Hilbert 16th Problem As A PDE Problem As A Physical Problem As A (Spectral) geometric Problem,As A Foliation Problem

Dear all who are interested in the second part of the Hilbert 16th Problem,or intersted in one of the subjects of the following area which is apparently unrelated to the second part of the Hilbert 16th:
are you agree to establish a special subforum on "Hilbert 16th problem,new interpretations"
??

Thanks
Ali Taghavi (alitghv@xxxxxxxxx)
> Dear Researchers
> Here Is The Letters Which I wrote 1 year ago to
> various specialist
> I Ask You To Review These Materials And Please Give
> Some Comments.
> My Recent Progress On unusuall aproach in Hilbert
> 16th problem is :
> <a
> href="http://www.arxiv.org/abs/math.DS/0408037";>http:/
> /www.arxiv.org/abs/math.DS/0408037</a>
>
> thanks
> Ali Taghavi
>
> *************************************
> * HILBERT 16TH PROBLEM *
> *************************************
>
> Can One Imagine a Possible Relation Between
> Hilbert 16th Problem and
> PDE,QM,NCG...(Or Other Apparently nonrelated
> area!)?:
>
> During 1900_2000 people Looked at Limit Cycles Just
> As Isolated Zero's
> of "Poincare map -Identity"
> What Other New Approach Could Be Considerd For
> Hilbert 16th Problem:
> 1)A PDE:Can One Solve Following analytic PDE on The
> Plane(Globally),If
> not
> How Many Obstruction (curve) Exist For This Global
> nonconstant Solutions:
> Let P and Q Be Polynomials Of Degree n Solve
> PDE:PU-x+QU-y=0 (Unknown
> nonconstant analytic U).I Have No Background In
> Atiyah Singer Index
> Theory But I Learn From
> Lecture
>
> "http://www.abelprisen.no/nedlastning/2004/popular_eng
> lish_2004.pdf" That
> This Theorem Deals with PDE,What Is An Appropriate
> Index
> Phenomena In Hilbert 16th Problem???
> 2)NCG:Assigning A C*-Algebra To A Given Foliation(An
> Algebraic Vec. Field
> On R^2 Define A Foliation,however This is a low
> Dimewnsional Case,But We
> Can Immerse It In Higher Dimension:Thus I Reformulate
> This Problem As
> Follow:
> How Many Compact leaves with nontrivial Holonomy
> Exist For Following
> integrable 2 dim foliation in R^4:
> Foliation Tangent To x'=P(x,y) y'=Q(x,y) z'=0 w'=0
> and x'=0 y'=0
> z'=P(z,w) w'=Q(z,w)
> Determine Uniform Upper Bound For The Numbewr Of Such
> Leaves In Terms Of
> Degree Of P and Q?(MY FORMER ADVISOR,Professor
> Siavash shahshahani,
> ENCOURAGED ME TO LEARN NCG FOR A POSSIBLE RELATION
> WITH HILBERT 16TH)
> 3) An Algebraic Vec. Field On The Plane Define A Flow
> On R^2 and then
> Naturaly a flow On C^Inf(R^2):What Is The
> NONcommutative Version Of
> Hilbert 16th Problem?
> 4)KAM theory: a finit parameter family of planar vec.
> field is actually a
> single vec. field inhigher Dimension(For Example In 4
> Dimension):In Some
> Case,Perhaps,Existence Of Infinit Number Of Invariant
> Tori Implies :"H(n)
> =Infinity for Some n"!
> 4)Quantum Theory:(Please See Glazek's paper"Limit
> Cycles In Quantum Theories" and Cetto's and de la
> pena's Paper "Is Quantum Mechanics A Limit CXycle
> Theory in Fundamental Problem In Quantum physics
> Kluwer Publ.1995
> PDF)
>
> 5)Riemannian Geometry:Assigining An appropriate
> Riemanian Metric That
> Solution Of A Given Algebraic Vec. Field Would Be
> Geodesic,Now looking
> at Curvature sign(The Story Of Curvature)
>
> For More Information on Hilbert 16th Problem Please
> Review My
> Note "Concerning Hilbert 16th Problem WO PDF,and My
>
> Questions "Hilbertlimitcycle16,pdf.)
>
> I Wait Hearing Your Opinion About Hilbert 16th
> Problem
>
>
> Best Regards
>
> Ali Taghavi
> PS:Can We Equipe The Phase Space Of Lienard System To
> A Riemanian Metric
> (Out Of Singularity) Such That Trajectory Of Vector
