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Okay, so I know MO has had a recent proliferation of this kind of question, and I know MO is not really for this type of question (though I suspect perhaps this is a phenomenon that is likely to repeat toward the end of every academic year...)- nonetheless I find myself cap in hand and hoping for some guidance.

Background

I wasted my undergraduate degree: following a fairly successful first year and an interest in pretty pictures, I found myself digging around in the region of complex dynamics and fell for it hard. As first loves go it was a great one- I swooned over Montel's theorem and cooed over the simple presentations of iterative dynamics gleaned from the uniformization theorem- but like all first loves; the detail of the thing did not surpass the idea, and pretty soon it had to end. I was disillusioned and reluctant to look for more fish in the sea- my work ethic dropped to zero.

I fell in love again: but too late- algebraic topology/ differential geometry hit me in my fourth year like a simplicial arrow from cupid's own bow but by this time, my grades blew and all the people I knew in the department were DS theorists. I got a 2:1 (for all you non-UK MOers- it's a degree class that's basically a rubber stamp with the word 'mediocre' on it).

I tried teaching school kids: not enough cohomology.

I've got myself a year, a jolly good library and a lot of determination: My aim being to produce something so intriguing/charming/advanced that someone will give me funding to do pure maths.

Question

So what, if anything, should I try to produce?

Specifically: Would I have to solve some grand unsolved problem? Would I get by with just a small one? If so, where would I find it? Perhaps even a complete set of excercises from an advanced book? A digest paper on a difficult topic? [If it helps my research interests are differential geometry, differential topology and gauge theory- but I'm flexible]

I am aware: That the above situation is my fault- and I would be grateful if you were restrained in your remonstrations. That the question, as stated, is highly subjective- but the opinions of research mathematicians is precisely what I am trying to gauge. That the answer may simply be: 'try some less prestigious universities'- in which case, fair enough- but I don't want to rule anything out just yet.

Thanks in advance for any help you can spare.

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Enrolling in a research-oriented masters degree might help your cause. –  supercooldave Jun 25 '10 at 12:02
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-1 for yet another roadmap question, +1 for actually providing some background and motivation, -2000 for the 2:1 = mediocre (will reconsider if you say which university). (But I'll call it even for the use of the description "jolly good".) –  Loop Space Jun 25 '10 at 13:48
    
@supercooldave- Have considered it but funding is likely to be an issue... :( –  Tom Boardman Jun 25 '10 at 15:11
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You've saved yourself with that last phrase. –  Loop Space Jun 25 '10 at 18:10
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I'll note, for American readers, Wikipedia says a 2:1 is the second highest "honors" you can get in the UK; it roughly puts you in the top half, but outside the top 10%. –  Ben Webster Jun 27 '10 at 7:31
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6 Answers 6

up vote 18 down vote accepted

I sympathize with your case. A 2.1 is really not bad. You shouldn't denigrate yourself and view your peripatetic interests as requiring redemption.

Taking on a big unsolved problem without guidance or the background of a PhD student seems doomed to fail. Locking yourself in a library with all the world's books is unlikely to produce anything of merit. I have never heard of a case of a student producing something of "intriguing/charming/advanced" and using that to gain graduate admission. The romantic, amateur heroic view of math is largely bunk as pointed out by Terry Tao.

There is still hope. I know that undergraduate research is less common in the UK, but I would expect that if you email lots of professors in areas of interest to you and basically offer yourself as cheap or free labor (undergrad student level), there is a good chance that you'll be taken on as an unofficial research student by someone. I know many cases of people in math and science using this sort of informal contact to start research projects that eventually develop into PhD positions. Making yourself known to a tenured professor who can write you a strong recommendation is probably enough to get you a PhD position somewhere (in the US, UK or Europe). It is unlikely that claiming to solve a big problem or do research on your own is going to be trusted by graduate committees. You need recommendations from people trusted in the academic community.

There are MO users who have taken a decade or more off from education and successfully started PhD positions at Princeton and other top research institues. Good luck.

