Timeline for Is it common for an undergraduate thesis in pure mathematics to prove something new?
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19 events
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| Sep 6, 2018 at 13:34 | comment | added | BCLC | Cameron Williams, given "pure mathematics operates at a level that is not very accessible for most undergraduates, even those doing research" can you help here How much knowledge is expected of a PhD applicant as compared to a postdoc or a research assistant? or here Emailing professors: Guidelines on choosing papers please? | |
| Aug 2, 2017 at 15:08 | review | Suggested edits | |||
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| Aug 15, 2015 at 9:28 | vote | accept | BCLC | ||
| Jul 31, 2015 at 18:33 | comment | added | Cameron Williams | @JackBauer basically just a Riemann sum in some sense. In every proof it was just used as a tool without any justification or intuition. That's the kind of thing I'm talking about: giving a survey and then really deconstructing in ways that make sense to a novice. | |
| Jul 31, 2015 at 18:32 | comment | added | Cameron Williams | @JackBauer Sort of like that, but not quite. More like: "I'm studying representation theory of locally compact groups but I'm an undergrad and this is pretty hard so I need to boil it down in easy to dissolve chunks. Here's a unique way I found to explain/understand something that not many people have discussed." As an analogy, I did a project on Haar measures and when showing that they exist, there is one object that seems rather.. unmotivated.. called the Haar covering number. No one ever describes what it is. I thought about it and thought about it for a while and realized it was.. | |
| Jul 31, 2015 at 18:13 | comment | added | BCLC | @CameronWilliams That sounds not only possible but also great! Could it be something like introducing topics in say advanced X to elementary X? Like introducing, maybe heuristically or maybe rigorously, some topic often skipped in elementary X (maybe calculus?) but discussed in advanced X (real analysis maybe?) ? | |
| Jul 30, 2015 at 22:49 | comment | added | Alexander Woo | @JackBauer - see the link that OswaldVeblen provided. All those papers were written by undergrads. Personally I coauthored an REU paper as an undergraduate, and my undergraduate thesis also had original results in graph theory, but I went into computer programming for a couple years and didn't publish before those results ended up (completely independently) as part of someone else's PhD dissertation. If you want details, e-mail me; I'm using my real name and can be easily found by Google. | |
| Jul 30, 2015 at 16:22 | comment | added | Cameron Williams | @JackBauer By surveys I mean they're expository rehashes of some topic in math. Often it isn't easy math by any means but it may be a topic that hasn't very many elementary introductions. The survey might act as a basic introduction for people looking to get into the topic. Often novices to a topic provide a very different and unique insight and can deconstruct ideas in simple ways (because they have to in order to understand it), so it provides a gentle exposition. | |
| Jul 30, 2015 at 16:18 | comment | added | BCLC | @AlexanderWoo Combinatorics? Elaborate further please. I cannot imagine any area of pure mathematics that is regularly contributed to by undergraduate research. | |
| Jul 30, 2015 at 16:16 | comment | added | BCLC | Thanks Cameron. Please expand on "As such, the theses are more like surveys of a specialized topic in mathematics." What do you mean by surveys? The heck they are discussing is precisely what I am wondering. | |
| Jul 30, 2015 at 0:14 | comment | added | Oswald Veblen | I do agree that most math papers are not of world-changing importance. But I think it is too much to hold undergraduate research to a hypothetical high standard that most papers by professional researchers don't meet. If an undergrad is author or co-author on a paper in a journal that a colleague also publishes in, I think we have to count that as genuine research by an undergrad. Random unpublished "senior project" type work is something else entirely, of course. | |
| Jul 30, 2015 at 0:09 | comment | added | Alexander Woo | Let's face it - most math papers, my own included, are pretty much uninteresting to anyone. They're written at least as much for the ancillary side benefits of research activity to education as for any importance in what they contain. I stand by my statement, even for most (not all!) papers produced at Duluth. (BTW, there is a lot of undergraduate research available in metric geometry - I suppose you might consider that combinatorics rather than geometry though.) | |
| Jul 29, 2015 at 22:31 | comment | added | Oswald Veblen | @Alexander Woo - I think it is important to distinguish between undergraduates working alone (who are indeed unlikely to produce much publishable work) versus undergraduates working in collaborations with faculty. For example, the well-known Duluth REU run by Gallian states they have over 200 published papers, in professional journals. These papers seem to be no more likely to be "uninteresting to anyone" than all the other papers in those journals :) See d.umn.edu/~jgallian/progbib.html | |
| Jul 29, 2015 at 22:28 | comment | added | Cameron Williams | @AlexanderWoo I was definitely thinking mostly of analysis, algebra, geometry, and set theory when considering pure math, but combinatorics definitely lends itself to undergraduate contribution. The others are almost impossible to get into as an undergrad unless you're effectively at a PhD level already but that is a very small portion of students. | |
| Jul 29, 2015 at 22:24 | comment | added | pjs36 | I fully agree with @AlexanderWoo (and, perhaps counter-intuitively, Cameron's Answer): I think undergrads can definitely do bona fide research, in combinatorics if nowhere else. But, it is probably is likely that most undergrads don't do original research. | |
| Jul 29, 2015 at 22:15 | comment | added | Alexander Woo | I think a large part of the difference here is subfield. It is very rare for an undergraduate to make a substantial contribution anywhere, or any contribution to a subfield requiring a large amount of background. On the other hand, it's not so unusual for undergraduates to be able to prove new results in many areas of combinatorics, even if these results are unlikely to be interesting to anyone except other undergraduates working on follow-up projects. | |
| Jul 29, 2015 at 21:09 | comment | added | paul garrett | Bingo. Exactly. Further, I think it is bad to promote the mythology that "undergrads can do meaningful research in mathematics" if only because it sets of unrealistic expectations, so that "everyone fails". That is, it does not help anyone to "assure" them that "they can do research while undergraduates", because most likely they will not, and this is not failure. And so on. For that matter, many graduate students misunderstand the degree of "originality/creativity" that will actually play a role in their thesis, since the bulk of the work is assimilation of known techniques... | |
| Jul 29, 2015 at 19:33 | history | edited | Cameron Williams | CC BY-SA 3.0 |
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| Jul 29, 2015 at 19:27 | history | answered | Cameron Williams | CC BY-SA 3.0 |