I am generally familar with the way academia works. I was in a PhD myself program once. However, we did lab based science and there was real hands-on work to be done every day often around the clock. It was hard enough finding something new in that field but you could keep busy all the time by just learning lab science much like a job but more demanding.
English Ph.D. candidates can always read and compare some obscure authors from the 17th century or something. Computer scientists can program some cutting edge thing that may not be completely new but you can see it. Astronomers can do large scale surveys of some unexplored region of space.
What do math students do for research? It seems pretty clear-cut that the result is some degree of right or wrong so they can’t fake it and their dissertations are short from what I hear? What do they do all day? The world needs people good in math as college professors and in high-tech fields but how do they keep them all busy finding brand new stuff? Not all of them can solve Fermat’s Last Theorem. For that matter, what do they do once they get to be a college professor at a 3rd tier university.
IANAM, but math is a subject that is potentially larger than all of literature by orders of magnitude. Not just new problems, but entire new fields of math are discovered all the time. There can be no shortage of problems to do dissertations on, since the entire duration of the universe can’t suffice to hold them.
The problem in fact might be that so many new fields and new problems and new variations exist in such esoteric and unexplored fields that finding enough people familiar enough with the subject to properly comment on it may be an issue.
But I’m positive that finding new stuff has never been a problem in math and will never be.
First link for math dissertations takes me to OSU, which maintains a listing.
Just like lab scientists, they spend their time learning the tools of their field. Those tools happen to be a little less concrete, but the process is otherwise identical. As for problems, they get those from our advisors like anybody else.
(For the record, computer scientists are much more like mathematicians than programmers. The goal of research is almost never a program.)
Of course, both computer scientists and mathematicians (especially applied mathematicians) often write computer programs as part of their research. But the program isn’t the whole goal of the research, the original theory is.
Just about every PhD advisor has a stack of two or three or four dozen research ideas that they wish they could devote more time to, if only they had another couple dozen more lifetimes. The problem isn’t coming up with ideas, the problem is securing funding and time.
Finding a topic for your dissertation was never a problem. The problem was figuring out which ones were doable in a reasonable time frame before you give up. There are always unsolved questions in every branch of math. The math faculty knows these questions and mostly likely have tried to answer them themselves, but lacked the time or energy. For a Ph.D. candidate whose final hurdle is the dissertation, the difficulty was not in finding a topic for the dissertation, but to find one that’s doable in a reasonable time frame along with an advisor who sets reasonable goals that can be achieved in order to graduate.
I think the OP should be qualified as “math that is not applied math”. Because the applied math bit is little different from computer science - you pick a scientific or technical problem and you write computational programs for it.
The bit about what math profs in 3rd tier universities do is funny. Not all work being done by profs in the more practical fields in higher tier universities is real exciting to read about either…
As a CS graduate supervisor, I would be stronger than that. The dissertation will typically only talk about results from any program written. In most cases the code is little more than a tool, no more interesting than an experimental scientist’s lab set up.
There are dissertations where new paradigms are investigated and implementation techniques are described (my thesis was a lot like that, as it delved down into operating system related areas) but the majority don’t. It is common to see theses that have not a single line of program code in them. Mine had one line. I still rather wish I had removed it
Computer science does sit on a funny divide between experimental and mathematical sciences. And work progresses from one extreme to another. As a science CS is a funny thing. For the most part it is close to impossible to make it fit the traditional Popperian mould of science. Many times it is more engineering than science. Mathematics doesn’t fit the Popperian mould either. Indeed many would posit that mathematics simply isn’t a science, it is what it is: mathematics.
Pretty much any dissertation needs to demonstrate two things, that the student has a grasp of what has gone before, and that they have made a contribution to the state of the art.
well, the “goal” is to solve the problem at hand. The programs and the math behind them are the tangible output of the process.
In any event, I think that applied math should be a much more understandable case in light of the OP. Given that Shagnasty is a programmer, “I want to make a faster running Simplex variant in so-and-so case than what has been achieved previously” should sound like a much more natural thing to consider than any of the upthread-mentioned problems in pure mathematics.
