, but it's only as debilitating as you allow it to be. As with many things, it's about maintaining perspective and realizing that the student down the hall with four theory papers this year is not who you need to be. Students develop into researchers at different rates and through different routes.

Yeah. It's important to maintain mental health. I know it will get worse for me as well, I'll have to figure out something to make it better.

[Reid]

This is very true and many hiring managers could probably care less if you even went to college (especially if you already have a work history in the field). This is why a lot of people lie on their resumes about their education. The student debt isn't worth it and I'm not convinced the quality of education received is either.

Huh, I'm not bold enough to lie on a resume, but it's really that common?

[Reverend Jones]

Like alco said, it would surely depend on how little the hiring manager cares about ancillary things that could in theory be red flags, apart from the positive reasons s/he subjectively assess as sufficient to make you as good a candidate as any in their mind.

For example, somebody with decades of experience is not even going to need to bring up their credentials, whereas if the job just requires a warm body and the employer is a large corporation, they have no way to filter out the would be psychopathic disgruntled employees or flakes except through background checks, which more or less means contracting out the task of investigating and verifying your entire employment history, including the supposed gaps you may have claimed to cover up being fired over and over.

[Jon`C]

Huh, I'm not bold enough to lie on a resume, but it's really that common?

Not in professional jobs at least, where education and employment background checks are standard. You aren't going to get very far pretending you have a CS degree for example.

[Ni]

I'm currently doing a PhD in software engineering and work at a university as a lecturer after having spent almost 10 years as a software developer prior to that. You've probably heard this a hundred times but: Don't do a PhD because you want a qualification or Dr. in front of your name. If you do this, you'll end up spending 4-6 years of your life for a qualification that gives you some incredibly niche knowledge that probably won't be meaningful to most employers. If you find a topic you're genuinely interested in and want to explore then go for a PhD, if you just want a higher salary at the end of it, you're much better off getting those 4-6 years experience in the industry you're targeting. I was lucky enough to get my funding paid as part of my job as well, if you're paying for it yourself you'll end up spending a fair amount of money depending on the institution.

A friend of mine has a PhD in Biochemistry, he was working in a yogurt factory packing boxes for nearly a year until he managed to get a job at a hospital, *and* he is required to do another masters related to the job he is doing while he's there (He'll have two masters and a PhD by next year!). Because he went down the route of Degree -> Masters -> PhD. When he finished his PhD he only had academic lab experience so was often overlooked for people with less qualifications but more relevant experience.


n.b. I live in the UK, the US system may be different and will be different in different industries.

[Reid]

I'm currently doing a PhD in software engineering and work at a university as a lecturer after having spent almost 10 years as a software developer prior to that. You've probably heard this a hundred times but: Don't do a PhD because you want a qualification or Dr. in front of your name. If you do this, you'll end up spending 4-6 years of your life for a qualification that gives you some incredibly niche knowledge that probably won't be meaningful to most employers. If you find a topic you're genuinely interested in and want to explore then go for a PhD, if you just want a higher salary at the end of it, you're much better off getting those 4-6 years experience in the industry you're targeting. I was lucky enough to get my funding paid as part of my job as well, if you're paying for it yourself you'll end up spending a fair amount of money depending on the institution.

A friend of mine has a PhD in Biochemistry, he was working in a yogurt factory packing boxes for nearly a year until he managed to get a job at a hospital, *and* he is required to do another masters related to the job he is doing while he's there (He'll have two masters and a PhD by next year!). Because he went down the route of Degree -> Masters -> PhD. When he finished his PhD he only had academic lab experience so was often overlooked for people with less qualifications but more relevant experience.


n.b. I live in the UK, the US system may be different and will be different in different industries.

Well I don't expect anything I study in math to be super marketable right out of the gate anyway, the PhD is purely because research is interesting.

