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What do undergraduate students in mathematics do for their thesis, if they have done one, besides expository or applied math?

I was thinking that the kind of research they do is something applied, say using math in social sciences or a problem in one of the less rigorous natural sciences, or discussing such a problem (that's what expository is, right?).

To me it seems something non-expository or non-applied is an original contribution to mathematics, something that PhD students do.

I attended some pure math undergraduate thesis presentations. I was quite surprised: Did they prove anything new? Never bothered to ask due to fear of looking stupid. Would it be out of the ordinary to expect an undergraduate proves something new? If they did not prove anything new, what the heck are they talking about?

It seems like if it's not new, they are giving a lecture. If it's new, that seems like a PhD-level accomplishment.

I mean, do math undergraduates frequently prove new things?

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    "Anything new" is rather broad. I myself proved "something new" in by bachelors thesis, in the sense that nobody answered that particular question rigorously before. Was it deep? Probably not. Could I have published it? I don't think so. Still, it was new. – Raphael Jul 30 '15 at 8:05
  • Depending on the country and the quality of the teaching, yes it is possible. If you have a Professor who gives you an actual problem knowing you have been taught the right modules/topics to investigate it, yes. If you have little teaching and then are told to pick a topic (as it happens in some places) then the chance is significantly lower. – DetlevCM Jul 30 '15 at 12:03
  • @DetlevCM I'm guessing there may be 1 for every dozens or hundreds. I was actually wondering about the average batch of undergrad math majors whose thesis is in pure math. My guess is in the first place not many math majors will do pure math in their thesis. So what about those who do? They actually try to prove something? What happens if they cannot prove that particular conjecture in a month after they come up with the proposal after a month? 2 months left in the semester. So what happens? – Jack Bauer Jul 30 '15 at 16:09
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    @JackBauer Yes, that Enigma - I only covered the rotors and left out the switchboard but the switchboard is the trivial part. Heck, I suspect I could write an implementation fairly easily nowadays having gotten better at programming since. (Side note, its not cracking Enigma, its encoding and decoding which is really trivial.) – DetlevCM Jul 30 '15 at 17:05
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    @JackBauer Something like that. Some researchers claimed a property of the things they worked with; it was crucial for their method to work, but they did not provide a proof. (I don't know if they could have.) I filled that gap. My advisor found it, and I had the luck that it was a reasonably scoped task that took mostly undergrad stuff plus some tinkering. (I think he (and I) hoped I'd happen upon a case where they were wrong, but they weren't.) – Raphael Jul 30 '15 at 21:28
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I'm going to disagree with Oswald. In my experience, undergraduate students do not often prove new things in pure math. I wouldn't even say master's theses often contain new results. There are a few main reasons for this.

Firstly, pure mathematics operates at a level that is not very accessible for most undergraduates, even those doing research. Undergraduates doing research are often well out of their depth and holding on for dear life. This can mostly be attributed to just not having enough time to get up to speed with what is considered modern mathematics. Most courses in mathematics at the undergraduate level are about math from 50-100 years ago (if not older).

Secondly, undergraduates do not often have the mathematical experience to know what the right plan of attack is when faced with an abstract and new problem and they may not know how to check their work thoroughly to make sure there are no major oversights or blunders. A lot of mathematics involves lateral thinking and it takes a lot of time to build those connections. The hardest part of a pure math PhD (in my opinion) is learning how to attack a problem no one has considered before. Standard techniques that others used may not be useful at all to you for one reason or another. An undergraduate won't have the creativity to navigate this kind of issue because the kind of creativity that is needed comes with a lot of experience. Even when an undergraduate student thinks they've proved something, the nuances of their argument likely will not be apparent to them. (This is especially true when it comes to functional analytic/measure theoretic arguments - the devil is in the details.) Thus a proposed proof may not even be close to being right.

Lastly, not many undergraduates in pure math do research because the gap they have to overcome between coursework and modern mathematics is pretty substantial. Those that make contributions in pure math are those that are very, very talented and have very thorough backgrounds (backgrounds that rival master's/PhD students).

Undergraduates in pure math are not expected to make contributions. That is not what research is about for them. Introducing an undergraduate to research serves a couple of different purposes: it introduces them to more advanced topics and it gives them a taste of what research is like so that they can make an informed decision about whether or not graduate school is right for them. As such, the theses are more like surveys of a specialized topic in mathematics. There is a lot of independent learning involved and there may be some unique examples, insights, and connections contained therein. They may not be presenting "original" work, but poster sessions are there to present what they've learned regardless of whether or not it was original. So yes, it is kind of like a lecture. They are undergraduates and far from being experts in their field.

