I am an avid reader of Wikipedia as there is always a lot to be learned. Surfing around I have found the article Yang-Mills existence and mass gap and the corresponding discussion page. Well, someone put out my name but this is not the real matter. A Russian mathematician, Alexander Dynin, presently at Ohio State University, was doing self-promotion on Wikipedia at his paper claiming to have found a solution to the problem. This is not published material, so Wiki Admins promptly removed it and started a discussion. By my side, I tried to make aware the right person for this and presently no answer come out. I cannot say if the proof is correct so far but, coming from a colleague, it would be a real pity not to take a look. Waiting for more significant judgments, I will take some time to read it.

This entry was posted on Thursday, June 17th, 2010 at 9:50 pm and is filed under Applied Mathematics, Physics, QCD, Wikis. You can follow any responses to this entry through the RSS 2.0 feed.
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21 Responses to Yang-Mills millenium problem solved?

As far as I understand the criteria for this ‘prize’ … one is required to provide a full axiomatic system from which the traditional picture may be derived. Without even looking at the paper, I suspect that this has not been done.

I am saying this only because I just had found his paper.
Nevertheless, even though our papers are done differently, in the end we came to practically the same main results.

is going to be published in August issue of IJGMMP
The task now lies in using these results for solving the problem of the quark confinement. My own preliminary investigation, based on the latest reviews on this subject,e.g. read 2 reviews by M.Shifman on arxiv.org , indicates that this problem can indeed be solved. Surely, everybody is entitled for his/her opinion whether such an opportunity is realizable.

Thank you for your comments pointing out your work. I would like to understand better the content of this work to become aware of the consistency of your conclusions with those I and Alexander Dynin obtain. Looking at your paper I can see a sequence of equivalences between models reaching in the end a spectrum the same of the Richardson-Gaudin model. So, please correct me if I am wrong, you conclude that quantum Yang-Mills theory has a mass gap being in the end reducible to such a model that it is a successful model by itself.

Now, the work I have done by solving directly the equations of the Yang-Mills theory, both classical and quantal, and that agrees with the conclusions obtained by different mathematical techniques by Alexander Dynin, shows that Yang-Mills theory reaches a trivial fixed point at infrared and the spectrum is that of free particles (a harmonic oscillator spectrum) with superimposed another harmonic oscillator spectrum as these particles would have a kind of internal structure (to be eventually understood with a higher level theory). Richardson-Gaudin model as discussed by you in your paper does not seem to fit the bill in this way. Please, could you help me to clarify my ideas about as you claimed that you arrive at the same conclusions obtained by Alexander Dynin (that are the same as mine)?

which are nonperturbative in their nature. For some reason or another his ideas were and still in much use in mathematics and much less in physics. Publication of two long papers http://arxiv.org/abs/hep-th/0610149 and

containing Floer’s ideas, to my knowledge, was left unnoticed by physics community, including you and Dynin. Since my results are nonperturbative, they cannot be compared straightforwardly with yours. As for Dynin’s results, my own and his are in agreement in the sense that in both cases we had obtained bosonic spectrum which has a gap between ground and first excited state. This superconducting conclusion is in accord with many-many less rigorous papers advocating the same superconducting picture for Y-M fields. As for my use of Richardson Gaudin model I have to state that exactly this model was used for description of superconductivity (e.g. read references in my paper) and, furthermore, it is obtainable directly from the combinatorics of the scattering data, e.g. see my paper

in accord with philosophy advocated originally by Heisenberg:
That is: given the combinatorics of excitation spectrum, restore the underlying microscopic model. I had written a separate paper about this methodology

In addition, my gap paper deals with gravity-Y-M correspondence in the way very different from that known as ADS-CFT correspondence. Because of this, not only the Y-M but gravity is involved in the gap issues and imposes very stringent constraints on the whole formalism of the Standard Model.

The point is that I and Alexander Dynin stipulated that our approaches take to an identical conclusion about the spectrum of the theory, at least in the infrared. These are different ways to arrive to an identical conclusion that, by itself, permit to learn a lot about Yang-Mills theory.

By your side, what conclusions could be drawn about the spectrum of the theory? While in your paper there is a claim about the gap existence nothing is said about this. I think you should agree that the work by Alexander goes more in-depth with respect to a simple conclusion of a gap being there.

