Background
Sometimes, I have ideas for research in mathematical subjects about which I don't know much. Let me describe an example to make it more concrete.
In his 2009 article "The Brachistochrone Problem for a Disk" by L. D. Akulenko, he describes along which curve a disk with radius $R$ rolls down in the shortest amount of time. The problem that was considered is a generalization of the classical Brachistochrone problem, which entails finding the curve of quickest descent for a point mass.
In Akulenko's article, the density of the disk is assumed to be uniform. I wonder: what if we start tweaking this assumption? We could consider, for instance, the following three variations:
(Please excuse my bad drawing skills in this case. All circles have radius $R$.)
Variation $(A)$ concerns a disk rolling down a curve, for which the density is uniform except at an extra point mass at the edge. The question becomes: how does this extra point mass affect what the curve of quickest descent looks like? How can this curve be described mathematically, also as a function of the magnitude of the extra mass?
In variation $(B)$, the extra mass is put in a pie slice in the circle, and in variation $(C)$, it is put in a vertical segment of it. For all of these variations, the question remains the same: what are the curves of steepest descent?
I do not possess the required background on the theory of dynamical systems and/or (partial) differential equations to solve this problems. At the same time, I believe these who does have sufficient knowledge in these areas might be interested in doing research on these topics. Especially since, I believe, these particular generalizations of the Brachistochrone problem have not been considered yet (for other generalizations, see for instance this paper by Gemmer).
I find the idea that others might build on the questions I've raised appealing, because it could mean you've indirectly added a little bit of extra knowledge to the existing body of work on mathematics. Moreover, you've aided people with finding a research topic that suits them.
Questions
- Are there journals that welcome instances of people sending their ideas for research projects (which they can't or won't delve into themselves) and publish them?
- Are there any repositories - either online or on paper - specifically dedicated to receiving and listing ideas for projects that researchers might consider investigating?
- Are there other things one might consider doing with such ideas? For instance, is it wise to send them to professors in relevant research areas?
Which is great and is something that naturally happens when you think about mathematics, especially at an early stage. You can either note down these ideas for a later time, or try to pursue them.
In the latter case, either you try to build up knowledge yourself or you need to find a collaborator. Forget about your options 1, 2. Maybe 3 could be feasible: sending emails. However you must be sure to include some content in your proposal. Just writing: "why don't we work together on blah" is not enough. You must include some ideas, share something nontrivial if you want to wake the other person's interest.
I have a friend who is very good at finding collaborators. He does not send many emails, though. He keeps an eye on mathematical production, he reads a lot. When he finds somebody who seems interesting to collaborate with, he approaches them, preferably in person: that's the reason why conferences, workshops and meetings are done.