Note that I am not saying that no undergraduate ever produces new results in pure math (there are some high school students that are better than most PhDs), but it is not a common occurrence and is not expected or considered the norm.

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    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... – paul garrett Jul 29 '15 at 21:09
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    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. – Alexander Woo Jul 29 '15 at 22:15
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    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. – pjs36 Jul 29 '15 at 22:24
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    @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 – Oswald Veblen Jul 29 '15 at 22:31
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    @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. – Alexander Woo Jul 30 '15 at 22:49
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The answers so far contain a yes and a no, so let me add a yes-and-no.

Undergraduates can - and often do - prove new things, but hardly ever anything of importance. It is up to the advisor to find an interesting question which is simple enough to serve as the topic of a thesis, but not yet dealt with in the literature. Different from a Ph.D., a bachelor or master thesis is heavily constrained in time, so as an advisor you should only give a topic if you are pretty certain that something can be done by an unexperienced researcher in short time. On the other hand just repeating the literature is boring for the student. One way to find good topics is to look at what is often referred to as folklore: Every textbook contains the theorem that X implies Y, and every expert knows that quasi-X already suffices, but noone bothered to write it up. This will most probably not be worth a publication, but proving a theorem not yet contained in the literature is motivating. Another simple method is looking at all the things you excluded from your own papers. If you worked out an example, but did not include it in a publication, you can let the student generalize it.

What you should not do is ask a student a problem you are really interested in. First the student will be frustrated, because the problem is too hard for him, then you will be frustrated, because you will spend much more time explaining things to him then you would need to find the results for yourself, and finally everyone is frustrated, because you find an answer and have to explain it to the student.

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    "Every textbook contains the theorem that X implies Y, and every expert knows that quasi-X already suffices, but noone bothered to write it up." Are there a lot of things like that just lying around? For example? – Jack Bauer Jul 30 '15 at 16:03
  • If someone proves a result, which only serves as a tool, the conditions are quite often too restrictive. For example, Hilbert space is used where reflexive Banach space suffices, or compact can often be replaced by countably compact. In number theory you can look at older paper using exponential sums, and see what improvements for the latter yield in the application. – Jan-Christoph Schlage-Puchta Jul 31 '15 at 16:59
  • Jan-Christoph Schlage-Puchta, "the conditions are quite often too restrictive", do you mean it wouldn't be of interest to many mathematicians anyway? – Jack Bauer Aug 5 '15 at 12:04
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    When talking about topics for a Bachelor or Master thesis, I think about problems which are open in the sense that they are not published, but solved in the sense that every expert in the area could immediately write down a proof. So I don't think these questions are interesting to other people. – Jan-Christoph Schlage-Puchta Aug 10 '15 at 7:51
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    @JackBauer Most of the work is in figuring out what these things are. If I had an example which I knew well enough to cite it here, probably somebody would have proved it. – Ben Webster Aug 11 '15 at 17:17
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Yes, undergraduates frequently prove new things, in the sense that every year there are new, publishable results proved by undergraduates. So, although a relatively small number of undergraduate math students participate in true "research", there are certainly students who are able to make nontrivial discoveries as undergraduates, and more than one might initially think. I have been at prestigious research schools and at anti-prestigious regional universites in the U.S.A. At every school I have been, there were undergraduates in mathematics with the aptitude for publishable research. The talent needed may not be "common", but it is certainly not "rare". The obstacles are primarily cultural, not intellectual.

The topic of undergraduate research has also been the subject of a question on MathOverflow, which makes for good reading.

For an example from personal experience: I recently published a peer-reviewed paper in what I consider to be a high-quality journal (and which is not in any way a "student" journal), with an undergraduate student co-author, who discovered the proof of one of the main theorems on his own between two of our research meetings.

Another example is the journal Involve, which is devoted to genuine student research. From their self-description:

Involve showcases and encourages high-quality mathematical research involving students from all academic levels. The editorial board consists of mathematical scientists committed to nurturing student participation in research.

Submissions in all mathematical areas are encouraged. All manuscripts accepted for publication in Involve are considered publishable in quality journals in their respective fields, and include a minimum of one-third student authorship. Submissions should include substantial faculty input; faculty co-authorship is strongly encouraged. In most cases, the submission (and accompanying cover letter) should come from a faculty member.

Involve, bridging the gap between the extremes of purely undergraduate-research journals and mainstream research journals, provides a venue to mathematicians wishing to encourage the creative involvement of students.