> Field Be Geodesics,
> Now We Can Try To Hear the Shap Of Such DRUM(Or
> Counting The Number Of
> Closed Geodesic)
>
> Regarding A Possible Relation Between Hilbert 16th
> Problem And NCG Please Review My Followiung
> Conversation To Professor Masoud Khalkhali:
> >From: Masoud Khalkhali <masoud@xxxxxx>
> >To: Ali Taghavi <taghavi@xxxxxx>
> >Date: 01/08/2004 10:00 AM
> >Subject: Re: non commutative geometry Quantum
> Mechanics and Limit Cycles
>
> ------------------------------------------------------
> --------------------------
>
>
> Dear Mr. Taghavi,
> Since these foliations have singularities (not
> foliations properly
> speaking!) one perhaps should use a variation of
> foliation C*-algebra
> (may be this is done by X. Wang alreday). I would
> suggest you look at A.
> Conne's book and see what he has been able to prove
> using operator
> algebra techniques. I vaguely remember some foliation
> people like S.
> Hurder were quite interested in his works so might be
> a good idea to
> look at his papers as well. Of course some L^2 betti
> numbers which are
> real valued can be defined with these techniques as
> well, but your
> problem is in a different direction. I have no idea
> about this map
> that you mention below . P.S. Is this Hilbert problem
> related to a
> conjecture of Poicare about central vector fileds ?
> (I guess, all
> solutions are closed locally at least)
> Sincerely
> Masoud Khalkhali
>
> Ali Taghavi wrote:
>
> >Dear Professor Khalkhali
> >Thank You Very Much For Your E-mail
> > Formulation of Hilbert 16th problem is not so
> complicated and no
> >background is required:
> >This Problem asks about the number of isolated
> closed leaf of foliation
> >correspond to a singular algebraic foliation of
> Plane with a polynomial 1_form:
> >Pdy-Qdx=0
> >(A Limit Cycle is a closed orbit with nontrivial
> monodoromy map.It Is
> >WellKnown That A Given polynomial 1_ form has
> only a finit number of
> >Limit Cycles But A Uniform Upper Bound for the
> number of such isolated
> >cycles Depending on Degree of P And Q is
> >Requreid,[open since 1900))
> >How ,study of C*_Algebra of Such Foliation Could
> Help to Working In
> >Hilbert 16th.I Wait Hearing Your more Clear
> Advise?
> >Another Question:
> >Whether The Following Map Is Surjective:
> >let X be An Analytic Vector Field On S^2 .Map X
> to Dimension Of its
> >Centralizer: Dimension Of Lie Algebra{All
> analytic Y that [X,Y]=0}
> >May You Review TheAttachment Note And InformMe Of
> Your Opinion about It
> >Best Regards
> >Ali Taghavi
> >
> >
> >-----Original Message-----
> >From: masoud@xxxxxx
> >To: Ali Taghavi <taghavi@xxxxxx>
> >Date: Sat, 3 Jan 2004 12:20:47 -0500
> >Subject: Re: non commutative geometry Quantum
> Mechanics and Limit Cycles
> >
> >
> >
> >>Dear Mr. Taghavi,
> >>It would be impossible to say there is no
> relation since there may be
> >>one
> >>between any two fields! On the other hand I'm
> not sure if I'm competent
> >>enough
> >>in dynamical systems to see a relation right
> now. In general it is a
> >>very good
> >>idea to approach old hard problems from a new
> point of view, in this
> >>case NCG.
> >>This field has helped a lot of other areas so
> may be that's the case
> >>here too!
> >>Sincerely
> >>Masoud Khalkhali
> >>
> >>Quoting Ali Taghavi <taghavi@xxxxxx>:
> >>
> >>
> >>
> >>>Dear Professor Khalkhali
> >>>My PHD thesis problem was "hilbert 16th
> problem"
> >>>But My Advisor encouraged me to take a
> course in Non commutative
> >>>Geometry!
> >>>what relation could be exist(Between
> these subjects)
> >>>Sincerely
> >>>
> >>>Ali Taghavi
> >>>
>
>
>
.

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