The real fun field to classify is AI. You want to do natural language processing? Congrats on becoming a pseudo-linguist! Natural language learning? Add Cognitive Development to your resume! That goal-based language ain’t doing what you want? A language design/compiler theorist is you! Want to do inferences and logic? Hope you like statistics! You want your computer to learn through observation how to play a game? Silly researcher, you need an environment to learn in, and a fun one so people will willingly teach your bot! A game designer is you!
Okay, so this applies to almost any research field, but since AI attempts to produce some arbitrary level of “intelligence” which encompasses so many complex things to work, it seems like it should be called a PhD in Jack of All Trades.
I have a PhD in theoretical CS. I did not have to do any programming at all during my PhD study, and could have got away with doing none. I did choose to write a program as a proof-of-concept (it took about a week and a half to hack together in Haskell) to put in my dissertation, but this was only because I really couldn’t be bothered writing out another chapter of theorems, definitions and examples. Writing about a program you’ve hacked together is much easier, and by that point I had enough material to write a thesis, but couldn’t be bothered starting. So I programmed for a bit.
My dissertation fell roughly half into applied mathematics and half into pure. I had one chapter on a new unification algorithm, proved it correct, and so on, that was aimed at addressing problems many programmers face when constructing compilers, or other metaprograms that have to handle abstract syntax with name binding. My other chapters/publications were on context-calculi and proof terms for incomplete derivations in type-theory. These were more theoretical and will probably never be implemented.
Most of this stuff was inspired by current problems in the field (for instance, the incomplete derivation stuff is a real problem for those implementing proof assistants like Coq) but is also of independent mathematical interest. More work was inspired by asking “what-if”, or thinking “wouldn’t this be cool”. I doubt my context-calculus will find any real application any time soon, for instance (though I can think of some applications that will probably never be realised).
The thing about graduate level math is that it is often so abstract that no non-mathematician can really have any idea what we’re doing.
To really dumb down the sort of research I did, I could explain it this way: Think about geometry. Try to consider what weird shapes could have as few “nice” properties as we might desire. Take one of the weirdest and hard to describe shapes that has certain nice properties; are there nicer shapes that have basically the same properties? My research showed that no shape that contains a T-shaped junction in it will have similar properties to something called the pseudo-arc.
But that means nothing to anybody not in the field, really. How do math people come up with new dissertation topics? Well, consider how hard all those graduate math classes are… and most people haven’t ever experienced one, so, that’s not a good place for me to start explaining. Anyway, at this point, math gets a lot more theoretical (for the most part and certainly in my field) than practical. And just as the theory gets weirder and harder, there are always new things to discover.
It mainly involves proofs, not just simple calculations. I know a lot of undergraduates seem to think that math=calculations and are really surprised when they run into a course involving proofs. And, yeah, not every proof is on the level of Fermat’s Last Theorem, in the same way that not every biomedical research paper involves finding the cure for cancer. Smaller results are still results.
What they’re trying to say is that at the highest levels, computer science and mathematics are essentially the same thing, with computer science being concerned with a subset of mathematics that are applicable to computer related things, such as algorithms, data theory, etc…
Both are all about proofs - in CS’s case, about proving where and when the algorithms do and don’t work, how fast they are, and some other things.
Not only that, but mathematicians working in other areas probably don’t know a lot about what you’re doing. I don’t think there’s any person on the planet who could understand all of the dissertation titles that Exapno posted.
That pretty much says it. My advisor had just written a paper and asked me to pursue some subsidiary points. I actually wrote an appendix to his paper, but the referee performed an appendectomy and that was never published. Meantime, I had made a silly conceptual error. He kindly explained to me why I was wrong. But I explained that in one case of his theory, what I assumed would happen did happen. He said that was special (to dimension 2). Then I did it for dimension 3 and, after a long painful computation (that involved squaring a symbolic expression consisting of 24 terms, which has, in principle, 576 terms (the individual terms did not commute). Exceedingly ugly. But that, along with some of the consequences, was my thesis. Five years later I did find a general approach that did it in every dimension.
But the problem does lie in knowing what is feasible. You can ruin a student by giving him a problem he can’t get started on. But there are surprises. One mathematician I knew very well once attempted to discourage a student from working on a problem that the advisor considered intractable. The student solved it and revived a subject that had just about died (finite group theory, if you want to know).
Even for things like the Fermat problem, before it was solved there were many ways of nibbling around the edges, some of which contributed to the final solution.
