Is this an appropriate MathOverflow question: can I ask for some intuition and "excitement" behind the Dynamical Mordell-Lang conjecture?

I am a math undergraduate at my local university, and I have decided to make it a habit to attend the weekly seminars that are given by the faculty.

Today, the talk was on the Dynamic Mordell-Lang conjecture, which I can safely say is a piece of mathematics on the 'frontier'. I was writing stuff as the talk was being given: basically, a long list of all the things said that I did not understand. At the end, I asked the speaker to provide me with a take-home message suitable for "my level" of current understanding ("how would you present your talk as a 2 minute movie trailer?"). He was very polite and gracious, and gave me the following trailer.

Let's say you have a set $$S$$. Inside $$S$$, you have a point $$x$$ where you start off at. Now, you also have a function $$f: S \rightarrow S$$ (is that how I correctly write "maps from something in $$S$$ to something else in $$S$$"?). So each time you apply $$f$$ to $$x$$, you land at another point inside $$S$$ ("duh"), but what if you keep landing inside a closed subset of $$S$$, $$Y$$? Not just once ("duh"), a few times ("hrm"), or a billion times ("hmmm"), but an infinite number of times?

Clearly, $$Y$$ must have some special relationship to $$f$$? Some special "structure"? What he was talking about then, is that he was involved in a proof for the fact that there is some number $$k \in \mathbb{N}$$ such that $$Z \subset Y$$ is invariant under the "$$k$$th iterate" of $$f$$, $$f^{(k)}$$ (i.e. $$f$$ applied $$k$$ times to $$x$$).

Okay, so this is what I remember of what he told me. He had stayed behind very late talking to someone else, so I didn't want to take more than a few minutes of his time even though he was so kind, so I am left with questions:

1. Does what I remember (see above) make sense?

2. What does it mean to be "invariant under the $$k$$th iterate"? Is it that $$f^{(k)}(Z) = Z$$?

3. If I am understanding 2) correctly, and the answer to 1) is positive, could you now help me provide a "soundtrack" to the "trailer" in order to get me really pumped for why this is a big deal? Like, it would be really nice to hear of an interesting question that may be, after a lot of work, be answered by such a conjecture.

Would it be appropriate to ask such questions on the main site? The piece of mathematics is cutting-edge, but in the end, it is just an "explain the intuition" type of thing...

• As someone who doesn't know the dynamic Mordell-Lang conjecture from a bar of soap, if this is an exciting frontier of research, then it is worth explaining in a way suitable for a broader mathematical audience.
– David Roberts Mod
Commented Sep 27, 2014 at 1:29
• Especially if answers give reasons why we think this may be true, or the nub of the difficulties in approaching it, and interesting applications. For the cognoscenti, here's a relevant example of a paper arxiv.org/abs/1401.6659
– David Roberts Mod
Commented Sep 27, 2014 at 1:30
• @DavidRoberts That is exactly the paper of interest -- Professor Ghioca was the one who I was lucky enough to speak with today! Commented Sep 27, 2014 at 3:30
• @David: I see that you are a strong advocate of liquid soap!
– Asaf Karagila Mod
Commented Sep 27, 2014 at 5:17
• @DavidRoberts Your first comment is an absolutely true statement. However it only partially helps answering the question, which is ultimately whether this is a good question for MathOverflow. Is explaining frontier of research questions (be in Dynamical Mordell-Lang or something else) to a broader mathematical audience a good use of MathOverflow and does it fit the format? That is not so clear to me, as it can be a bit too broad and open ended. Commented Sep 27, 2014 at 13:13
• @Felipe MO has been so used in the past by eg Tim Gowers (well, recent frontiers, rather than current). It would necessarily be tagged [soft-question], and so some I'm sure would frown on it.
– David Roberts Mod
Commented Sep 28, 2014 at 0:51
• Perhaps it would be worth editing this meta post to mention that the question is now on the main site: mathoverflow.net/q/182434/22971 Commented Oct 4, 2014 at 7:20