Dear Marco, please, again keep in mind the differences between the perturbative and nonperturbative approaches to the problem. Incidentally, please take a close look at the book by A.Ushveridze “Quazi-exactly solvable models in quantum mechanics”

It has a very detailed discussion about the Richardson-Gaudin model but, in addition to yours and Dynin’s results which this model is certainly capable of reproducing if the parameters are assigned properly, it can be (and was already)used in other instances of nuclear and condensed matter physics where combinatorics is analagous to that observed in high energy physics. This circumstance makes it a universal paradigm for all kinds of exactly and almost exactly solvable quantum mechanical models exibiting gap in their spectrum. Again, please, do not jump off the hook by ignoring Floer’s approach to Y-M on one hand and combinatorics of the problem on another. Everybody and his brother in mathematics uses Floer’s approach to Y-M fields! Last but not the least, while my results can be (and shall be)used to solve quark confinement problem (thus relating it to the gap problem as anticipated) I am having a hard time to see how this result will emerge from your or Dynin’s results. May be you can shed some light on this topic for me.
Warmest regards,
Arkady

The point is quite simple and it appears to me that you are not answering. Dynin uses a non-perturbative technique that, in the proper limit, is able to reproduce my results at least from a qualitative point of view. If you look at Dynin’s paper you will see that the author does a clearcut statement about the spectrum of the theory that I indeed see in the infrared through perturbation techniques. Dynin and mine claims are really strong and imply a deep understanding about the way Yang-Mills theory should behave. This is surely better than just claiming to have a mass gap also by an operational point of view. So, let me state clearly my question about your approach:

What is your conclusion about the full spectrum of the theory? Is this in agreement with lattice data?

Dear Marco,
you are really surprising me! Should I doubt about goodness of Dynin’s work, you would not even know about my existence. I would not make a comment on your blog in the first place. Yes, I know that his work matches with yours and yes, I know about agreement with lattice calculations and so on. But, by doing my work I was not looking at the whole spectrum because I was thinking about the existence of a gap as such. However, by fiddling with parameters of the Richardson-Gaudin model it should be possible to reproduce the whole spectrum. Hence, please, do not try to say whose results are deeper and whose are not…After all, I am having now 165 downloads at the ads database

while staying much longer at the same database. But, I am not making a big deal out of this fact since, I believe, that we all had made a significant progress in the gap problem and, perhaps, as the next step, it is of interest to figure out how the obtained results match each other. This is interesting problem for somebody who is thinking about earning his/her PhD, but NOT immediately for me. I am much more interested currently in the quark confinement problem and I can see already what to do in this regard from my side but I cannot see yet this from yours or Dynin’s side and you are ignoring my invitation to be a little bit more specific in this regard. In my case, the Abelian reduction causes some uses of the already known results to arrive at final destination….I suspect, that the number of downloads indicates that my paper points into the right direction by also helping somehow to solve the problem of quark confinement. This problem holds the key to the complete understanding of hadron physics. I also suspect that the gap problem by itself is interesting and should be considered as solved now but, regrettably, it is mainly of academic interest. E.g. see how many good results people were able to obtain with or without exact solution of this problem

The confinement is much more subtle issue as far as I can understand things at this stage of my development :-). Please, correct me on this.
Warmest regards,
Arkady

I would like to remember that in your initial comment you claimed (with respect to Dynin’s paper)

Nevertheless, even though our papers are done differently, in the end we came to practically the same main results.

So, we can conclude without any doubt that your original statement was false.

I think you are late also for QCD. It is since 2008 that I partecipate to conferences and publish papers showing the low-energy limit of QCD with my approach.

Dear Marco, I had provided my detailed answers to your latest questions. However, my answers were erased. This is not good. Although I hate to do so, but if my reply is not going to be restored, I will be forced to write it again in about a day or so. If again it is going to be erased, I am not going to participate in this noncence. Sorry.
Arkady

As you may know I live in Italy. This implies that in some hours when you are awake I am sleeping instead. My comment area is moderated. This means that I decide what should appear in my blog and what should not before publication. This means that when you write your comment this does not appear immediately but wait until I do not moderate it. So, I am convinced that when you posted your answer, WordPress warned you about this.