One thing that undergraduates are unlikely to have is the breadth of knowledge that is expected for PhD recipients. Particularly in mathematics, PhD students are examined in a range of subjects, and are expected to have mastered large parts of the undergraduate curriculum. Undergraduate research often involves learning enough about one particular area to prove new theorems. The student still needs to spend time learning other areas to have the knowledge expected of a PhD.

The real key for undergraduates who are looking to do publishable research is to find a collaboration with a good faculty mentor. Independent research by undergraduates is indeed quite rare (in fact, the majority of mathematics papers currently published have two or more authors - even experts benefit from collaboration). The MathOverflow thread linked above has more advice from other mathematicians.

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    Thanks Oswald. Your example is kind of strange. WOuld your undergraduate co-author even have the opportunity to do such if not for knowing you? – Jack Bauer Jul 29 '15 at 18:41
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    Perhaps I should have said: I heard a PhD is like an original contribution or something. Doesn't proving something new kind of amount to an original contribution? Again, I understand this may seem stupid. – Jack Bauer Jul 29 '15 at 18:42
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    I wouldn't say undergraduates frequently prove new things, especially not in pure math. A small number of math undergraduates do serious research and even fewer make major contributions to the work. Most undergraduates hardly have the mathematical chops and insight to make major contributions simply due to lack of enough exposure. – Cameron Williams Jul 29 '15 at 19:06
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    I respectfully disagree. I'm saying that you're way over-inflating how successful undergraduate students are and my guess is that it's because you've worked with some very successful ones. My point is that on average, so very few that actually do research make contributions. Hell, successful PhD students maybe end up with only one or two papers by the time they're finished. – Cameron Williams Jul 29 '15 at 19:47
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    So far, I've worked with three (sets of) students at a non-selective school, resulting in three peer-reviewed papers that, in their journals, are indistinguishable from any other research. The students all met the usual standards for co-authorship. (This record is partially because, as a researcher, I know enough to pick math problems where we are likely to find publishable contributions.) When I was at prestigious research schools, I saw even more math majors who would have been able to work on publishable research as undergrads. As I wrote, the issue is much more culture than aptitude. – Oswald Veblen Jul 29 '15 at 21:51
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I can tell you my experience as I am currently writing an undergraduate thesis (though as a summer project).

I am an undergraduate student in mathematics currently doing a summer « introduction to research » internship. I'm studying probability theory.

As a first year student, about half of my time was spent solidifying my mathematical background in probability, measure theory and analysis. I also spent quite a lot of time studying specialised articles, and finally I applied the general theory I studied to a specific problem, where I did prove something « new », while very closely following other published results. On the way there, I also proved a few lemmas, that, while not of general interest, are « new » and interesting to me.

Clearly, undergraduate students are not expected to find groundbreaking results of general interest. However, they can contribute to mathematics by summarising and gathering related results from multiple articles, applying new theories, finding examples, etc.

A word of advice.

You should not aim for great discoveries, but rather simply try to do your own mathematics. Ask yourself a lot of « stupid » questions and find their answers. That's how you'll end up with a few small new results. Make sure you can grasp the big picture of your field of study, that you look at it from a critical standpoint and that you understand the issues that motivate it.

Do math undergraduates frequently prove new things?

Yes. But not great things, and sometimes things that might already be known to experts (but not widely accessible). I think that it is good enough for an undergrad to prove things that are new to him/her and her classmates/advisor/etc.

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Some of the implicit premises of these sorts of questions, or the implicit premises in responses to the question, are really the issue. I would heartily agree that undergrads of all "calibers" should "be in the room" when something resembling "live" mathematics is being discussed. But/and this is most meaningful when we look at the falseness, artificiality, and sterility of the typical undergrad curriculum: it's fake and moribund, with no immediate room for anyone to do anything at all, and no hints about reality, either. Ghastly, yes. But that does not immediately entail a sort of "opposite", that novices need know very little to make meaningful contributions. Raw cleverness has already been exercised, quite systematically, for some hundreds of years (thousands?). People have learned useful things, and to not know these is to not know how to change a tire, or a light bulb, or a furnace filter, or open the door. Not that the usual curriculum helps much, either, I agree! But that does not mean that basic operational skills (involving occasionally subtle mathematics, literally, here) are irrelevant. Getting outside the degenerate "school math" thang is excellent... but thinking that that means "we don't need to know anything!" is obviously silly... even if appealing. "Complicated".

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