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I applied to various US Pure Mathematics PhD programs for entry this fall and the responses from my top preferences were not favorable, though some of my lower preferences were. I also have an offer to do "Part III" at the University of Cambridge, which is a 1 year masters degree via coursework (although there is an essay component worth about 1/6th of the assessment). I'm considering the option of accepting this, and then re-applying to the US schools with the hope that my application has improved and I get accepted into one of my higher preferences.

My question is: How do US Mathematics departments view the Part III program, and if I do it then will it have a good/bad/no effect on my subsequent application?

Some things to consider: The Part III program is intended as preparation for a PhD degree, unlike many Masters degrees that are intended as terminal degrees. Is this well known?

An essay is part of the course - it usually involves giving a unified exposition of several recent papers on a certain topic. Though the content of the essay can be quite advanced, it often does not contain much original research.

The course is from October '14 to June '15, while applications to US schools are due around December '14. This means I will not have had enough contact with Cambridge faculty to get a letter of reference from them. The marks for all the subjects taken are released around June '15, so I won't be able to include any of those marks in my application either. However in the time between now and October I will continue work with my undergraduate supervisor, which should improve my main letter of reference at least a little. My field is algebraic geometry, and my supervisor has said that it would be better for me to go through Hartshorne in the next ~6 months rather than forcing myself to research just yet. So I won't be getting any original research done by the time I apply again.

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I think Part III is quite common amongst US students nowadays, and is thus well-known to graduate admissions committees. Probably the effect on your application will be good, since if nothing else it shows you are serious about studying mathematics; a school can also reasonably expect you to be considerably better prepared coming after a year at Part III than when you finished your BA. So if one of your issues was a weak curriculum as a BA student (as opposed to poor grades or weak letters), then Part III could help quite a bit. On the other hand, I wouldn't count on a dramatic change in your graduate school admissions, in part because as you note, you won't have your grades or a strong recommendation from Part III in hand when you apply for graduate schools the next time around. I wouldn't worry about the fact that Part III is not research based; obviously doing research before starting graduate school is great, but most graduate schools in the US are not really expecting you to have done much in the way of it beforehand, or to be even close to ready to start when you enter.

I think if money is no object, then Part III is probably as good as anything else you might do for having a strong graduate application in the coming cycle. EDIT: I should probably also say that a matter of substance (as opposed to application strategy), Part III is probably on average better than spending a year as a grad student at a random respectable grad program in the US, since it will bring you into contact with a wide variety of other students and ideas.

However, I think you should weigh that next to the possibility of starting graduate school at one of your lower preferences, and then trying to transfer after a year, or when you get a master's degree. There's no guarantee this will work, but the same can be said of Part III. If you've been offered funded admission at a respectable school in the US, and would have to pay your own way at Part III, I would look hard at whether you think it's worth the money.

EDIT: I wouldn't count on the "glamour" of Cambridge itself to have a strong effect. I honestly don't know how selective Part III is (maybe someone who knows can comment.), so I wouldn't rely on assuming that admissions committees will consider it as such. There's some psychological "band-wagon" effect where getting one prestigious position reinforces getting others, but it won't work if the substance isn't there. Getting a BA from Harvard or MIT is helpful for getting into graduate school (if you have a strong record) because an admissions committee is more confident that getting an A in math class at Harvard really means something, and that a professor at Harvard has a lot of experience with talented undergraduates and thus can speak with some certainty about what it takes to succeed in graduate school. So, if you went to Part III, got good scores on your exams and got a strong recommendation from a professor there, that could strengthen your application a lot as a "second opinion" reinforcing the recommendations and grades you have from your BA. However, as discussed, those would only be available for the fall 2016 admission cycle, not 2015. One possibility is to see about accepting one of your safety schools with a deferment to go to Cambridge, going to Part II in 2014-5, going to the safety school in 2015 and applying for transfer in 2016 or 2017 if you're unhappy there.

One piece of information we're lacking is what your "lower preferences" are (don't put too much stock in USNW rankings, but they help to be concrete in a discussion like this). It makes a big difference whether we're talking about a school in the rank ~25 (like UCSD), ~50 (like UVA) or ~100 (for example, South Carolina). In the former case, I'd say it's a waste of time to try to try to move up unless you go and are miserable, whereas in the latter it makes a certain kind of sense.

