Timeline for Elliptic operators corresponds to non vanishing vector fields
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| Jun 29, 2019 at 20:24 | comment | added | Ali Taghavi | @LiviuNicolaescu may I ask you to read the update version of this question? | |
| S Jun 27, 2019 at 10:03 | history | bounty ended | CommunityBot | ||
| S Jun 27, 2019 at 10:03 | history | notice removed | CommunityBot | ||
| S Jun 19, 2019 at 9:00 | history | bounty started | Ali Taghavi | ||
| S Jun 19, 2019 at 9:00 | history | notice added | Ali Taghavi | Draw attention | |
| Jun 19, 2019 at 8:57 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jun 19, 2019 at 8:50 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Oct 26, 2018 at 10:09 | comment | added | Ali Taghavi | @JoonasIlmavirta I found the Sibreg _Witten equation very similar to your idea. Can one find some relations in this contex? Thanks again for your interesting comment physicsoverflow.org/41645/… | |
| Jun 3, 2017 at 6:43 | comment | added | Ali Taghavi | after about 13 years i am still wonder to find an appropriate operator associated to a vector field whose index or other quantities can be used to count the number of limit cycles | |
| Jun 3, 2017 at 6:41 | comment | added | Ali Taghavi | and the last version of the following mathoverflow.net/questions/164059/… | |
| Jun 3, 2017 at 6:40 | comment | added | Ali Taghavi | Please see the very interesting comment by Lukas Geyer math.stackexchange.com/questions/1163800/… | |
| Jun 3, 2017 at 6:38 | comment | added | Ali Taghavi | @DeaneYang because the codimension of the range $X^2$ is infinite in case of existence of at least two attractors(either limit cycle or singularity | |
| Jun 3, 2017 at 6:14 | comment | added | Ali Taghavi | @DeaneYang Thank you for interesting suggestion $X^2+ \epsilon \Delta$ It seems that the index is unbounded when epsilon goes to zero. | |
| May 15, 2017 at 13:51 | comment | added | Ali Taghavi | @SebastianGoette I sincerely thank you very much. I just realize that there are non elliptic operator which are Fredholm. I confess that I did not pay good attention to your valuable comments. Thanks a Lot. i hope that it works for dynamical interpretations for a vector field. | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 14, 2015 at 16:24 | comment | added | Ali Taghavi | @SebastianGoette thank you for useful information. | |
| Nov 13, 2015 at 7:19 | comment | added | Sebastian Goette | @AliTaghavi For the last ten years, Bismut has been working on so-called hypoelliptic operators related to the geodesic spray on $T^*M$, see [math.u-psud.fr/~bismut/]. This is a manifold version of the Fokker-Planck equation, and it gives Fredholm operators. Maybe, you can set up similar operators on $M$ itself under nice assumptions on $X$? | |
| Nov 11, 2015 at 21:42 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| S Aug 24, 2015 at 7:15 | history | bounty ended | CommunityBot | ||
| S Aug 24, 2015 at 7:15 | history | notice removed | CommunityBot | ||
| Aug 17, 2015 at 22:01 | comment | added | Ali Taghavi | @DeaneYang Do you mean the index of $\Delta+\epsilon X^{2}$ depends on $X$ and $\epsilon$? If yes, what is the mistake of the following argument. $\Delta=\epsilon X^{2}$ is a path of fredholm operator hence the index is fix. | |
| Aug 16, 2015 at 18:25 | comment | added | Deane Yang | A couple of quick little comments: 1) The operator $\Delta + \epsilon X^2$ depends not only on $X$ but on the Riemannian metric used to define $\Delta$. 2) Perhaps a better thing to look at is $X^2 + \epsilon^2 \Delta$ and ask what happens as $\epsilon \rightarrow 0$? | |
| S Aug 16, 2015 at 5:52 | history | bounty started | Ali Taghavi | ||
| S Aug 16, 2015 at 5:52 | history | notice added | Ali Taghavi | Authoritative reference needed | |
| Aug 6, 2015 at 8:50 | comment | added | Ali Taghavi | @PaulSiegel can I ask you to introduce me a precise reference which need the minimum background. | |
| Aug 6, 2015 at 8:31 | comment | added | Ali Taghavi | @DeaneYang According to my previous comment is the index of $\Delta +\epsilon \partial^{2}/\partial X^{2}$ independent of $X$ and $\epsilon$, hence $0$? So this operator is useless from dynamical view point? | |
| Aug 5, 2015 at 15:33 | comment | added | Deane Yang | Ali, I assumed that you wanted the PDO to be first order and the top order term to be $X$. | |