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'I have never heard of a case of a student producing something of "intriguing/charming/advanced" and using that to gain graduate admission.' Well, there's always Alexandre Grothendieck ( ams.org/notices/200409/fea-grothendieck-part1.pdf ), who arguably gained notice by discovering measure theory on his own... not that I am advocating such a career path for the OP. :p –  Willie Wong Jun 25 '10 at 12:22
    
@ Justin- thanks! +1 That sounds like exactly what I should be doing and it hadn't even crossed my mind :) Might still leave the question open for a bit though in case there's another interesting perpective floating about... –  Tom Boardman Jun 25 '10 at 15:37
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I've made things clear in my own (similar) answer, but anyway: (i): A Masters degree in the UK only takes one year, and is very valuable, much more valuable than independent study (even if you learn the same things! The qualification is the key thing). I actually think this is equally as important as getting good recommendations from professors (if you want EPSRC funding, which you do). I think it's well worth doing an M.Sc. before starting the Ph.D. (ii): As other posters have said, what's great in the US might not be very good for the UK; I think this is a case in point. –  Zen Harper Jul 20 '10 at 1:54
    
In line with Willie Wong's comment, there's also the case of Ritabrata Munshi. According to p. 8 of math.uchicago.edu/~may/PAPERS/MunshiFinal2.pdf, his proof of the Nullstellensatz helped his case with J. P. May (and possibly also at Princeton, where he ended up). –  L Spice Dec 25 '10 at 17:11
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???!!!

My answer is similar to Justin Curry's answer, except that I am specifically talking about the UK, and advocating a more formal approach.

Speaking as someone who does have a Ph.D. from England, and knows people who were given Ph.D. funding from EPSRC despite relatively poor first degrees (and also knowing someone who was given 3 years' funding to start a Ph.D. a second time, after dropping out of his first Ph.D. in the first year - EPSRC may be more generous than you think, although it's probably getting worse):

I believe your plan is totally wrong and worse than useless. The best that can happen is you spend one year learning a bit of stuff, and rediscover a few known results (which won't count for much; rediscovery is useless for funding purposes, unless your method is totally different and better than known ones). Far more likely, you will waste another year.

The first thing is, no matter how much you might think you know now, you must know that you know NOTHING! Without proper, regular guidance from a research supervisor, your plan is GUARANTEED to fail, unless you're a genius or very lucky (preferably both). You simply CANNOT do good, worthwhile, original research yourself without a supervisor. You should not even be talking about research at all now, that's what Ph.D.s are for!

Assuming you want to stay in the UK, as far as I can see the answer is simple.

A 2:1 from Warwick is not that bad.

Your best chance is to do a one-year M.Sc. (or equivalent) at the best university you can (try, e.g., the Cambridge Part III, which has different funding rules from most others). If it goes well, funding for a Ph.D. will be no problem. If it goes badly, well, ask the question again next year...!

A good M.Sc. will more than make up for your 2:1 (which, as I said, is not so bad; a 2:2, on the other hand, would leave you in a very tricky situation!)

You should do this IMMEDIATELY!!! You still should have enough time to enter in September 2010 if you hurry.

If you can't get funding to do the M.Sc./Part III or whatever, you'll just have to borrow money and pay for it yourself with student loans. If you're not willing to do this, as a LAST RESORT you should live near a good university and sneak into the lectures without registering with the university. This is (I believe) perfectly legal, since UK university lectures are still open to the public, as long as you don't disturb anyone. You will almost certainly be unnoticed in Oxford or Cambridge (since everyone will just assume you're from a different College), or large universities where many students don't know each other.

If you ask the Maths Department very nicely, they'll probably let you use their library (and if you pay, they definitely will). But they almost certainly won't let you sit any exams or have any help/tuition unless you pay. You could make private tuition arrangements with Ph.D. students or even university staff; this would probably be much less expensive than registering properly with the university, if you only have a small number of specific courses to look at.

Like I said, doing some M.Sc. lectures without the actual final qualification is a very poor solution, but it's still better than your proposed plan.

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I'll just reiterate what Justin Curry said with a bit of personal anecdote.

Making yourself known to a tenured professor who can write you a strong recommendation is probably enough to get you a PhD position somewhere (in the US, UK or Europe).

This is basically how I got into a PhD program (with a fellowship, even!). My undergraduate GPA was 2.7, and although I did well in some of the more serious math classes, my transcripts were bad enough that I imagined I could get outright rejected from many places without much other consideration. I had done no research either. So I stuck around for an extra year, took some graduate classes, and most importantly got to know a few professors pretty well. It was mainly a matter of me making an effort to interact in class, and to go to office hours and ask questions. I don't mean questions like "I'm stuck on the homework, can you help?" but rather on strengthening results, or related ideas, something I'd read, etc. I also had gotten to know one professor via independent study.

