any distribution is really a topological object.
I find this interesting, but I like to apply things to a specific example so I'm sure I understand it. Suppose I give you the following distribution of measurements of two variables (units are GeV, not that I suppose this matters):
1.80707 0.148763 1.87494 0.151895 1.86805 0.140318 1.85676 0.143774 1.85299 0.150823 1.87689 0.151625 1.87127 0.14012 1.89415 0.145116 1.87558 0.141176 1.86508 0.14773 1.89724 0.149112
What sort of topological object is this, or how do you go about treating it as one? Presumably you can think of these points in mD-deltaM space as being two-dimensional vectors. N-vectors are a group under addition, and if I understand the definition correctly they are also a Lie group. But I confess I don't understand how this is important; I'm never going to add together two events, the operation doesn't make any sense. If a group lives in a forest and never actually uses its operator, does it still associate, close, identify, and invert? (I further observe that although 2-vectors are a group, the second variable in this case can't go below 0.13957 for kinematic reasons; the subset of actual observations is not going to be closed or invertible.)
I'm not sure what harmonic analysis is; I might know it by another name, or do it all the time and not realise that's what it's called. Could you give an example?
My attempts at putting LaTeX notation here didn't work out, so I hope this is at all readable.
I would not call the data you gave me a distribution. I think of a distribution as being something like a Gaussian; some function f where, if I keep collecting data, and I take the average sum of powers of that data, it looks like the integral over some topological group of that function.
so: lim n->\infty sum.{k=1}^n g(x.k,y.k) = \int_{R^2} f(x,y)g(x,y) dx ^ dy for any function g on R^2
usually rather than integrating over R^2, I would be integrating over SU(2)...
In response to falenas108's "Ask an X" thread. I have a PhD in experimental particle physics; I'm currently working as a postdoc at the University of Cincinnati. Ask me anything, as the saying goes.
This is an experiment. There's nothing I like better than talking about what I do; but I usually find that even quite well-informed people don't know enough to ask questions sufficiently specific that I can answer any better than the next guy. What goes through most people's heads when they hear "particle physics" is, judging by experience, string theory. Well, I dunno nuffin' about string theory - at least not any more than the average layman who has read Brian Greene's book. (Admittedly, neither do string theorists.) I'm equally ignorant about quantum gravity, dark energy, quantum computing, and the Higgs boson - in other words, the big theory stuff that shows up in popular-science articles. For that sort of thing you want a theorist, and not just any theorist at that, but one who works specifically on that problem. On the other hand I'm reasonably well informed about production, decay, and mixing of the charm quark and charmed mesons, but who has heard of that? (Well, now you have.) I know a little about CP violation, a bit about detectors, something about reconstructing and simulating events, a fair amount about how we extract signal from background, and quite a lot about fitting distributions in multiple dimensions.