| Aug 5, 2015 at 7:07 | comment | added | Ali Taghavi | @JoonasIlmavirta any way your first comment on this question was very interesting for me and that comment is a motiviation to this home genus second order operator. | |
| Aug 5, 2015 at 7:04 | comment | added | Ali Taghavi | @LiviuNicolaescu Is the index of the operator in my previous comment independent of $X$? | |
| Aug 5, 2015 at 6:53 | comment | added | Joonas Ilmavirta | @AliTaghavi, that is always elliptic but I don't know about the index off the top of my head. | |
| Aug 5, 2015 at 5:27 | comment | added | Ali Taghavi | @JoonasIlmavirta What about $\Delta +\partial^{2}/\partial X^{2}$? Is its index independent of choosing a vector field?Is it always elliptic? | |
| Aug 5, 2015 at 5:20 | comment | added | Ali Taghavi | @DeaneYang Thanks for your very interesting comment. I had the same problem with your comment as I explained above.Can I ask you to more explain on your last statement. | |
| Aug 5, 2015 at 5:19 | comment | added | Ali Taghavi | @PaulSiegel Thanks for your very interesting comment. I do not know why I did not realized your comment. I did not received any announce for your comment. | |
| S Jul 12, 2015 at 16:55 | history | bounty ended | Ali Taghavi | ||
| S Jul 12, 2015 at 16:55 | history | notice removed | Ali Taghavi | ||
| Jul 7, 2015 at 22:27 | comment | added | Deane Yang | I just want to reiterate what Liviu Nicolaescu has already said: Any PDO defined using only the vector field $X$ and no other PDEO is never elliptic. The best way to understand this is to look up the most general definition of an elliptic partial differential operator and test it against examples such as $\nabla_X$, $[X,\cdot]$, and any other example you can think of. Any PDO defined using only $X$ is essentially an ODE along the integral curves of $X$. If such an operator is Fredholm, it is due to the global dynamics of the operator and not a local property of the PDO such as ellipticity. | |
| Jul 6, 2015 at 12:48 | comment | added | Paul Siegel | I don't know enough to turn this into an answer, but you might have a look at some of the stuff that symplectic geometers and low dimensional topologists are doing. Some of their most powerful tools - like symplectic homology and Heegaard-Floer homology - are based on index theory for nonlinear Fredholm operators, and there are deep connections with dynamical systems. | |
| Jul 6, 2015 at 12:42 | answer | added | Paul Siegel | timeline score: 10 | |
| S Jul 6, 2015 at 10:45 | history | bounty started | Ali Taghavi | ||
| S Jul 6, 2015 at 10:45 | history | notice added | Ali Taghavi | Draw attention | |
| May 12, 2015 at 16:41 | comment | added | Ali Taghavi | is the second part of this note | |
| May 12, 2015 at 16:40 | comment | added | Ali Taghavi | @ChrisGerig But what is the error of the following argument which shows the codimension is "2"?: Let $g$ be an smooth function with $g(N)=g(S)=0$ then $D_{X}(f)=g$ for $f(x)=\int_{-\infty}^{+\infty} g(\phi_{t}(x))dt$ where $\phi$ is the flow of the vector field.$ Since the singularities are hyperbolic, then this integral is well defined(converges). In this note I tried to extend this idea to count the number of limit cycles as a fredholm index. However there is a gap in this not but a true (and weaker) version... | |
| Mar 4, 2015 at 18:08 | comment | added | Chris Gerig | Yes; I think the spherical harmonics won't be in the image, because they can't be integrated (when solving for the corresponding function in the domain). For example, in order for $D_X(f)=\cos\phi$ you need $f=\sin^2\theta\ln|\sin\phi|$ which blows up at $\phi=0,\pi$. | |
| Mar 4, 2015 at 8:52 | comment | added | Ali Taghavi | @ChrisGerig Are you considering the smooth function? That is : do you believe that the codimension is infinite if we consider the operator on smooth functions? | |
| Feb 24, 2015 at 22:55 | comment | added | Chris Gerig | @AliTaghavi, I think the cokernel is infinite-dimensional. | |
| Feb 24, 2015 at 21:29 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Feb 24, 2015 at 17:58 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Feb 24, 2015 at 17:46 | history | edited | Ali Taghavi |
