TITLE: Why is this theorem (about $L(P(\omega_1))^V$ and $L(P(\omega_1))^{V[G]}$) nice?
QUESTION [9 upvotes]: I was recently told that the following (due to M. Viale) is a nice theorem:

Suppose there are arbitrarily large supercompacts, and $\mathrm{MM}$ holds in $V$.  Let $G$ be generic for a proper forcing and $V[G] \vDash \mathrm{MM}$.  Then $L(P(\omega_1))^V$ is elementarily equivalent to $L(P(\omega_1))^{V[G]}$.

My question (borne of ignorance, not skepticism) is:

Why is this theorem nice, and how does it fit into the bigger picture?

Some slightly-more-specific questions that refine my main question are: Do the hypotheses of this theorem often come up in natural settings?  What's the upshot of the conclusion?  Is it that proper forcing which preserves $\mathrm{MM}$, leaves the theory of a small but not-that-small chunk of the universe unchanged?

REPLY [11 votes]: Let me give a brief answer as the author of the mentioned theorem. I shall first say that (as you might imagine) I was very pleased to see that my theorem raised your interest. Now, as mentioned by Joel in his latest post, your statement of the theorem is not correct. However
there is a typo also in Joel's post, namely the conclusion of the theorem holds for the Chang model $L([Ord]^{<\omega_2})$ and not for the Chang model $L([Ord]^{<\omega_1})$ as posted by Joel.
For this Chang model the result is due to Woodin and was already known, it appears for example in
Chapter 3 of Larson's book on the stationary tower forcing. To complement the very good answer Joel has already given you, I invite you to read the many survey papers related to the philosophical position subsumed by this theorem, here is a sample list:
Woodin's two short papers for the "notices of the AMS" where he exposes the basic ideas behind the generic absoluteness results for $L(\mathbb{R})$:

Woodin, W. Hugh (2001a). "The Continuum Hypothesis, Part I" (PDF). Notices of the AMS 48 (6): 567–576. http://www.ams.org/notices/200106/fea-woodin.pdf. 
Woodin, W. Hugh (2001b). "The Continuum Hypothesis, Part II" (PDF). Notices of the AMS 48 (7): 681–690. http://www.ams.org/notices/200107/fea-woodin.pdf.

Several of Peter Koellner's papers available at his webpage:
http://www.people.fas.harvard.edu/~koellner/
Most of them contains a long introductory part which motivates and explains very carefully and plainly the ideas at the heart of the $\Omega$-logic approach to absoluteness result.
It has to be noted that the kind of solution to the continuum problem prospected by my theorem follows Woodin's view as exposed in the papers on AMS notices. Currently Woodin has pursued a different approach towards the solution of the continuum problem that leads him to prospect a view of the universe (Ultimate $L$) which is radically different from the one given by MM.
Finally in case you are interested there are in my webpage several slides of talks I gave on this and related subjects, as well as a proof of the theorem you have mentioned in your question:
http://www2.dm.unito.it/paginepersonali/viale/index.html
(unfortunately my university is currently changing the websites locations, so for some time you may have troubles to consult it....)