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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 iterant" 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 iterant"? 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...

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    $\begingroup$ 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. $\endgroup$ – David Roberts Sep 27 '14 at 1:29
  • $\begingroup$ 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 $\endgroup$ – David Roberts Sep 27 '14 at 1:30
  • $\begingroup$ @DavidRoberts That is exactly the paper of interest -- Professor Ghioca was the one who I was lucky enough to speak with today! $\endgroup$ – user89 Sep 27 '14 at 3:30
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    $\begingroup$ @David: I see that you are a strong advocate of liquid soap! $\endgroup$ – Asaf Karagila Sep 27 '14 at 5:17
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    $\begingroup$ @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. $\endgroup$ – Felipe Voloch Sep 27 '14 at 13:13
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    $\begingroup$ @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. $\endgroup$ – David Roberts Sep 28 '14 at 0:51
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    $\begingroup$ 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 $\endgroup$ – Benjamin Dickman Oct 4 '14 at 7:20
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Refering to part 3:

As someone who has been curious about this exact subject before, I would love to see this question posted. There are frequently heuristic questions on mathoverflow, and while this can cause some controversy some of the most upvoted questions are at their core asking for intuition. As such, asking for the reasons why a subject is exciting and draws people is in my opinion very valid.

On a related note, there is a certain stigma in asking "why" a subject is exciting, as such answers are inevitably subjective, and often the answer is "because i like working on it" or, in the case of students, "my advisor told me to work on it". As such, people feel embarrassed and/or defensive when presented with such a question about their work. However, such questions often have very GOOD answers, and so I think the question should always be asked. At worst, there is no good answer readily available, and it shall remain unanswered.

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