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| Feb 24, 2015 at 17:26 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 7, 2014 at 19:25 | comment | added | Ali Taghavi | @LiviuNicolaescu Now I reviewed your definition of symbol in your lecture on Atiyah singer index theorem.I have a question: you spend at least one or two pages to define the symbol. it seems that you take it very difficult. why you do not simply define the symbol as in the Nakahara book or in a paper of of Atiyah?(As I wrote in the above comments). What is the advantage of your complicated definition? Is there a real application and a motivation for this complication? | |
| Oct 7, 2014 at 9:18 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 7, 2014 at 8:49 | comment | added | Ali Taghavi | @LiviuNicolaescu According to your comment "On a compact manifold a partial differential operator is Fredholm if and only if it is elliptic" what is the error of the following statement:Let $f:S^{2}\to \mathbb{R}$ be a morse function with exactly two critical points a minimum at S and a maximum at N(for example $f(x,y,z)=z$). Now consider the gradient vector field $X=\nabla f$. Now I think the derivational operator $D_{X}$ is a non elliptic operator on $C^{\infty}(M)$ which is a fredholm operator(of codimension 2 and the kernel is one dimensional). Am I mistaken? | |
| Oct 6, 2014 at 17:13 | comment | added | Ali Taghavi | @LiviuNicolaescu I enter the chat but I am not sure that it works. Did you received my message? | |
| Oct 6, 2014 at 17:01 | comment | added | Ali Taghavi | @LiviuNicolaescu for a covector $\Xi$ and a section $s$ $P(D)(\Xi,s)=D(1/n!fs)$ where $f$ is a function with $df=\Xi$. are you meaning some more complicated definition of ellipticity? | |
| Oct 6, 2014 at 17:00 | comment | added | Liviu Nicolaescu | Let us continue this discussion in chat. | |
| Oct 6, 2014 at 16:55 | comment | added | Ali Taghavi | @LiviuNicolaescu I think it is a linear map $D$ on $\Gamma E$ for which principle symbole is an invertible bundle morphism on $q^{*} E$ where $q:T^{*}M\to M$ is the natural projection. The definition of principle symbol is written clearly in Nakahara. Are you meaning some other complicated definition of ellipticity? | |
| Oct 6, 2014 at 16:50 | comment | added | Liviu Nicolaescu | @ Ali You first need to understand well the meaning of ellipticity before you attack more complicated issues. | |
| Oct 6, 2014 at 16:47 | comment | added | Liviu Nicolaescu | You can find this in Theorem 5, Chapter IV of R. Seeley's memoir Topics in pseudo-differential operators. 1969 Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) pp. 167–305 Edizioni Cremonese, Rome | |
| Oct 6, 2014 at 16:35 | comment | added | Ali Taghavi | @LiviuNicolaescu and what type of results on non elliptic on non compact without boundary? | |
| Oct 6, 2014 at 16:33 | comment | added | Ali Taghavi | @LiviuNicolaescu thank you. May you give a reference? | |
| Oct 6, 2014 at 16:30 | comment | added | Liviu Nicolaescu | On a compact manifold a partial differential operator is Fredholm if and only if it is elliptic. | |
| Oct 6, 2014 at 15:44 | comment | added | Ali Taghavi | @LiviuNicolaescu what type of nonelliptic operatores are known to have finite fredholm index? | |
| Oct 5, 2014 at 0:10 | comment | added | Liviu Nicolaescu | @ Ali Yes, the index is zero. | |
| Oct 4, 2014 at 20:53 | comment | added | Ali Taghavi | @LiviuNicolaescu another question: Am i right to think that the index of every operator $D_{X}+\Delta$, the same operator which proposed byJoonas, is zero where $D_{X}$ is the derivation operator | |
| Oct 4, 2014 at 20:50 | comment | added | Liviu Nicolaescu | @ Ali Compute the principal symbol and notice it is not invertible. It vanishes for any covector $\xi$ such that $\xi(X)=0$. | |
| Oct 4, 2014 at 20:19 | comment | added | Ali Taghavi | ... the following question mathoverflow.net/questions/182139/… | |
| Oct 4, 2014 at 20:18 | comment | added | Ali Taghavi | @shu thanks for your information on Witten Laplacian. My reason that I assumeed the nonvanishing condition is the following; Some times ago a researcher suggested me to consider the operator $D(U)=PU_{x}+QU_{y}+i(QU_{x}-PU_{y})$ as complex diff operator associated with vector field $P\partial_{x}+Q\partial_{y}$. This operator is elliptic at non singular points of $X$. This situation was my main motivation to consider the following question..... | |
| Oct 4, 2014 at 20:08 | comment | added | Ali Taghavi | @LiviuNicolaescu Thanks for the comment.where is a proof of this statement?is it elementary?What about the algebra of diff operatores on $\Gamma TM$ generated by $\nabla_{X}$ or $[X,.]$?I seach for an operator associated with $X$ such that some interesting quantity of this operator can count the number of attractors. | |
