Is there research for mathematicians?

[Naranja]

I am not a math or science person, but I'm trying to learn so please forgive my lack of knowledge.

Is there research in math? Are there boundaries to math that can be pushed out? Studying the universe, it's obvious that we don't know everything yet. There is plenty to discover for physicists. But is there more to discover for mathematics? If a large donation was made to a university mathematics department endowment, would it have less of a positive impact than giving it to physics?

I read in the press about a mathematician solving an old problem every once in a while, but I never hear of one inventing a new type of math. Has all the math we need been discovered or invented?

 

[pwsnafu]

Let me put it this way. Universities have a research paper quota that academic staff must satisfy. If there was nothing left to research there would be no pure math departments. Obviously that is nonsense.

but I never hear of one inventing a new type of math

My PhD was in time scales calculus, an area which did not exist 30 years ago.

 

[homeomorphic]

It's really more like the opposite situation where there's so much that there's not that much point to it because there's too much for anyone to be able to manage and keep track of it very well. Anything new tends to be like a drop in the bucket. Usually, when they solve old problems, they have to invent a lot of new math in order to do it. I think there are something like 10,000 math papers published every year. So, actually, I think there is a rather severe over-supply of new math being discovered/invented. Maybe more quality over quantity would be good. Even Leibnitz complained that there were too many papers 300 years ago, and the problem was nowhere near as bad then. Part of the reason I lost interest in math research after my PhD was that there was so much math being done that I don't see the point to adding any more to it. I feel like it just makes the mess bigger to add more to it. I'd rather spend my time trying to make the existing math a little more comprehensible.

The thing about math research is it's very cheap compared to a lot of other research. We don't have to build billion dollar particle accelerators, etc. So, in some ways, you get a lot of bang for your buck when you invest in math, but on the other hand, I think the vast majority of the math that is being done will have little or no practical application in the foreseeable future. A lot of it is pursued only for its own sake.

Rather than throwing money at it, I think we just need to revive the mathematical culture. It seems pretty dysfunctional to me right now, and I'm not alone in that opinion.

 

[MathematicalPhysicist]

How exactly do you want to revive it when professional mathematicians have the the "publish or perish" monkey on their backs?

 

[homeomorphic]

How exactly do you want to revive it when professional mathematicians have the the "publish or perish" monkey on their backs?

Beats me. There's a lot of stuff you can do, even with publish or perish. There's no need for people to give those silly incomprehensible talks directed at the whole math department, when only a few specialists can understand them. People can just knock it off with that, if they choose to. I felt like I was really exercising my non-conformist muscles when I gave talks that actually made sense, but it shouldn't be that way. I had first year grad students, not just specialists in my area, tell me they understood my talks--what a concept! But if you are going to fit in, you have to make it a rapid-fire overview of the subject that probably isn't the best use of everyone's time.

 

[MathematicalPhysicist]

Beats me. There's a lot of stuff you can do, even with publish or perish. There's no need for people to give those silly incomprehensible talks directed at the whole math department, when only a few specialists can understand them. People can just knock it off with that, if they choose to. I felt like I was really exercising my non-conformist muscles when I gave talks that actually made sense, but it shouldn't be that way. I had first year grad students, not just specialists in my area, tell me they understood my talks--what a concept! But if you are going to fit in, you have to make it a rapid-fire overview of the subject that probably isn't the best use of everyone's time.

I had several talks which I gave in my Msc (in seminars and courses).

Some talks were just big mambling (also there was a course (which me and another one attended) where the lecturer started literally saying "bla bla bla..." which made me laugh out loud), usually the talks that were in analysis were more discernible, the talks in topology and dynamics were big mambo jumbo.

The problem when you deliver a lecture you need to knw what your crowd already knows and not to repeat it, since if you do, there's not enough time to get to the point.

The only really good way to learn stuff is by your own with a good text or talking in forums like this where you are judged by your reasoning alone and not by your prestige as it seems to be in this community or every human community. in this regard I see the WWW as the savior for math research. The problem is that you still need to make living, even if you still want to make everything in math crystal clear. I don't believe that in academic circles it's possible, you really need to be a maverick to do it.

 

[homeomorphic]

The problem when you deliver a lecture you need to knw what your crowd already knows and not to repeat it, since if you do, there's not enough time to get to the point.

Yes, you need to know the crowd, but I'm not so sure that it's that important to get to the "point", unless it's a very specialized seminar. I think it's more important for the audience to get the best use of their time. I think it's better to err more on the side of repeating things than to lose people. And I wish people would focus more on the motivation and why they think their subject or results are important, rather the results themselves. I think I reached the point where I qualify as meeting the lowest common denominator of math knowledge for colloquium purposes, but most of the talks were still not nearly as enlightening as they could have been.

Talks are a small part of math, but they are one thing that can be improved fairly easily, I think.

 

[SteamKing]

Is there research in math? Are there boundaries to math that can be pushed out? Studying the universe, it's obvious that we don't know everything yet. There is plenty to discover for physicists. But is there more to discover for mathematics? If a large donation was made to a university mathematics department endowment, would it have less of a positive impact than giving it to physics?

The most highly publicized math research result of recent times was the announcement that Andrew Wiles had proven Fermat's Last Theorem:

http://en.wikipedia.org/wiki/Wiles'_proof_of_Fermat's_Last_Theorem

The effort reportedly took Wiles seven years to complete, and the proof itself is over 150 pages long. Even after its initial publication, other mathematicians examining the proof found a significant flaw, which required some additional work to correct.

