Wikipedia:Reference desk/Archives/Mathematics/2013 March 27

= March 27 =

Mathematics research
I know that faculty in research-university mathematics departments are expected to engage in research like other kinds of academics, I know that there are some mathematicians who are primarily researchers, a former roommate who got a math Ph.D. in topology explained somewhat to me about his thesis, and I've even read this page. None of these things, nor all of them put together, helps me understand — what do mathematics researchers typically research? Surely the guys who set out to prove famous theora are rare; are the average researchers simply creating obscure/unique problems and then solving them? As a historian, I can't really relate very well, since inventing new or modified theoretical approaches (do they even exist in mathematics?) to already-known data doesn't seem possible in math as it does in history, and presumably there aren't frequent discoveries of new concepts or data as there are when we discover lost documents in archives or when we analyse recent events. Nyttend (talk) 02:38, 27 March 2013 (UTC)


 * I went to graduate school in math before switching to neuroscience in the late 1980s. One of the things that motivated me to switch was a sense that it was very difficult to find anything to work on that felt important. The largest group of new and important problems, I believe, relate to computers in one way or another -- algorithms, numerical methods, etc. When it comes to pure math, it all seems to have become very obscure and idiosyncratic. A symptom of that is that most papers in math journals are incomprehensible to 99% of other mathematicians. I don't mean the lay public, I mean other professional mathematicians. The main thing that keeps pure mathematicians in business nowadays is that they are needed to teach algebra and calculus to undergraduates. Looie496 (talk) 02:52, 27 March 2013 (UTC)


 * — List of unsolved problems in mathematics — 79.113.209.59 (talk) 03:17, 27 March 2013 (UTC)


 * But the things on that list are notable problems; if all the mathematicians concentrated on them, we'd have far fewer proofs published, since they'd all be working on the same things. These are the problems I meant when by referring to the guys who set out to prove famous theora.  Nyttend (talk) 03:24, 27 March 2013 (UTC)


 * I hope you won't be offended if I point out that the singular is "theorem" (from Greek θεώρημα). There is no Latin word "theorum" to have "theora" as its plural.  Perhaps you were making a joke and I've missed the point?    D b f i r s   16:45, 27 March 2013 (UTC)


 * Just because one does not succede in finally proving or disproving something, does not mean that one cannot discover several interesting facts along the way, some of which may even have practical application. — 79.113.209.59 (talk) 03:28, 27 March 2013 (UTC)


 * Totally sympathise with Looie - I was much the same. First class honours, highly respected by my class mates, even won a couple of prizes, competing against former Olympiad students. But the bottom line is, that took a lot of effort, and when I got right down to it, I wasn't that good. I was never even looked at for the Olympiad when I was in school. To do something exciting, you have to be really good. For my part, I was doing a PhD in robust statistics before switching to IT. The work is essentially a bit obscure, and perhaps not exceptionally relevant - this was my fear, a job having no relevance outside of academia. Statistics is somewhat separate to maths proper, but robust stats is a very good example of what a mathematician might do. You find some class of functions and analyse their behaviour as alternatives to conventional statistics, in terms of dealing with outliers. Imagine looking at a huge, complex set of data, and inspecting it visually, to look for outliers (eg. the outliers could be due to a measuring device that is faulty). If you have a computer to spot the outliers, your job is easier, and if I can work out how to develop algorithms that are more efficient, and better (practically or mathematically, with the emphasis squarely on the practical), I could get a phd. But to put it mildly, it bored the hell out of me. So there are lots of practical problems to invent better solutions for. If you are looking for really heroic stuff done by non-heroic people, check out Classification of finite simple groups. IBE (talk) 08:19, 27 March 2013 (UTC)

Mathematicians solve problems mathematically. Most of the time their methods are known in the literature, but some times it happens that the solution is new, and then it is mathematical research. That has happened to me a couple of times, and it was great fun. I don't know how other mathematical researchers feel, but I doubt that they are bored. Bo Jacoby (talk) 08:55, 27 March 2013 (UTC).


 * I don't know if it sounded wrong when I spoke of my own boredom, but just in case, certainly a lot of PhD maths could be very interesting, but finding the interesting bits that are new is time-consuming. As I found with my honours, when you get some steam up, it gets fascinating, but even with the hons dissertation, this took so long. I would imagine this could happen more with a PhD - there is new stuff to be done, but it takes a while getting into it. IBE (talk) 10:25, 27 March 2013 (UTC)


 * Thats where your PhD supervisor comes in. He or she will know where the problems are. A lot of the task of getting a good PhD is finding the right supervisor. I was lucky in finding one which matched my talents with an emphasis on geometry and computers. Whilst the actual PhD itself my not have been earth shaking it was novel and I found it fascinating at the time. It did lead to other research jobs and gave me many useful skills. --Salix (talk): 11:11, 27 March 2013 (UTC)


 * You say "as a historian". Does that mean you work at a college or university? If so, there are probably dozens of research mathematicians within a mile of you right now! Ask one for a brief chat over coffee, or attend one of their symposia if you want to learn about how mathematical research works, and what types of problems they are working on. Off the top of my head, I'll say that much of pure mathematics (by volume) is incremental work. Most "big problems" have smaller facets that can be tackled by "normal" mathematicians. Often, "big" results are actually just consequences of many smaller lemmas that have been proved (as in the finite simple groups example listed above). Sure, mathematics is increasingly specialized, but that is true of most modern academic practice. I don't think math is any more compartmentalized or mutually unintelligible than physics or chemistry. As for "new techniques", remember, many "results" in math actually provide new approaches to other problems. So new angles of attack become available all the time. In my opinion, it is certainly not true that the "main thing that keeps pure mathematicians in business nowadays is that they are needed to teach algebra and calculus to undergraduates" --Sure, teaching undergrads is part of the job (and is a large source of income for the department), but again, that is true for e.g. cosmologists and biologists as well. At least mathematicians have some way to earn their keep, in that there is massive demand for undergrad math instruction. Many a humanities grad student has envied the plentiful TA-ships available for math grad students! Also, there are many types of research that are not funded by market forces (e.g. history), so it seems weird to try to single math out in this way. Finally, some opinions terminology: there is very little "data" in pure math. We have assumptions, axioms, relations, lemmas theorems etc., but these don't really fit the definition of data (if you think they do, that's fine, but that's not how mathematicians talk about it). And I'm pretty sure "theora" is rarely (if ever) used as a plural of "theorem" by working mathematicians. Consider "Sylow theorems" with ~57k Ghits, and "Sylow theora", which has zero. (and would it even decline that way? It's from "theorema", which seems like a standard first-declension feminine to me...) SemanticMantis (talk) 16:47, 27 March 2013 (UTC)


 * If you really want to use Late Latin words instead of English, then the correct plural is "thoremata" "theoremata".   D b f i r s   16:57, 27 March 2013 (UTC)


 * Dbfirs, isn't it "theoremata" ? Bo Jacoby (talk) 20:30, 27 March 2013 (UTC).


 * Oops! Silly typo.  Corrected now. Those who live in glass houses shouldn't throw stones!    D b f i r s   21:21, 27 March 2013 (UTC)


 * And those who live in grass houses shouldn't stow thrones. StuRat (talk) 21:31, 27 March 2013 (UTC)


 * And if we're being really correct, we should probably fess up to our heritage and say it's Greek (not that it wasn't also LL), and third declension neuter. IBE (talk) 22:56, 27 March 2013 (UTC)


 * A Good Lemma is Worth a Thousand Theorems Count Iblis (talk) 23:30, 27 March 2013 (UTC)