[ragna]

A little background info first: at MIT currently, but in a past life I worked at McBain Consulting Group and before that I quit my PhD program because I went in without thinking through exactly why I wanted to do this ****. If you are pursuing a PhD because "research is interesting" (I thought so too), you are in for a world of hurt. It will very likely get ****ing brutal for you, and when that time comes, when the sun is setting on your 20s and you realize you're still living like a filthy undergrad but sunk cost fallacy and atrophied professional/social skills makes you scared ****less to leave, your casual interest in "research" is not going to sustain you through it and you will end up like this guy: https://en.wikipedia.org/wiki/James_...mass_murderer). Also, as I understand it, math is one of those disciplines for which you either "have it" or you don't, and at this point you probably know where you stand. Unless of course you're secretly a Putnam winner or something, in which case you are smarter than everyone in this forum combined and should go shoot an email to Jim Simons at RenTec.

Lastly, some tips and realities: 1) if you're thinking of becoming a professor, you pretty much have to go to a top school and do a ****ty postdoc after you graduate, and 2) if you're thinking of getting an industry job, look through the school's graduate program employment statistics and more specifically, the employment records of grad students advised under the PI you wish to slave under. That will set some expectations - grad students even at MIT have a harder time than you'd think finding jobs, so it's a very real possibility that by the time you graduate, you will be forced to do a postdoc because you couldn't find a job.

[Reverend Jones]

math is one of those disciplines for which you either "have it" or you don't, and at this point you probably know where you stand

I think Reid "has it" if he's figured how to read and write proofs and has developed some intuition for the general roadmap of the subjects he's interested in pursuing. If you have the grades and GRE scores to get into grad school and are able to hold your own in a grad course, it really just comes down to keeping up the endurance and obsession that math demands. Experience with the mechanics of research (get in the habit of reading mathscinet abstracts) is essential too. And yeah, ask around and make friends so you find out early the do's and dont's w.r.t duties like T.A.ing, e.g. calculus and spending all your time grading and failing to keep up with classes, exams, and research. There's a lot to juggle, but also don't shoot yourself in the foot by setting up an impossible pace for yourself. Anything you can do to get ahead is going to really reduce your stress levels here.

He definitely has his work cut out for him, though, if he wants to pass has qualifiers on the first go. Don't procrastinate on this and fail the exams more than once or twice, or you get kicked out.

[Jon`C]

Theorem proving is an uncomfortable synthesis of choosing the right problems, having past exposure to the exact combination of ideas that will permit a proof, and the dumb luck to eventually find the proof you want in an infinite space of dead ends.

Feynman's Algorithm pretty much describes it, in my experience.

[Reverend Jones]

Disclaimer: I am not a graduate student and only have an undergraduate degree in math.

Theorem proving is an uncomfortable synthesis of choosing the right problems, having past exposure to the exact combination of ideas that will permit a proof, and the dumb luck to eventually find the proof you want in an infinite space of dead ends.

This is totally true to my knowledge, but I think there are some mitigating factors.

I would hazard to guess that a huge benefit of a graduate education in mathematics comes from spending enough time in the weeds that you eventually develop some very good heuristics, examples (and a whole bunch of counterexamples) that help you evaluate the plausibility of a given statement without too much work (in fact, in analysis, most of what you do involves getting good enough estimates). In certain subfields of math, this is even stronger; for example, in geometry, my understanding is that there are a great deal of statements you can make which are obviously true or false, simply because a false theorem would contradict what is a highly interconnected lattice of theorems that intuitively fit together.

Buuuuut, we all know that believing something is true is not the same as proving it so. So yeah, expect to waste a ton of time running into dead ends, and it only gets worse as you tackle harder problems. But you'll still get to learn a lot of cool stuff in the process, right? (And once you're a professional mathematician, you'll be the one inventing the cool theoretical stuff, regardless of the success in actually proving the theorem you sought.) It also helps that the theoretical core of modern mathematics is highly interconnected (especially to the extent that algebra makes an appearance), so you really do get a lot of intuition just for free as your knowledge compounds. The key idea you'll need to prove your theorem (sometimes) might simply require applying something you already know in a more exotic setting (even if you don't have some obscure missing link to make it actually work out). Your advisor will also have a ton of papers for you to read, and you'll start to see the connections. As you advance in your career, you're going to want to be super social and absorb as many good ideas as possible, as you never know which obscure technique you'll need to borrow to get your result.

