I don't know the answer, but +1 for good use of XOR.

Will, thanks for the partial answer regarding critical points of $M_n$. Regarding the characterization of submanifolds of $\mathbb{R}^n$ for which the first coordinate is a Morse function, see mathoverflow.net/questions/102956/… and set $M = \mathbb{R}^n$ and let $\mu$ be the function which extracts the first coordinate...

@Robert: Does the next line say "And if you reverse orientation on every other day then in fact you shall only march around the city once..."?

Adding on to Alexandre's answer: although I am not sure about the origins of the term "completely invariant", the only person that I know of who regularly uses this term is Stankewitz. See, for instance, arxiv.org/abs/math/9810090 and also rstankewitz.iweb.bsu.edu/numcomp.pdf where the term is defined in the background sections. If you can't find a cite-able original reference, maybe you can send him an email and ask where he saw first it.

If you have been in graduate school for a while, then you probably have a designated advisor. If this is the case, and if this advisor can be trusted, then you should definitely discuss your situation with him or her. On the other hand, if you are just starting out, there is no shame in having a publication, whatever you do now will simply be superseded by your thesis work later on. I'm not saying you should put your name on something you are ashamed of... I'm saying it matters less than you think if you are just starting grad school.

Ryan, two further questions about your recent edit: 1. how did you triangularize the point cloud, and 2. did you try another probability distribution function whose sub-level sets are homologically similar to the Gaussian one, eg a Poisson distribution? That is, can you tell the Gaussian and Poisson barcodes apart even in 1D?

I'm surprised no one has asked this yet, but: how many sample points? If you have only one sample point, the barcode is not goint to be terribly hard to describe. Perhaps by asymptotic behavior you mean "let the sample size go to infinity" at which point generically nothing survives for too long. In short, I don't see a sample size invariant answer to your question that is also interesting. What do you have in mind?

Thanks for the book reference, Igor, but which part contains an answer to this question?