I think that before putting out such comments a moment to think and understand is essential. Mostly from a person that showed to be an incompetent about computer science.

is OK then, we are certainly in the different schools of thought. Who is late and for what is late only future might tell. Should your results be useful for others and lead to some next steps, most likely, your paper would be much better read..It is obvious that you do not care about results of Floer, Donaldson, Atiyah and other mathematicians on Y-M theory.
Warmest regards,
Arkady

I’m sorry, maybe I haven’t followed the latest developments about the Yang Mills Theory, unexpected things in life do surface once in a while.
However, I believe that the Yang Mills Theory has been solved by the use and solution of Feynman diagrams. As far as I know, the theory has not been solved through conics algebra.
If the conics solution is the Millennium Problem then Alexander Dynin solution is not what is required. Moreover, if the Yang Mills Millennium problem is not the problem of conical solution, and Alexander Dynin solution is correct as per the problem description, then maybe the conics problem should be added to the Millennium problems!

If blogs serve any purpose, then, those who write comments should try to stick to facts. For instance, it is being said above that Alexander Dynin is a RUSSIAN PHYSICIST. Well, it takes only a second to look up into MATH.SCI NET in order to see that he is respectable mathematician, always was and always will be. If the quality of the rest of information on any science blog is of the same quality, then what is the purpose of blogs: to search for diccidents ;-), to figure out who is your friend? Or what else?
I wonder, why without any noise(“help”) from ANY blogs the gap paper by A. Kholodenko had accumulated 329 downloads as of today, July 24th , 2012

What is the role of blogs in such a process?
People who are sitting in all these respectable places will never ever admit that somebody outside their clan is capable of producing something of value. What else is new about this?! So, with blogs or without, life goes on as usual……And always will go this way….
In regard to conics algebra. Why not if you are so sure about how the Y-M problem should be solved, you provide us with some exact references for a change! As far as the way you are expressing yourself: “the Yang Mills Theory has been solved by the use and solution of Feynman diagrams” What is this? Use of Feynman diagrams upfront is good only if one is expanding theory near trivial vacuum. But why the trivial vacuum is the right initial state? What means “solution of Feynman diagrams”? These are NOT equations, these DIAGRAMS which require some standard calculation, necessarily perturbative (in such a case you are having an answer BEFOREHAND when you are making such a computation). This surely is not proving a thing….

I can understand your frustration. But let me state a couple of facts about what you claims. Firstly, your links are not good as just the one by Dynin provides the correct diagram of downloads. Please, note that my paper to consider in this case is this.Secondly, I fear that the number of downloads is not the right indicator for a scientific publication. What really matters here is the number of citations. In your case this number amounts to 4 that is not such a high number taking also into account a pair of self-citations. This fact is clearly stated at NASA ADS site and, indeed, people at arXiv avoid always to make this number known to authors and for very good reasons.

In this post I called Alexander a physicist but the reason is that, at that time when I became aware of him through Wikipedia, I did not know his role in the community. This post is already quite old and I should update it. Please, be aware that I have a lot of papers reviewed on MathSciNet but this does not qualify myself as a mathematician. I am also a reviewer for this journal but again I do not claim to be a mathematician.

The role of blogs is essential as a vehicle of information for all the community. This blog is read by a lot of my colleagues and also much more authoritative than me. But the dynamics for a published idea to emerge can follow quite strange ways. A blog can be just one of them.

indicating that I am having 36 publications and 65 citations
Microsoft Corp. had made a ranking of mathematicians using as a guide the ratio of citations/publications WITHOUT COLLABORATORS

You are right. I have confused it with Mathematical Review that is indexed there.

I think you are somewhat confused. I am not looking for a competition with you about ranking, h-index and all that. This is really silly. I am talking about the paper you claim is the most downloaded. For this it is important the number of citations as you are claiming a MIllenium Prize I guess. Your paper has just 4 with 2 self-citations. I let you discover about mine.

As far as I understand the criteria for this ‘prize’ … one is required to provide a full axiomatic system from which the traditional picture may be derived. Without even looking at the paper, I suspect that this has not been done.

Impressive paper. It certainly looks legit and comes from someone who does not appear to be cranky. If there are flaws in the proof, they are subtle.

Hmmm … yes, the paper does look very interesting. This will test my memory, lol.

As an alternative to the Dynin paper, I came up with my own (posted much earlier than Dynin’s)

http://adsabs.harvard.edu/abs/2010arXiv1001.0029K

I am saying this only because I just had found his paper.

Nevertheless, even though our papers are done differently, in the end we came to practically the same main results.

For everyone who is interested, I would like to say that my paper

http://adsabs.harvard.edu/abs/2010arXiv1001.0029K

is going to be published in August issue of IJGMMP

The task now lies in using these results for solving the problem of the quark confinement. My own preliminary investigation, based on the latest reviews on this subject,e.g. read 2 reviews by M.Shifman on arxiv.org , indicates that this problem can indeed be solved. Surely, everybody is entitled for his/her opinion whether such an opportunity is realizable.