But maybe I am getting too far ahead here, since all this assumes more experience than you'll have starting out in grad school. You probably will waste a lot of time in research in the beginning, but OTOH your homework won't be about open problems, but about things which you'll have a ton of practice wrangling with by the end of the course.

"An expert is a person who has found out by his own painful experience all the mistakes that one can make in a very narrow field." --Niels Bohr

[Reverend Jones]

OTOH, after grad school, all bets are off. From my experience at a big state school, the permanent faculty more or less came from very strong doctoral programs in each of their respective subfields, if not top schools like MIT, Berkeley, Princeton, Chicago, etc.

But would it be the worst thing in the world if you were "forced" to take a high paying industry job (rather than prove theorems and teach the rest of your life), after having a blast learning a bunch of cool math and meeting cool people?

[Reverend Jones]

There's one more thing I want to address here, which is the idea that one ought to be far above-average intelligence to do mathematics:

Also, as I understand it, math is one of those disciplines for which you either "have it" or you don't, and at this point you probably know where you stand. Unless of course you're secretly a Putnam winner or something, in which case you are smarter than everyone in this forum combined and should go shoot an email to Jim Simons at RenTec.

I think this doesn't apply to math, unless you absolutely won't settle for anything less than eventually becoming a professor at a very good research university.

I think what you said probably does apply to physics and finance. I was originally a physics major, and frankly sucked at it. My peers were wickedly smart, and the professors seemed to encourage competition. My mathematical methods of physics professor was unable to define his strange use of terminology he was using in a lecture on integration that he picked up from quantum field theory without realizing it. The field is almost completely dominated by men--it's a similar crowd as electrical engineering.

Mathematics is totally different. It is true that there are somewhat less (American) women in graduate school (many of them seem to choose to become school teachers right after college), significantly more international students, and in general much smarter people. On the other hand, I know more than a couple undergraduates who do not fall into this category at all. Many were girls, and few were hyper-competitive or anti-social. Their intuition for the subject did not seem incredible either, and sometimes their questions seemed to me to reveal a certain naivete.

What they did do, is relentlessly extract information from the professor by asking questions. And they did undergraduate research. Not necessarily very inspired research, but it was solid, because they kept at it, and they used their advisor to full advantage. These skills are probably much more important than some kind of inner fire fuelled by an inferiority complex. Maybe physics is different.

I'll just leave this passage from Terry Tao here, from his article, "Does one have to be a genius to do maths?"

As long as you have education, interest, and a reasonable amount of talent, there will be some part of mathematics where you can make a solid and useful contribution. It might not be the most glamorous part of mathematics, but actually this tends to be a healthy thing; in many cases the mundane nuts-and-bolts of a subject turn out to actually be more important than any fancy applications. Also, it is necessary to “cut one’s teeth” on the non-glamorous parts of a field before one really has any chance at all to tackle the famous problems in the area; take a look at the early publications of any of today’s great mathematicians to see what I mean by this.

In some cases, an abundance of raw talent may end up (somewhat perversely) to actually be harmful for one’s long-term mathematical development; if solutions to problems come too easily, for instance, one may not put as much energy into working hard, asking dumb questions, or increasing one’s range, and thus may eventually cause one’s skills to stagnate. Also, if one is accustomed to easy success, one may not develop the patience necessary to deal with truly difficult problems (see also this talk by Peter Norvig for an analogous phenomenon in software engineering). Talent is important, of course; but how one develops and nurtures it is even more so.

It’s also good to remember that professional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest “score”, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications. For these tasks, mathematics needs all the good people it can get.