| Oct 4, 2014 at 9:31 | comment | added | shu | why you assume $X$ is non vanishing? If not, for example $X=-\nabla f$, where $f$ is a Morse function. There is a Laplacian-like operator, called Witten Laplatian, related the stable/unstale cells of the dynamical and the cohomology of maniflold. | |
| Oct 3, 2014 at 20:20 | answer | added | Yuri Bakhtin | timeline score: 7 | |
| Oct 3, 2014 at 14:28 | comment | added | Joonas Ilmavirta | Consider the Hamiltonian function $H:T^*M\to\mathbb R$ defined by $H(x,p)=\frac12g_{ij}(x)p^ip^j+X_i(x)p^i$. The corresponding Hamiltonian flow on the cotangent space is a dynamical system corresponding to $X$. If $X=0$, you get geodesics. A simpler dynamical system corresponding to $X$ is the system $\dot x=X(x)$ on $M$, but this does not look "as elliptic" as the Hamiltonian one. Do you want the dynamical system to live on $M$ itself rather than $T^*M$? The dynamical viewpoint would be clearer if you gave the dynamical system (if you have one). | |
| Oct 3, 2014 at 12:59 | comment | added | Liviu Nicolaescu | The $C^\infty(M)$-algebra of scalar differential operators generated by $L_X$, the Lie derivative along $X$, contains no elliptic operator if $\dim M>1$. | |
| Oct 3, 2014 at 12:41 | comment | added | Ali Taghavi | @JoonasIlmavirta As another example of dynamical interpretation for certain diff operator see the Veku theorem in .(Topology and analysis, the Atiyah-Singer index formula and gauge-theoretic physics, by B. Booss and D. D. Bleecker) | |
| Oct 3, 2014 at 12:22 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 3, 2014 at 12:06 | comment | added | Ali Taghavi | @JoonasIlmavirta By dynamical interpretation, I mean , for example"The number of attractors of $X$" In the second part of this note I explained some thing related to this concept arxiv.org/abs/1302.0001. In this note I tried to make a remedy for non elliptic ness of the derivational operator. But what do you mean by "Hamiltonian flow associated with the operator"? | |
| Oct 3, 2014 at 4:50 | comment | added | Joonas Ilmavirta | I'm not exactly sure what you mean by a dynamical interpretation, but my first intuition is to consider the Hamiltonian flow associated with the operator. The flow associated with the Hamiltonian is the geodesic flow (a free particle), and the extra term adds a force described by the vector field to the equations of motion. I am not familiar enough with the Fredholm index in this context to be able to say anything meaningful about it. | |
| Oct 2, 2014 at 22:31 | history | edited | Ali Taghavi |
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| Oct 2, 2014 at 22:30 | comment | added | Ali Taghavi | @JoonasIlmavirta do you think there are some dynamical interpretation for the fredholm index of the elliptic operator which you proposed? What about if the laplacian correspond to a metric which has some compatibility with $X$? By compatibility I mean some situation like thihttp://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsgeodesiable-flows: | |
| Oct 2, 2014 at 22:25 | comment | added | Ali Taghavi | ...in mathoverflow.net/questions/164059/… | |
| Oct 2, 2014 at 22:25 | comment | added | Ali Taghavi | @JoonasIlmavirta Ah! I realy thank you very much for your comment. As you said $\Delta +D_{X}$ is a an elliptic operator.Now my next question is "what is the dynamical interpretation for the fredholm index of this ellitic operator". My main motivation: The codimension of the range of derivational operator is an upper bound for the number of limit cycles of $X$. But two difficulities: D is not elliptic the second $D$ is not fredholm in the case of existence of a limit cycle surrounding a non resonance singularities. Thanks again for your interesting comment. I wrote a related motivation in | |
| Oct 2, 2014 at 22:13 | comment | added | Joonas Ilmavirta | The Laplacian plus the operator you mention is an elliptic operator associated with the vector field. Do you want the operator to be of first order or to depend linearly on the vector field? Any additional assumptions would help. | |
| Oct 2, 2014 at 22:06 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 2, 2014 at 21:58 | history | asked | Ali Taghavi | CC BY-SA 3.0 |