Dear Arkady,

Thank you for your comments pointing out your work. I would like to understand better the content of this work to become aware of the consistency of your conclusions with those I and Alexander Dynin obtain. Looking at your paper I can see a sequence of equivalences between models reaching in the end a spectrum the same of the Richardson-Gaudin model. So, please correct me if I am wrong, you conclude that quantum Yang-Mills theory has a mass gap being in the end reducible to such a model that it is a successful model by itself.

Now, the work I have done by solving directly the equations of the Yang-Mills theory, both classical and quantal, and that agrees with the conclusions obtained by different mathematical techniques by Alexander Dynin, shows that Yang-Mills theory reaches a trivial fixed point at infrared and the spectrum is that of free particles (a harmonic oscillator spectrum) with superimposed another harmonic oscillator spectrum as these particles would have a kind of internal structure (to be eventually understood with a higher level theory). Richardson-Gaudin model as discussed by you in your paper does not seem to fit the bill in this way. Please, could you help me to clarify my ideas about as you claimed that you arrive at the same conclusions obtained by Alexander Dynin (that are the same as mine)?

Cheers,

Marco

Dear Marco,

the first part of your conclusions is correct. In my work I was driven by ideas of A.Floer

http://en.wikipedia.org/wiki/Andreas_Floer

which are nonperturbative in their nature. For some reason or another his ideas were and still in much use in mathematics and much less in physics. Publication of two long papers

http://arxiv.org/abs/hep-th/0610149 and

http://arxiv.org/abs/0803.3302

containing Floer’s ideas, to my knowledge, was left unnoticed by physics community, including you and Dynin. Since my results are nonperturbative, they cannot be compared straightforwardly with yours. As for Dynin’s results, my own and his are in agreement in the sense that in both cases we had obtained bosonic spectrum which has a gap between ground and first excited state. This superconducting conclusion is in accord with many-many less rigorous papers advocating the same superconducting picture for Y-M fields. As for my use of Richardson Gaudin model I have to state that exactly this model was used for description of superconductivity (e.g. read references in my paper) and, furthermore, it is obtainable directly from the combinatorics of the scattering data, e.g. see my paper

http://arxiv.org/abs/0810.0250

in accord with philosophy advocated originally by Heisenberg:

That is: given the combinatorics of excitation spectrum, restore the underlying microscopic model. I had written a separate paper about this methodology

http://arxiv.org/abs/hep-th/0608117

In addition, my gap paper deals with gravity-Y-M correspondence in the way very different from that known as ADS-CFT correspondence. Because of this, not only the Y-M but gravity is involved in the gap issues and imposes very stringent constraints on the whole formalism of the Standard Model.

Hi Arkady,

The point is that I and Alexander Dynin stipulated that our approaches take to an identical conclusion about the spectrum of the theory, at least in the infrared. These are different ways to arrive to an identical conclusion that, by itself, permit to learn a lot about Yang-Mills theory.

By your side, what conclusions could be drawn about the spectrum of the theory? While in your paper there is a claim about the gap existence nothing is said about this. I think you should agree that the work by Alexander goes more in-depth with respect to a simple conclusion of a gap being there.

Please, could you elucidate about?

Cheers,

Marco

Dear Marco, please, again keep in mind the differences between the perturbative and nonperturbative approaches to the problem. Incidentally, please take a close look at the book by A.Ushveridze “Quazi-exactly solvable models in quantum mechanics”

http://books.google.com/books?id=u4jv1bydQXMC&dq=ushveridze&source=gbs_navlinks_s

It has a very detailed discussion about the Richardson-Gaudin model but, in addition to yours and Dynin’s results which this model is certainly capable of reproducing if the parameters are assigned properly, it can be (and was already)used in other instances of nuclear and condensed matter physics where combinatorics is analagous to that observed in high energy physics. This circumstance makes it a universal paradigm for all kinds of exactly and almost exactly solvable quantum mechanical models exibiting gap in their spectrum. Again, please, do not jump off the hook by ignoring Floer’s approach to Y-M on one hand and combinatorics of the problem on another. Everybody and his brother in mathematics uses Floer’s approach to Y-M fields! Last but not the least, while my results can be (and shall be)used to solve quark confinement problem (thus relating it to the gap problem as anticipated) I am having a hard time to see how this result will emerge from your or Dynin’s results. May be you can shed some light on this topic for me.