As an undergraduate I had always been hesitant to talk to professors outside of class mainly because I felt that whenever I had a question, I just hadn't tried hard enough to answer it myself. Reasonable or not, that sort of attitude will not get anyone's attention, and you will miss out on a lot of ideas if you just try to learn everything by yourself. So talk to people, and don't be afraid of not knowing something. If you're smart enough to do a PhD and are putting in a legitimate effort, the professor will pick up on it. And they will have something to say about you in a letter beyond simply "_ took my class and got an A."

My advice: find a professor at a local university who is doing stuff that you are interested in. Go talk to them and see if you can meet with them once every week or two weeks for an independent study course. I've found that most seem to be fairly receptive to the idea. It's important to make two points: first, that you are serious about doing math, and second, that you are mathematically mature enough to handle independent study without being too much of a burden (in terms of time spent) on the professor. From what you wrote I imagine you are fine on both counts.

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By independent study course are you referring to meeting periodically with a professor without being enrolled at their (or any) university? –  Matthew Willis Jun 26 '10 at 5:54
    
That was what I was suggesting, although in my case I was enrolled as a student. –  Erik Davis Jun 29 '10 at 5:59
    
This might be great for the US, but I don't believe this advice is so good for the UK. –  Zen Harper Jul 20 '10 at 1:43
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I really have no idea how much of an option this is in the UK, but to a US student in a similar position, I would recommend enrolling in a masters program at a respectable university and working hard, and then trying to transfer somewhere research-focused. I know a couple of people for whom this path worked (for example, one who got a masters degree at San Francisco State University and then transfered to Berkeley). This provides an opportunity to show that you are serious, to get some good recommendations and at least carries some possibility of having financial support.

(Of course, I should credit supercooldave for suggesting this in comments)

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For the UK, this makes no sense: basically all UK universities with respectable Masters courses also have good Ph.D. programmes; you can't have one without the other. –  Zen Harper Jul 20 '10 at 1:45
    
Um, isn't this exactly what you recommend in your answer? I wasn't advocating aiming to go someone non-research based. It's just that sometimes that's where you can get in. –  Ben Webster Jul 20 '10 at 8:22
    
This is similar but subtly different to my answer. You said "transfer somewhere research-focused", and this is meaningless in the UK because practically EVERY university in the UK with a proper masters degree programme is research-focused (it's almost if and only if). If I understand correctly, you're recommending going to universities which offer worthwhile masters degrees in Mathematics, without concentrating on research, and so are therefore easier or cheaper or both to get into. Such places simply don't exist in the UK. –  Zen Harper Jul 22 '10 at 13:52
    
A side remark: practically all UK universities are basically the same price regardless of quality (at least for home students, it might be different for international students); e.g. Cambridge has the same tuition fees as the University of ...... (fill in the blank appropriately!) –  Zen Harper Jul 22 '10 at 13:57
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Ok,I don't have much time to chime in here,but I'm so moved by Tom's post,I feel compelled to.

Tom,my career has been a long road of disappointment and agony. My prime years were spent caring for cancer stricken family-particularly a father who died slowly of prostate cancer for 18 years before mercifully passing 2 years ago. To make a very long story short,it dragged me from a double honors standing in mathematics and biochemistry to an aged, sickly graduate student in pure mathematics barely squeaking by.My career suffered a major last setback when I was forced to take courses for a year and a half to maintain my health insurance,which lead to subpar performances that dragged my GPA down to barely 3.3 from a nearly 4.0. I now need to take my oral exams,get my masters' degree and try and work for a year to support my elderly mother and myself for a year before entering whatever PHD program will take me.

My point is I haven't given up either and you're absolutely right that in published research lies your salvation. I think Erik's given you the best career advice of all the responses so far. The right connections combined with both strengthening your background and diving into journal reading and writing can turn your fortunes around quickly for the best.

But it's absolutely critical you don't founder for 2-3 years doing this.

My advice?

1) Write up a career plan on a calendar with dates and set goals for yourself. Don't set unrealistic ones.For example,don't print out a list of the Millenium Problems,pick one and say that's your ticket to Harvard. (Well,anything's possible,but don't set that as a goal!)

2)Comb the internet for unsolved problems and try and find some that look like they can be attacked reasonably quickly and with some success. Combinatorics and number theory are loaded with these. Start reading significant papers that aren't over your head,get yourself a black notebook and start scribbling. And make sure the notebook never gathers dust no matter how little you write in it per day!

