This is a bit of a soft and subjective question, but can someone give me some examples of mathematical topics or areas which have no or very little research done about them? I love reading math texts that discuss unconventional or little-known topics. Of course, some topics are not researched so much because they are not fruitful areas of research. I am looking for things that aren't researched so well, but which in your opinion deserve to be.
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8If you're looking for areas that have very little research done in them, then they will be known and understood by very few people because of that. There are so many unexplored lines of inquiry that this question seems almost limitless in scope. – user829347 Jan 30 '22 at 01:30
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1If such topics exist a reason that they are unexplored could be: useless or not interesting. – Ryszard Szwarc Jan 30 '22 at 05:35
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@Ryszard Szwarc that is a logic fallacy and isn't true... always there something new, like new methods or problems that starts news areas of research... and sometimes they are rised from previous forgotten solutions of other problems that results useful in other math areas... almost no new research area starts being important, they rise as snow balls until they becomes a trend. – Joako Jan 30 '22 at 16:54
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@Ryszard Swarc as example, in the 70s an engineer of Bell Labs in its free time, just of because scientific curiosity, take an old rule of thermodynamics and apply it to telecommunications, his name was Claude Shannon and through entrophy he is now the father of information theory... an one of his students, John Larry Jr., discover a probabilty property known as Kelly proportion, quite unknown, that right know is rising among finance researchers as a requirable property for portfolio optimization since it does not rise alone from Markowitz Theory, and soon it will be wide known even in physics – Joako Jan 30 '22 at 17:23
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Problems that have been under-explored, but are liable to succumb to attack if only someone would bother, were routinely identified by Paul Erdős; no doubt some exist today too. Whole topics/areas may be thinner on the ground. – J.G. Jan 30 '22 at 20:03
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FINITE-DURATION SOLUTIONS TO DIFFERENTIAL EQUATIONS
I don´t know how "unknown" this it is, but I am stack on it and I have found too little to my surprise about it, and the topic is finite-duration solutions of differential equations (meaning this, that the solution itself behaves as having a final time from where its becomes exactly zero forever on).
Thinking about a scalar (one-variable) second order differential equation with solutions of finite-duration (as every simple classical mechanics system should be I think), so far I have found that:
- the differential equation, to be able to sustain finite-duration solutions, it necessarily must be a non-linear differential equation, and also,
- since the solution becomes zero for a non-zero measure set of compact points, the finite-duration solution can´t be analytical in the whole time domain (maybe, piece-wise, but I don´t really know it - thinking here like in a crop version of a bump-function $\in C_c^\infty$), discarding every possible solution through Power Series like Taylor expansions.
The only proper paper I found explicitly investigating them is Finite Time Differential Equations by V. T. Haimo (1985), but it looks like are restricted only for the time near the solution becoming zero (but, nevertheless, is highly interesting).
I have found this really interesting and surprising, actually finite-duration solutions should be the most used kind of functions at least in classical physics (in my humble opinion), but I have already discarded every linear differential equation and classical functions which can be described by power series (so, everything I have saw in engineering!!!), but I don´t know why, it looks like nobody know much about this topic (actually the publication was done from a research group that works for the army, so maybe extensions of the research are still classified, but who knows).
So far, I have no idea of How a finite-duration solution looks like, neither an example to share, only finding numerical representations of them so far - no idea if they can even been described in closed-form.
Hope you join me trying to figure it out! (here).
Joako
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1The wave equation supports solutions a bit like that, if you plug in compactly supported initial data. This is called "finite speed of propagation". – Giuseppe Negro Jan 30 '22 at 13:23
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@Giuseppe Negro so far I now, wave equations solutions are vanishing at infinite and not properly finite-duration solutions (at least for scalar/one-variable functions), as example a soliton solutions is described through a $\text{sinh}()$ function... for higher dimentions, I am not sure if they stand finite-durations solutions, actually I have recently made that question here,... (continue in next comment) – Joako Jan 30 '22 at 16:33
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@Guiseppe Negro (...), but since its temporal characteristics are described through an $\mathbb{R}^{n+1}$ model parametrization instead as an $\mathbb{R}^{n}$ function, I am not sure if they can be compact-supported in the time variable, given that they do can be it in the space-coordinates as is beatifully shown in one of the answers (maybe they get restricted in the $x_i-ct$ variable, but I still looking for that answer). – Joako Jan 30 '22 at 16:34
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@Guiseppe Negro ...also, in the temporal variable a compact-support solution would look like it just appear and dissapear in the void, and since wave eq. aren't nonlinear differential eqs. I think they can't stand this kind of behavior, but because of the $\mathbb{R}^{n+1}$ description, I don't know if what is said in the cited paper still stands (they review only the scalar scenario)... I don't know how to extend their explanation into higher dimensions neither (is quite advanced math for my actual knowledge). – Joako Jan 30 '22 at 16:42
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Logically, research results can only be published once they have been achieved. But you could collect the published current and proposed research topics of all mathematical working groups and find out the topics on which only very few working groups are currently researching.
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Could an automated theorem prover be possible that extends all mathematical objects and theorems to the most general cases, looks for interrelations between them and proposes new research topics?
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It struck me that the question of existence of solutions in given function classes for equations and inverse functions has so far only been answered insufficiently and seem to be not pursued in general.
These problems can be used i.a. for presenting the solutions in closed form.
For elementary inverses of elementary functions, we have the theorem of [Ritt 1925] that is proved also in [Risch 1979]. It can simply be extended to partial inverses. For solutions of equations in the elementary numbers, we have the theorems of [Lin 1983] and [Chow 1999]. I recently found that both problems are interrelated.
I propose to extend these mathematical problems from the elementary functions to other function classes: https://mathoverflow.net/questions/320801/how-to-extend-ritts-theorem-on-elementary-invertible-bijective-elementary-funct
Only a few working groups seem to be working on the problem of Topological Galois theory for inverses (see Kohavanskii, [Belov-Kanel/Malistov/Zaytsev 2020]). This method should be generalized from individual functions to whole function classes. $\ $
[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
[Khovanskii 2014] Khovanskii, A.: Topological Galois Theory. Solvability and Unsolvability of Equations in Finite Terms. Springer 2014
[Khovanskii] Khovanskii, A.: Topological Galois Theory
[Khovanskii 2019] Khovanskii, A.: Topological approach to 13th Hilbert problem
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