Warmest regards,

Arkady

Dear Arkady,

The point is quite simple and it appears to me that you are not answering. Dynin uses a non-perturbative technique that, in the proper limit, is able to reproduce my results at least from a qualitative point of view. If you look at Dynin’s paper you will see that the author does a clearcut statement about the spectrum of the theory that I indeed see in the infrared through perturbation techniques. Dynin and mine claims are really strong and imply a deep understanding about the way Yang-Mills theory should behave. This is surely better than just claiming to have a mass gap also by an operational point of view. So, let me state clearly my question about your approach:

What is your conclusion about the full spectrum of the theory? Is this in agreement with lattice data?

Cheers,

Marco

Dear Marco,

you are really surprising me! Should I doubt about goodness of Dynin’s work, you would not even know about my existence. I would not make a comment on your blog in the first place. Yes, I know that his work matches with yours and yes, I know about agreement with lattice calculations and so on. But, by doing my work I was not looking at the whole spectrum because I was thinking about the existence of a gap as such. However, by fiddling with parameters of the Richardson-Gaudin model it should be possible to reproduce the whole spectrum. Hence, please, do not try to say whose results are deeper and whose are not…After all, I am having now 165 downloads at the ads database

http://adsabs.harvard.edu/abs/2010arXiv1001.0029K

on my gap paper while Dynin’s paper has only 13

http://adsabs.harvard.edu/abs/2009arXiv0903.4727D

while staying much longer at the same database. But, I am not making a big deal out of this fact since, I believe, that we all had made a significant progress in the gap problem and, perhaps, as the next step, it is of interest to figure out how the obtained results match each other. This is interesting problem for somebody who is thinking about earning his/her PhD, but NOT immediately for me. I am much more interested currently in the quark confinement problem and I can see already what to do in this regard from my side but I cannot see yet this from yours or Dynin’s side and you are ignoring my invitation to be a little bit more specific in this regard. In my case, the Abelian reduction causes some uses of the already known results to arrive at final destination….I suspect, that the number of downloads indicates that my paper points into the right direction by also helping somehow to solve the problem of quark confinement. This problem holds the key to the complete understanding of hadron physics. I also suspect that the gap problem by itself is interesting and should be considered as solved now but, regrettably, it is mainly of academic interest. E.g. see how many good results people were able to obtain with or without exact solution of this problem

http://books.google.com/books?id=oglLYvtPyfAC&printsec=frontcover#v=onepage&q&f=false

The confinement is much more subtle issue as far as I can understand things at this stage of my development :-). Please, correct me on this.

Warmest regards,

Arkady

Dear Arkady,

I would like to remember that in your initial comment you claimed (with respect to Dynin’s paper)

So, we can conclude without any doubt that your original statement was false.

I think you are late also for QCD. It is since 2008 that I partecipate to conferences and publish papers showing the low-energy limit of QCD with my approach.

Cheers,

Marco

Dear Marco, I had provided my detailed answers to your latest questions. However, my answers were erased. This is not good. Although I hate to do so, but if my reply is not going to be restored, I will be forced to write it again in about a day or so. If again it is going to be erased, I am not going to participate in this noncence. Sorry.

Arkady

Dear Arkady,

As you may know I live in Italy. This implies that in some hours when you are awake I am sleeping instead. My comment area is moderated. This means that I decide what should appear in my blog and what should not before publication. This means that when you write your comment this does not appear immediately but wait until I do not moderate it. So, I am convinced that when you posted your answer, WordPress warned you about this.

I think that before putting out such comments a moment to think and understand is essential. Mostly from a person that showed to be an incompetent about computer science.

Marco

Yes, this was a computer glitch. Now I know this. No problems. Viva Italia !

Arkady

Dear Marco,

if to have 3 reads on your the most significant paper

http://adsabs.harvard.edu/abs/2010NuPhS.207..196F

is OK then, we are certainly in the different schools of thought. Who is late and for what is late only future might tell. Should your results be useful for others and lead to some next steps, most likely, your paper would be much better read..It is obvious that you do not care about results of Floer, Donaldson, Atiyah and other mathematicians on Y-M theory.

Warmest regards,

Arkady

I’m sorry, maybe I haven’t followed the latest developments about the Yang Mills Theory, unexpected things in life do surface once in a while.