Obviously, a proof of this length is not something one happens to stumble upon while in the shower. It took a lot of cogitating and some extremely abstract mathematics for Wiles to develop his proof.

Is there a proof which is so short that it can be written in the margin of a book, as Fermat himself reportedly claimed? That is still a tantalizing mystery, but such a proof appears unlikely to be discovered or devised, even after more than 300 years of effort by countless mathematicians.

 

[AlephZero]

The effort reportedly took Wiles seven years to complete, and the proof itself is over 150 pages long.

That's tiny, compared with the classification of finite simple groups:

The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups

 

[rasmhop]

New types of math are invented all the time, but they are usually sub areas of the very broad areas you know about. No one just works with algebra or founds a new field of that magnitude anymore, that sort of low hanging fruit has for the most part been picked. Many of these fields are so technical and advanced that to even understand what they deal with can be hard for a layperson. Let me give you some examples related to my own work:

• Scheme based algebraic geometry was founded in the 1960's (by Grothendieck and his collaborators like Serre and Deligne) and completely revolutionized the field and many related fields like algebraic topology and algebraic number theory.
• Motivic homotopy theory was founded in about 1998 (by Voevodsky). Very simply put it tries to study the objects of algebraic geometry, but while considering objects as the same if they can be bended or stretched into each other (think of how you might transform a circle into a square, the technical term is homotopy).
• Derived algebraic geometry was founded in 2000 (by Jacob Lurie). Lurie takes Grothendieck's approach of defining the fundamental objects of algebraic geometry as a pasting of rings and generalizes it to a context where we allow stretched and bended versions of rings.
• Chromatic homotopy theory grew out of work with Morava K-theory, formal groups and stable homotopy theory. I don't think there was a defining paper, but it seems to have emerged as a distinct area of study somewhere around 2007-2008. Basically it is a way to look at a very hard problem (currently widely considered practically impossible to ever fully understand) and break it into easier chunks which we can deal with, but which only tell us part of the story. The chromatic level 0 is easy to deal with (it is just ordinary cohomology), chromatic level 1 is hard but we have tools to deal with it (complex K-theory basically represents chromatic level 1), chromatic level 2 is very hard but there is a lot of focus on trying to understand elliptic cohomology which basically represents this level, chromatic levels above 2 are currently close to impossible to say much about due to how complex they are, and what we really want is chromatic level infinity where complex cobordism lives. This and the previous topic are very hot topics these days, and in particular in the Boston area (Harvard + MIT) there is heavy interest in them.
• Highly structured ring spectra (aka Brave New Algebra). IIRC it was Peter May and Mike Mandelll who first studied this via S-modules, later other versions like orthogonal spectra and symmetric spectra emerged. I don't recall when this was published, but sometimes in the late 1990's I think. The nuances and what the field really is about is still being worked out. The basics idea is that we can replace the notion of a space by certain structured objects called spectra which in some nice ways allow for a coherently associative multiplication among other nice properties. It is both an active area of research and a fundamental tool for every algebraic topologist or geometer. It was basically thought impossible for such a theory to be possible, but it turned out to be possible and very useful.
• Global equivariant stable homotopy theory. I believe it was first introduced in 1986 with Lewis-May-Steinberger's comprehensive equivariant stable homotopy theory book, but has seen little serious work until some people, most notably Stefan Schwede, started looking at it seriously around 2012 with the machinery of orthogonal spectra. We basically llook at traditional stable homotopy theory, but introduce actions for all compact Lie groups at once in a compatible manner.
• The theory of derivators was first suggested by Grothendieck in an unpuplished manuscript (pursuing stacks) in 1983. Due to Grothendieck leaving the mathematical community and cutting off all contact about this time, not much came of it. Some people are picking it up these days, in particular Moritz Groth (who is a postdoc) made a nice expository article on derivators which seems to have really encouraged others to look into this area again. Basically it is a different way to approach some algebraic topics, and in relation to the previous fields I mentioned it is very small, but that does not mean it doesn't exist.
• Topological modular forms and elliptic cohomology theory was to a large extent founded by Edward Witten in the late 1980's (apparently they have some relation to string theory, but I'm not a physicist so I don't know about that aspect of it). It has received much renewed interest in the last 5-10 years though, where the tools of highly structured ring spectra, chromatic homotopy theory, and others are available to better put these ideas into their right context and to deal with them more effectively.
• Higher category theory. I don't know if there is really a point at which this was founded, but over the last 20 or so years this field has really come into its own. We still do not really seem to know what exactly it is, but we have a good idea and many smart people are working on trying to tackle higher categories seriously.
• Goodwillie calculus is basically about trying to understand what linearity of a map means, when space and map are taken to be extremely general objects (like stable (infty,1)-categories). It was founded by Goodwillie shortly before 1990. I wasn't around then so I don't know what people thought about it, but generally when people first hears about it they can't believe that it is possible and would yield something which can produce useful results, but it turns out to be quite possible.

These are just some examples from the top of my head which have appeared near the intersection of algebraic topology and algebraic geometry. To a large extent it ignores many very significant developments which can not be described as a new type of math.

To get some feel for the volume of math being developed check out arxiv.org where preprints are posted. In 2013 26785 preprints were posted there (in 2014 we are up to 14165 so far). Some of these are small articles, but many are 50+ page articles developing serious new mathematics. It is less than say physics, but only by something like a factor of 2.

EDIT: I should say that I only have a very superficial understanding of certain of the things I have mentioned, so I may have made erroneous or misleading statements, but the underlying points that these are all new areas of mathematics still holds.