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:

Does what I remember (see above) make sense?

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

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...

exactlythe paper of interest -- Professor Ghioca was the one who I was lucky enough to speak with today! $\endgroup$