However, I believe that the Yang Mills Theory has been solved by the use and solution of Feynman diagrams. As far as I know, the theory has not been solved through conics algebra.

If the conics solution is the Millennium Problem then Alexander Dynin solution is not what is required. Moreover, if the Yang Mills Millennium problem is not the problem of conical solution, and Alexander Dynin solution is correct as per the problem description, then maybe the conics problem should be added to the Millennium problems!

If blogs serve any purpose, then, those who write comments should try to stick to facts. For instance, it is being said above that Alexander Dynin is a RUSSIAN PHYSICIST. Well, it takes only a second to look up into MATH.SCI NET in order to see that he is respectable mathematician, always was and always will be. If the quality of the rest of information on any science blog is of the same quality, then what is the purpose of blogs: to search for diccidents ;-), to figure out who is your friend? Or what else?

I wonder, why without any noise(“help”) from ANY blogs the gap paper by A. Kholodenko had accumulated 329 downloads as of today, July 24th , 2012

http://adsabs.harvard.edu/cgi-bin/nph-ref_history?refs=AR&bibcode=2011IJGMM..08.1355K

and that by A.Dynin-58

http://adsabs.harvard.edu/cgi-bin/nph-ref_history?refs=AR&bibcode=2009arXiv0903.4727D

and that by M.Frasca-52

http://adsabs.harvard.edu/cgi-bin/nph-ref_history?refs=AR&bibcode=2009MPLA…24.2425F

What is the role of blogs in such a process?

People who are sitting in all these respectable places will never ever admit that somebody outside their clan is capable of producing something of value. What else is new about this?! So, with blogs or without, life goes on as usual……And always will go this way….

In regard to conics algebra. Why not if you are so sure about how the Y-M problem should be solved, you provide us with some exact references for a change! As far as the way you are expressing yourself: “the Yang Mills Theory has been solved by the use and solution of Feynman diagrams” What is this? Use of Feynman diagrams upfront is good only if one is expanding theory near trivial vacuum. But why the trivial vacuum is the right initial state? What means “solution of Feynman diagrams”? These are NOT equations, these DIAGRAMS which require some standard calculation, necessarily perturbative (in such a case you are having an answer BEFOREHAND when you are making such a computation). This surely is not proving a thing….

Dear Arkady,

I can understand your frustration. But let me state a couple of facts about what you claims. Firstly, your links are not good as just the one by Dynin provides the correct diagram of downloads. Please, note that my paper to consider in this case is this.Secondly, I fear that the number of downloads is not the right indicator for a scientific publication. What really matters here is the number of citations. In your case this number amounts to 4 that is not such a high number taking also into account a pair of self-citations. This fact is clearly stated at NASA ADS site and, indeed, people at arXiv avoid always to make this number known to authors and for very good reasons.

In this post I called Alexander a physicist but the reason is that, at that time when I became aware of him through Wikipedia, I did not know his role in the community. This post is already quite old and I should update it. Please, be aware that I have a lot of papers reviewed on MathSciNet but this does not qualify myself as a mathematician. I am also a reviewer for this journal but again I do not claim to be a mathematician.

The role of blogs is essential as a vehicle of information for all the community. This blog is read by a lot of my colleagues and also much more authoritative than me. But the dynamics for a published idea to emerge can follow quite strange ways. A blog can be just one of them.

Marco

Marco, please, make sure that Math Sci.Net is NOT a journal!

http://www.ams.org/mathscinet/

As a member of AMS I am a reviewer for Math.Sci Net for many years.Your record there is

http://www.ams.org/mathscinet/search/author.html?mrauthid=316757

indicates that you are having 25 publications and 6 citations….Mine record is

http://www.ams.org/mathscinet/search/author.html?mrauthid=101120

indicating that I am having 36 publications and 65 citations

Microsoft Corp. had made a ranking of mathematicians using as a guide the ratio of citations/publications WITHOUT COLLABORATORS

http://academic.research.microsoft.com/RankList?entitytype=2&topdomainid=15&subdomainid=8&last=10

I am ranked as 15th from the top for past 10 years Why?

I am letting you to decide and also to find yourself in this list

You are right. I have confused it with Mathematical Review that is indexed there.

I think you are somewhat confused. I am not looking for a competition with you about ranking, h-index and all that. This is really silly. I am talking about the paper you claim is the most downloaded. For this it is important the number of citations as you are claiming a MIllenium Prize I guess. Your paper has just 4 with 2 self-citations. I let you discover about mine.

I will not publish comments by you anymore.