Timeline for Which completion of the configuration space of $n$ distinct points in $\mathbb{R}^d$ is better suited for numerical analysis?
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| Jul 17, 2021 at 9:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jun 17, 2021 at 21:13 | comment | added | Malkoun | I think I agree with @DanPetersen's bet. This was my first impression too, though I needed to learn more to be able to be convinced one way or another. Now I am not so sure what to do with this post. It did generate an interesting discussion and the related research project can possibly start now. I thank you all, D.P., A.M. and hm2020. | |
| Jun 17, 2021 at 8:45 | comment | added | Andrea Marino | 3. The direction of collision and the mutual quotients $s_{ijk}= \frac{|x_i -x_j|}{|x_k-x_j|} $ --> you get the Fulton-Mac Pherson spaces. This is the structured version of the question: what kind of boundary do you want? It seems like you definitely want to keep track of the direction of collision, so that you can define the derivative of a collided configuration. I suggest you try to write a wanna-be approximation formula and you understand which quantities you are using! :) | |
| Jun 17, 2021 at 8:42 | comment | added | Andrea Marino | I agree with @DanPetersen, that's a wonderful reference. Since I have spent much time understanding compactifications of configuration spaces I would be glad to help, but I actually didn't understand the question (in the edit). What formula do you want to be able to write? There are many variants of Fulton Mac Pherson, and the question I think one should answer first is: which data do you want to keep track of in a collision? Some examples: 1. No data --> you get $(\mathbb{R}^m) ^n$ 2. The direction of collision --> Konsevitch spaces (contd) | |
| Jun 16, 2021 at 9:39 | comment | added | Dan Petersen | I would be slightly more optimistic about the Fulton-MacPherson compactification; more specifically, the version due to Kontsevich (the "real oriented Fulton-MacPherson compactification"), rather than the algebro-geometric version you can find in the paper of Fulton and MacPherson. A useful reference is Sinha, "Manifold theoretic compactifications of configuration spaces". | |
| Jun 16, 2021 at 9:38 | comment | added | Dan Petersen | That said, with your updated motivation I would bet good money that the Hilbert scheme is not what you're looking for - the local structure of the Hilbert scheme is horrendously complicated, and it seems you are anyway not looking to exploit the algebraic structure (such as via Gröbner methods). | |
| Jun 16, 2021 at 9:38 | comment | added | Dan Petersen | @Malkoun By "analytic topology" I mean the following construction. If $X \subseteq \mathbb A^n_{\mathbf R}$ is an affine real algebraic variety, then $X(\mathbf R)$ is a closed subset of $\mathbf R^n$ and inherits a topology from the usual euclidean topology. This topology on $X(\mathbf R)$ is independent of choice of affine embedding, and for a general real algebraic variety $X$ one can define a topology on $X(\mathbf R)$ by working locally on affine charts of $X$. I am just describing the real version of the usual complex analytic topology on a complex algebraic variety. | |
| Jun 16, 2021 at 8:51 | comment | added | user122276 | @Malkoun - The above claim is nonsense - The relevant topology to consider is the Zariski topology. | |
| Jun 16, 2021 at 1:04 | history | edited | Malkoun | CC BY-SA 4.0 |
added 1770 characters in body; edited title
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| Jun 16, 2021 at 0:13 | comment | added | Malkoun | @DanPetersen, how is the natural topology defined on the set $\mathcal{I}_n$ please? I have been reading on the internet about Hilbert schemes of points, and I have learned a lot, but still do not know what is the natural topology on it. If you could give an online reference perhaps, or write a few comments, it would help me and perhaps other readers. | |
| Jun 15, 2021 at 15:45 | comment | added | Dan Petersen | The set $\mathcal I_n$ is the set of real points of a finite type scheme over $\mathbb R$ (the Hilbert scheme of points). As such it has a natural topology, the analytic topology (the Zariski topology is not relevant). I claim that this is the only natural topology to consider. When $d>2$ this will not be a manifold, as you indicated, and the answer to your question is "no". Infinite-dimensional manifolds seem to be a red herring here. | |
| Jun 15, 2021 at 12:36 | answer | added | user122276 | timeline score: 1 | |
| Jun 15, 2021 at 12:11 | history | asked | Malkoun | CC BY-SA 4.0 |