Timeline for Discrete Morse function from smooth one
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Apr 4, 2013 at 16:10 | comment | added | Michał Kukieła | See also the recent survey , also by Benedetti arxiv.org/abs/1212.0885 | |
| Oct 22, 2012 at 13:51 | vote | accept | Niles | ||
| Oct 19, 2012 at 4:53 | answer | added | Vidit Nanda | timeline score: 4 | |
| Oct 19, 2012 at 1:25 | comment | added | John Shareshian | Maybe the paper of Bruno Benedetti found at arxiv.org/pdf/1010.0548v4.pdf would be of interest to you. | |
| Oct 18, 2012 at 20:52 | history | edited | Niles | CC BY-SA 3.0 |
going to think
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| Oct 18, 2012 at 20:33 | comment | added | Patricia Hersh | My paper "On optimizing discrete Morse functions" has a result on how to deduce information about cell incidences among critical cells from the data from a larger triangulation and a discrete Morse function having these as its critical cells. If you really have software to do that, maybe the paper will help a little bit too. | |
| Oct 18, 2012 at 20:32 | comment | added | Liviu Nicolaescu | Here is a special case of your question which is already nontrivial: is the sublevel set $\{f\leq c\}$ connected? | |
| Oct 18, 2012 at 20:25 | comment | added | Niles | Oh, my "strategy" for finding critical points of $f$ consists of some software I'm working on. I think (exploiting some details of $f$), that the triangulation is manageable (with software). The problem of determining the boundary maps in the discrete Morse complex then looks like a problem about paths on a graph. As long as the triangulation doesn't get out of control, existing algorithms should be able to handle that. | |
| Oct 18, 2012 at 20:17 | comment | added | Patricia Hersh | Actually, what Liviu writes sounds very reasonable to me. I had been focusing on your saying you had limited knowledge of the flow and so had thought you'd still have trouble finding a discrete Morse function with your desired critical cells and just be creating artificial, extra work for yourself. | |
| Oct 18, 2012 at 20:13 | comment | added | Ryan Budney | If you want to use discrete Morse theory as a computational tool you'll have to triangulate your domain appropriately to adapt it to your Morse function. The problem is there aren't (to my knowledge) any really sophisticated algorithms for this, so you'll get a huge triangulation manageable only with software. Off the top of my head I don't know any robust, rigorous software that can deal with discrete Morse functions on "large" triangulations. Googling around a bit I don't find anything but perhaps I don't know what to look for. | |
| Oct 18, 2012 at 20:03 | comment | added | Niles | Hmmm... @Patricia Hersh and @Ryan Budney, are you saying that the information contained in Forman's discrete Morse complex is equivalent to that of the smooth Morse complex, and therefore I can't hope to compute the former without knowing the latter? | |
| Oct 18, 2012 at 19:53 | answer | added | Liviu Nicolaescu | timeline score: 6 | |
| Oct 18, 2012 at 19:51 | comment | added | Ryan Budney | I also think you'll need more than discrete Morse theory. So from the sounds of it you know the groups in your chain complex, just not the connecting maps. If your chain complex isn't too large, frequently one can compute the homology by slowly building-up the level-sets of $f$, one critical point at a time. If you wanted to try and get some brute-force insight via computation you might want to consider intersecting $C$ with a regular lattice, and sending that lattice to JavaPLEX for it to compute the homology of that, then check to see if your answer is stable under subdivision. | |
| Oct 18, 2012 at 19:46 | history | edited | Niles | CC BY-SA 3.0 |
no pathologies
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| Oct 18, 2012 at 19:37 | history | edited | Niles | CC BY-SA 3.0 |
update about what I'm trying to do
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| Oct 18, 2012 at 19:30 | comment | added | Patricia Hersh | If I understand what you are asking, this sounds like a situation where Forman's discrete Morse theory would not have any advantage over traditional Morse theory. | |
| Oct 18, 2012 at 19:28 | comment | added | Niles | Oh, I wasn't aware of Bestvina-Brady -- thanks for the pointer. I was thinking of Forman, but a cursory look indicates that Bestvina-Brady may work for me. | |
| Oct 18, 2012 at 19:23 | comment | added | Ryan Budney | Could you perhaps tell us something about what you'd like to use the output discrete Morse function for, what you'd like to do with it? It might help people find a more useful answer for you. | |
| Oct 18, 2012 at 19:13 | comment | added | Misha | Niles: You should specify which form of discrete Morse theory you are interested in, since there are at least 2: Bestvina-Brady's and Forman's. | |
| Oct 18, 2012 at 18:02 | history | asked | Niles | CC BY-SA 3.0 |