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Should more universities and colleges offer degrees in mathematical research?

I am in the process of incorporating a non-profit college and I am considering offering a degree in mathematical research. However, it would have to be original research, e.g. a new proof or a new formula of some kind.

Is original research in mathematics something that can be done by anyone with some education and talent in mathematics, or is it something far too difficult for your average mathematician?

EDIT

The target student for this university would not be a young person wanting to learn mathematics, they should go to a traditional university. The target student is people like you guys on math.stackexchange. Anyone who already has good knowledge of math and who wants to expand this knowledge through researching something that will (1) be guaranteed to be published by a university affiliated journal, (2) get a degree, up to a Ph.D., by doing something in their own time that they both enjoy and adds to the world's knowledge-base.

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Why would you separate study of mathematics from mathematical research? There is no college that I know of that has a "mathematical research" degree separate from the mathematics degree, at the undergraduate level, although some programs will facilitate individual research. Graduate work almost always involves research, on the other hand. –  Thomas Andrews Jun 6 '13 at 17:07
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I see this as appealing only to very highly motivated students. In a North American context, such students will have undergraduate research opportunities at first-rank universities. –  André Nicolas Jun 6 '13 at 17:08
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I would also expect that the number of problems which can be solved, as well as understood, with a basic knowledge of maths, is incredibly small (although probably non-zero), and that of these problems most remain open because nobody cared enough about the answer to settle them. What level would the students be at when they start? If they don't already have a degree in mathematics, that would probably be more useful than something requiring them to produce original work. –  Matt Pressland Jun 6 '13 at 17:08
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I have no idea how you could teach anything in a "research-only" university. It's certainly possible to require original research to graduate in mathematics, but research-only seems to imply something more than that. –  Thomas Andrews Jun 6 '13 at 17:11
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Presumably, this college would have some guidance, otherwise it is not clear what it provides other than a piece of paper at the end. Still, it seems extreme in a pointless way. It seems like an arbitrary division, not one made not for pedagogical reasons but for either fiscal or ideological reasons. –  Thomas Andrews Jun 6 '13 at 18:03

1 Answer 1

up vote 14 down vote accepted

Firstly, a bit of an aside, which nevertheless seems relevant: many of "the people like you guys" on Math.SE are either mathematical Ph.D.s, or Ph.D. students. Several of them are academic staff at universities or similar institutions.

Secondly, regarding the initial question on whether more universities should offer degrees in mathematical research: this is what the Ph.D. is. Experience shows that it is difficult to engage in successful mathematical research without a strong undergraduate background in mathematics, and without mentoring by faculty who are themselves successful mathematical researchers. This is why the Ph.D. is a graduate degree, and why universities compete for strong researchers to make up their faculty.

I think the question "Is original research in mathematics something that can be done by anyone with some education and talent in mathematics, or is it something far too difficult for your average mathematician?" sets up a false dichotomy. I don't think that original research is something that can easily be done by anyone with some education and talent in mathematics; a good academic environment and good advising is essential. On the other hand, it is not far too difficult for your average mathematician: most university mathematicians are engaging in original research. But they are not students! They were students in a Ph.D. program, and most of them did find original research very difficult at that time --- but they developed their talent for it over their years of study, got their Ph.D., and (in most or at least many cases) continued to mature as a researcher as their career progressed. (For example, it is pretty unusual for very recent Ph.D.'s to advise Ph.D. students, because they are too new at the research game themselves; the wherewithal to successfully advise students is something that develops as one's own ability as a researcher matures.)

In conclusion, I don't really understand what you are envisioning/asking about that is different from the traditional Ph.D. program.

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Hi Matt, thank you very much for your reply. Yes, it sounds like you are right that for mathematics there seems to be no real advantage to doing a research only degree. I guess if this is the case then the only advantage, which I didn't mention, of my including of a mathematics research degree would be that the university I am forming is free - and so it would allow someone to get a degree, if they already had the knowledge, without spending money. Are you saying these people do not exist: people who do math as a hobby/passion and are capable of original research but currently have no degree? –  Alasdair Jun 7 '13 at 4:55
    
@Alasdair: Dear Alasdair, Surely some exist, but not many, I would guess. One thing to remember is that (at least in the U.S.) is that students who are accepted to math Ph.D. programs are paid by the university, rather than paying (in extreme constrast to the situation with undergrad degrees in the U.S.), so at least those with strong undergraduate degrees in mathematics can hope to attend a math Ph.D. program in the U.S. at no direct cost to themselves. To really answer your question, one would have to know more about the precise goals of your university, whether it is intended to ... –  Matt E Jun 7 '13 at 10:43
    
... involve physical attendance of students at the institution or is (for example) going to be an online university (in which case you have access to a much broader range of potential students), who the faculty will be who direct the research, and so on. (Having students do trivial research with no connection to mainstream mathematical research would certainly be possible, but I don't know if it would be acceptable to you. You talk about publishing in a university affiliated journal, which is obviously a much lower bar than publishing in a mainstream research journal.) Another thing to ... –  Matt E Jun 7 '13 at 10:47
    
... think about is that (again in the U.S.) there is a fairly large enterprise in the undergraduate mathematical worlds consisting of R.E.U.'s (Research Experiences for Undergraduates) which are summer programs for mathematics undergrads to participate in in which they do research (the REU will be run at some particular university, and directed by members of the mathematical academic staff there). These are usually quite competitive in their selection process, and are notoriously difficult to run successfully because of the difficulty of involving students who are still learning undergrad ... –  Matt E Jun 7 '13 at 10:49
    
... material in a meaningful research project. They often also involve some courses and other instruction, to help bring the participants up to speed in the research area being considered. Students don't get credit towards their degree for attending an REU, but it is one way of burnishing their CV for when they apply to Ph.D. programs. Finally, at least in the U.S. there are accreditation bodies which certify whether a university's program and degree satisfy certain minimum standards. If you don't intend for your university degrees to meet these standards (or some similar ones) ... –  Matt E Jun 7 '13 at 10:54

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