3) Find prestigous professors you either know or think are approachable and try and make connections with them. For example,I'm a religious attender of Melvyn Nathanson's number theory seminar this summer after attending a research level additive number theory course with Dr.Nathanson last year. He's a warm and brilliant man and so far,that seems to be working well for me. I hope to get him to agree to do research with me after delivering a few talks this summer and fall at the seminar. Most exciting-I'm going to try and take Dennis Sullivan's advanced topology seminar this fall and hope that leads somewhere.

You are definitely not doomed. But your GPA is a black stain on your academic soul,don't kid yourself. You now need to show people it doesn't accurately reflect your talent. This will be a much harder road to travel then a 16 year old prodigy taking honors analysis at Harvard his freshman year,no question. But this is not a road to nowhere unless you let it be.You just need to make smart choices from here on out and don't give up.

Good luck!

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See my comment up there. He was not talking about his GPA. A 2:1 at a British university is hardly a "black stain" on someone. And trying to do serious research before enrolling in grad school is putting the cart before the horse. It almost never results in much of anything. Its main use is giving people an opportunity to see if they like problems that take a couple of months to solve rather than a couple of hours. –  Andy Putman Jun 26 '10 at 14:57
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Mr L, a 2:1 is a degree class, not a GPA (whatever that is). See en.wikipedia.org/wiki/Degree_classes for an explanation. –  Robin Chapman Jun 26 '10 at 18:42
    
@Robin GPA=Grade Point Average. In the US,it's approximately as follows: 4.0 =A/A+, 3.5 = A-/B+ 3.0 = B/B- 2.5= C+/C 2.0 =C/C-.Any lower-well,you get the idea. In the US,any lower then a 3.5 is considered subpar for the top graduate programs and they laugh and toss your application in the trash without extenuating circumstances-which could mean any number of things depending on the individual student history and program being applied to. –  Andrew L Jun 26 '10 at 19:59
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@Andrew L : Aren't you still a masters student? I suspect that the most useful advice for someone trying to get into a PhD program would come from people who have either advised students who had gotten into such programs, or at least gotten themselves admitted. I'm not trying to denigrate your advice; rather, I'm trying to suggest that you have a limited perspective and thus should be a little less vehement about telling people what they should do. –  Andy Putman Jun 27 '10 at 22:44
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Following on from Andy Putman's comment: it should probably also be observed that the tertiary education systems differ greatly between England (which is where the original poster seems to be based, and where he got his 1st degree) and North America. In particular, "reasonably prestigious program" sounds a bit odd to this relocated Brit. –  Yemon Choi Jun 29 '10 at 0:38
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Differential Geometry and Differential /Algebraic Topology at first is a wide field. The main problem will be that you have to read the books and work the examples. Simply by reading (personal experience!) you won't have any chance to reflect the problem unless you are some sort of a genius. But as was pointed out before this is a rather romantic view of mathematicians althought there are certainly some exceptions from the general case. Secondly, I don't think that a big unsolved question can be solved offhand by someone with no PhD background knowledge. Or even if someone has that knowledge: mathematics is not made by persons sitting alone in a poorly lightened chamber with tons of papers and thousands of pens. No, it is made by people going to conferences and symposia or what else. The great theorems in our times mostly (I know there are exceptions like Perelman) deduced and proved in a team. Consider for instance the Atiyah-Singer Index Theorem. As the name already suggests, it was made up by two of the brightest mathematicians of our times. The situation in physics is quite the same.

BUT: If you really want to conduct research and not only read mathematical texts because it seems like an alternative to thrilling novels to you AND you already have a college degree, enroll in a graduate program. If you're motivated enough and succeed in showing the profesors that, I think you will have no problem with nor being accepted. As was already pointed out, there are many examples of people who enrolles 10 years after completeing their bachelor's degree in a graduate program. And they were successfull overall.

But I don't like the idea of giving someone something to conduct research about. Research is all about finding your personal interest and your personal style. And again, it's completely unrealistic, in my opionion, that big problems are solved overnight by people who are not directly involved in the mainstream mathematics (at least officially). Although, admittedly, there are exceptions (Perelmann, e.g., but even he completed his mathematical studies before that).

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Nice answer, but by now I'd rather expect a "how it worked out" retrospective by the OP... –  darij grinberg Dec 5 '11 at 17:42
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