What topics in mathematics have no or very little research done about them, so far?

Question

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.

Answer 1

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).

Answer 2

1.)

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.

2.)

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?

3.)

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. $\ $

[Belov-Kanel/Malistov/Zaytsev 2020] Belov-Kanel, A.; Malistov, A.; Zaytsev, R.: Solvability of equations in elementary functions. Journal of Knot Theory and Its Ramifications 29 (2020) (2) 204-205

[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

[Khovanskii 2004] Khovanskii, A.: On solvability and unsolvability of equations in explicit form. Russian Math. Surveys 59 (2004) (4) 661–736

[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

[Ritt 1922] Ritt, J. F.: On algebraic functions which can be expressed in terms of radicals. Trans. Amer. Math. Soc. 24 (1922) (1) 21-30

[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50

[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759

[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90