LATER EDIT: Incidentally, yes, given that your undergraduate grades are already "baked in," the main things you can still hope to change are your letters, and also your GRE scores if those were bad.

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From your answer, I suppose then the more important thing for me is to secure a stronger letter in the next 6 months. I will have to weigh the financial options. One last question: You say a potential benefit of taking the course is that it shows I'm serious about studying mathematics. I was wondering, does it also impress admissions people in any way? For example, if the first thing one sees is that the applicant is from Harvard, you might think there is a decent chance the applicant is a strong student. Does Part III have a similar effect? Thank you for your answer Ben. –  Craig Mar 11 at 7:36
    
@Craig I added more about your question to my answer. –  Ben Webster Mar 11 at 10:57
    
"I honestly don't know how selective Part III is" -- it's fairly selective: dpmms.cam.ac.uk/~twk/PartIII.pdf, "... normally require First Class Honours from students coming from other British universities ... you should be able to profit from Part III if you are in the top 10% of mathematicians graduating in your country". I guess the questioner's previous unfavourable PhD responses already had the same information that Cambridge used to accept him for Part III. If they rejected due to lack of Master's-level study, then Part III is exactly that by international standards. –  Steve Jessop Apr 14 at 15:55
    
@SteveJessop I actually read that earlier. I don't read "you should be able to profit from Part III if..." as the same as "we will only admit you to Part III if..." as synonymous. Of course, I'm sure it isn't all that easy to get into Part III, but I wouldn't take it as a sign that somebody has a really strong file, like I would someone getting into, say, Harvard. That was also meant to suggest that other people may not know how to interpret it and thus won't put a lot of weight on it. –  Ben Webster Apr 17 at 10:22
    
@BenWebster: fair enough, it's all about how people interpret it after all, not how selective the course really is. But given what I know of Cambridge and the language of UK universities, I would not expect a course to admit someone if, in the opinion of the admissions people, they don't meet the criteria to benefit from it. Strictly speaking those criteria are stated as sufficient but not necessary anyway, so it's ambiguous however you cut it :-) "Normally require First Class honours" is an indicator, but of course you can prove that to the PhD program with or without Cambridge's say-so. –  Steve Jessop Apr 17 at 11:03

Part III is pretty widely known, including in the US. Many of my fellow students who sought PhD positions there were successful; though, of course, they were students with strong academic records to begin with. Something to keep in mind, though, is that not only will you not have had sufficient faculty contact to get a proper reference by the American application deadline, and not only won't you have any results by then: you won't have even a record of your ongoing classes. There's no official course registration, and you only declare your exams in the spring. Until then, the academic section of your transcript is literally blank. So, unless you're prepared to wait an extra year to apply for your PhD, the only boost Part III can give your December applications is on its reputation alone.

Side note: the essay is not mandatory, though it is highly recommended. In addition to giving you the closest faculty contact you're likely to have - thus making a reference letter possible - and being genuinely good practice at synthesizing complex research, the Part III courses are brutal. Being able to replace one of them with the essay is a gift. Also on the topic of reference letters: an academic reference commenting on your coursework as a whole is usually obtained from your Director of Studies (a fellow at your college) rather than from any particular course instructor you had. But there's no rule or anything, so if you were to get to know of your instructors and did well on the exam, there's nothing stopping you from getting a letter if he or she is willing.

The lack of primary research is a bit of a treat, to be honest. If you're doing a PhD in mathematics or theoretical physics, you will generally need to take some advanced courses before your research can really get off the ground. Being able to tackle some of those courses with intense focus for a year is a nice alternative to trying to get through them while also trying to justify your research pay, taking qualifying exams, and juggling TA duties. So, yes, you are paying your own way (unless you manage to get a scholarship) but, in fairness, you aren't really doing anything that would justify anyone paying you to be there.

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Thanks for your answer. You are correct in that many benefits are robbed by the timeline. I'll consider the option of waiting an extra year to apply for my PhD. –  Craig Mar 11 at 7:38

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