I don’t think you RC. Unless you’re defining “group theory” to be some more narrowly focused subset of all the math research topics typically classified as various flavors of group theory.
Or unless you’re telling us that you know the solution to the Andrews-Curtis conjecture, the second Burnside problem, the amenability problem for Thompson group F, and a whole bunch of other unsolved problems in group theory. In which case, damn, dude, you really need to get those papers submitted.
In some areas of science progress will slow for sad reasons. For example, knowledge of the wide range of variation in human languages is increasingly unrecoverable: Nowadays, over 3% of the world’s languages go extinct every decade.
Upthread the human eye was mentioned as something not fully understood. What about magnetoreception? It's widely thought that the magnetic compass many birds possess resides in their eyes. If some humans have, as some suspect they do, a magnetic sense it probably resides in their eyes also.
IIUC, human eyes have significant quantities of cryptochromes with unknown function. Those chemicals are related to, but distinct from, the cones' sensory pigments, and are implicated in magnetoreception.I can guarantee that mathematicians will never say that, either for math generally or for any field with math, as long as somebody is willing to pay mathematician’s salaries. I do believe, speaking as someone with an M.S. in math, that there are many sub-areas within the field where the useful research has basically been completed, and much of what’s being done now is just frivolous noodling with no real-world applications.
Granted, some mathematicians are doing useful work. Computational and applied math certainly produces useful results.
But to take graph theory, which is what I intended to do my Ph.D. on until I quite. When graph theorists are asked what are research is good for, we are expected to say that it applies to computer networks, as any network has various components (vertices) and links between components (edges), and thus a graph consisting of vertices and edges can represent a computer network. But many graphs, and many research problems about graphs, simply don’t correspond to any computer network or anything else that exists in the real world. There are certain subjects that are known to be “cancers”, i.e. they lead mathematicians to write hundreds or thousands of peer-reviewed papers that may all be true, but are mostly useless. Graceful graphs and the snake-in-a-box problem are two examples.
I think he’s referring to the classification of finite simple groups. It was a major achievement to finish this.
Though even at that, it’s not exactly “done.” The proof is tens of thousands of pages, and mathematicians are working on reducing this (it will still take 5,000 pages).
Group theory is full of open problems, even if you just restrict to pure group theory (as opposed to areas like representation theory, Galois theory, etc.). Just in the areas I’m personally familiar with, there are a bunch of open problems regarding growth rates in f.g. groups, variants on the word problems in hyperbolic groups, and the notorious Andrews-Curtis and Kaplansky conjectures. Maybe you’re thinking of the classification of finite simple groups in particular, which is pretty much done at this point. (I had a professor in grad school who was a prominent figure in that program. Despite not being that old yet, he stopped taking on new students because he thought there wasn’t anything left to do in his field. Probably for the best, given the state of the job market in academic math.)
Yeah, you’re right - that’s what I was thinking of. :smack:
I don’t disagree; but pure mathematicians often don’t know and sometimes don’t care whether what they are studying will ever have real-world applications. In pure math, that’s not what makes the difference between “important” or “interesting” or “useful” lines of research vs. “frivolous noodling.”
Arithmetic solved? Pffft, I saw, pffft!
Just read the Wikipedia article on Fürer’s algorithm (2007) for integer multiplication. If you want to multiply really large numbers* you don’t use the grade school method.
While Fürer’s is not really practical compared to Schönhage–Strassen (1971), people have been tweaking it. The article cites a paper from way back in 2018. (I remember that year like it was yesterday.)
History of math. They have sliced and diced the history of calculus as fine as it is going to go. When I pointed out 30 years ago that the founders of my specialty were still alive, but getting on in years and should be interviewed, they were not interested and those founders have since died. The second generation is now dying off. Several have died earlier and two died last summer and there are only three of us left. As far as I am concerned, the history of math has hit a dead end and should be deep sixed.
I was going to link to the Mantis shrimp and found
Nature’s Most Amazing Eyes Just Got A Bit Weirder.
I can’t really think of anything that addresses the OP though.
I wonder about things like that too. When I took the Classics in college, the sylabus and lesson plans were the same mimeographed sheets the teacher was likely using … well, 20 years prior to 20 years ago when I took it, you know, when people mimeographed instead of photocopying things.
Seriously, the Aneid, the Odyssey? What’s left to learn? Don’t tell me I’m dumping on archeology and linguistics, yes there’s still discoveries to be made. But what is known, what we have copies of, once translated into a Modern language – how many more conclusions need scholarly study?
Um, how is this unfortunate loss of information in history of mathematics due to insufficient oral-history research an indication that “there is nothing left to discover” in the subject? I would think it would indicate exactly the contrary. That is, there’s much more left to discover, e.g., on the history of recent mathematics, and more research needs to be done to discover it. This is true in a vast number of areas in history of mathematics, especially in non-European branches of the subject.
Even the history of calculus is by no means thoroughly known. There are dozens if not hundreds of early modern works on calculus that haven’t been studied yet by historians. Sure, the broad outlines of the major developments in history of calculus such as we teach in introductory courses are pretty well established. But, again, the broad outlines of almost every subject that we teach in introductory courses are pretty well established, which says nothing about whether there’s still work left to do on the research level.
Again, just because the content of introductory courses for novices hasn’t changed much in decades doesn’t necessarily mean that the subject itself has “nothing left to discover”.
I’m not sure what you’re driving at here. Are you saying that there are too many different translations of the Aeneid or the Odyssey?
Classical controls?
You’re kidding - right?
Enormous amounts are being learned all the time. To take one small area I know something about, airfoils for laminar flow at low Reynolds numbers are seeing regular innovation. Also winglets. Active boundary layer control. Riblets.
How about progress in modeling complex flow with computational fluid dynamics? Adaptive airfoils?
I’ll stop there, but this list could be enormous.
Human anatomy, perhaps? OTOH, now we have the human genome to figure out.
Again, depends how you’re defining that subject. There are boatloads of things still unknown about the human anatomy, many of which have been asked and argued about here on the Dope!
I think we pretty much know where everything is; knowing what it does or why it’s there? Yeah, some of that warrants further research.
Gross anatomy is pretty well known. But once you get down to the cellular level, there’s plenty to be discovered, such as the details of the neural circuitry in the brain.
There is still a lot to learn about WWII.
I’m on my phone so links are hard, but there are numerous books recently which have uncovered new material. Tully and Parshall’s Shattered Sword shoe that the accepted account of the Battle of Midway was wrong. Eri Hotta’s Japan 1941: Countdown to Infamy gives new insight to the actions leading up to the war.
There is still plenty to uncover about that conflict.