Dear Michael,

My point was not that there was an isomorphism, but that there might be some sort of relationship.

In addition I am wondering whether dessin d'enfant can be used to look at a general type of problem. You illustrate how Bell's theorem can be realized this way. I am interested in looking at whether this can be used to look at a general class of SLOCC groups. If they can't be found equivalent according to nilpotency on their Cartan subgroups then there is a polynomial invariant under that group which separates them.

Cheers LC

Dear R. J. Tang,

Thanks for you post. The Monstrous Book I am reading has only two letters but infinitely many words that can be arranged in 97239461142009186000 chapters (coset classes). The number of symmetries in the Book is about 10^54. Some theoretical physics expect a better understanding of the physical world thanks to the Book. The physical world is also a human creation however.

Regards,

Michel

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  • Post #58

Dear Lawrence,

This is the type of application we can discuss. Until now, I focused on dessins due to their relationship to quantum geometries and contextuality as in my [12] and [17], now I mentioned in the essay the link to most sporadic groups, there are plenty of other applications, some have to be discovered. Cheers.

Michel

Dear Michel,

Thank you for a very enlightening and enjoyable essay! I really like how you combined rigorous analysis with a very entertaining narrative, and it's quite a thought provoking educational resource. It reminds me of genres along the lines of Flatterland. You presented a very intriguing discussion on Bell's theorem and the moonshine group, and your remark on self referencing was the perfect conclusion. I give this now the highest rating.

Thank you for your very stimulating comments and ideas on my essay (if you haven't yet rated my essay, I ask that you please take a moment to do that). I responded recently to your post and your comment inspires me to particularly revisit time entanglement and incompleteness from the perspective of gate dynamics. (And thank you for the time entanglement of connecting my past and present essays :) ) I look forward to seeing your further work and correspondence in the future. Take care,

Steve Sax

Dear Steven,

I am glad that you found my reading of your excellent essay useful. Our discussion shows how much a 'correct' interpretation of what is going on in a physical experiment depends on the 'correct' maths. I am enthusiastic in your view that Goedel's incompleteness is (at least partially) related to the classical language and that the QM language is helpful on that matter, and similarly for the issue of self-reference.

As a clever physicist, I am sure you are also sensitive to the ongoing work about rubidium and the CNOT gate where entanglement between ligth and atoms has been established, e.g.

http://www.cos.gatech.edu/news/Researchers-Report-First-Entanglement-between-Light-and-an-Optical-Atomic-Coherence

I am also happy that you were not frightened by my (may be too ambitious) topic and I thank you for your high mark. I already rated your essay highly at the time I studied it.

My best regards,

Michel

Dear Vincent,

I have not much to add to what you superbly wrote about my dialogue. When you write "that physics is by definition mathematical, therefore mathematics has to be efficient" I agree and we are quite close to "Science and Hypothesis" that you also quote at several places.

The group concept: absolutely yes in the Grothendieck's expanded meaning. Quantum groupoid (in Wise's essay): not sure and may be this can be falsified.

A remark: why is it so difficult to find the maths of biology (including at the basic level of DNA and proteins) ?

Why is maths so close to physics? Despite so many essays, I don't consider the mystery is lifted, may be the key is in neurophysiology, ant colonies, human sociology. I like Bach-y-Rita's work.

Thanks for your time.

Michel

Michel,

Towards the end of your essay you discuss Rademacher sums, and matters related to the modular functions and integer partitions. There is the results of Brunier and Ono which give an exact computation of the integer partition. The integer partition function is a way of looking at how a set of N quantum states can be partitioned on an event horizon of a black hole. This would then suggest that these categorical constructions are of some utility in that type of problem.

In connection with what I just wrote I have been looking at large N states on an event horizon as SU(N). The Young's tableaux of eigenvalues and 8-fold periodicity reduces this to a simple topological matter. It would seem that this has some categorical correspondence with integer partitions and the Rademacher sum.

The stabilizing the square with Q(sqrt{2}) also suggests some sort of connection with the PR box or Tsirelson bounds.

Cheers LC

Dear Michel,

Wonderful essay! I enjoyed reading it very much, especially the part about Bell's theorem, and my rating reflects that.

Best of luck in the contest.

Mohammed

Dear Michel,

This is a great presentation of a large web of new discoveries and conjectures that are a very exciting part of the Langlands program. For me, your essay is a powerful and comprehensive update about the current status of the direction and objectives of the program, something that isn't really easy to come by. I too think that these similarities are very promising for the future of physics, the tantalizing flavor of their existence being a strong motivation for current and near-future research. Hopefully time will show the true value of modular forms; I am sure they have a great deal to tell us.

Wish you best of luck in your research and in the contest!

Alma

Dear Alma,

I spent a few hours this morning reading your essay and preparing a non trivial comment for you. You will have a very good comment and appreciation from me by the end of the day. May be this is an instance of distant entanglement between brains at least not a pure coincidence. My thoughts during the two hour fast walk I just had was about rivalry between the two hemispheres that you might be call Phys and Math., a kind of quantum superposition that collapses one side or the other depending on context.

Have a good afternoon.

Michel

    Haha, I see! I think we were reading each other's essays at the same time. It is at the very least a nice and amusing coincidence :) I am impressed by and thankful for the time you spent reading my essay and I can't wait to hear your thoughts on it. I will post back to your page to let you know when I've answered.

    Have a good afternoon too!

    Alma

    Dear Michel,

    It brings me great joy that a scientist of your caliber has found things to appreciate in my essay. For your comment alone and it was well worth participating into this contest.

    To me it is natural to speak about modular forms because in my opinion they are the Langlands program, first and foremost. The most famous achievement of the program lies with modular forms. I don't think they are forgotten, but probably very difficult even for skilled mathematicians. Moonshine is not often mentioned today much like the prime gap was not in focus before Zhang made his breakthrough.

    I too find interesting the way humans are capable of working with complex categories instead of exhaustive search to push knowledge further. There are many things we don't understand in detail about how our minds and brains operate and to be honest, it wouldn't be very surprising if quantum effects were found at the scale at which neurons operate. Regarding knowledge itself, another essay in this contest made me think the other day about how new ideas are generated. If knowledge can be modeled as information points in a network, a new idea may be thought of as the minimum number of information points needed to deduce a new piece of the puzzle, as related to the complexity dimension of the concept that needs to be deduced and occurs as a phase transition. Since you considered the cognitive ability in your work, it would be of great interest to me to know your thoughts and your approach to the subject.

    Many many thanks for your words! You made my day!

    I realize I didn't include any contact information that is visible of my profile, so I am adding my personal address here alma.ionescu83@gmail.com

    My warmest regards and my profound appreciation!

    Alma

    *I replied to you on my page and I am posting this on your page as well.

    Bonjour Michel,

    A wonderfully original, enjoyable, well written and possibly important essay. Now resorting to speed reading, and not a mathematician I infer the latter as much from your previous work as from the glimpses gained where I penetrated.

    I consider the essay's high position to be well warranted and my score will support it. More important I hope I can persuade you to read my own, which I'm convinced identifies the confounding trick behind QM and the physical mechanism reproducing the complex plane/spherical co-ordinate complementarity and apparent non-local state reduction. A collaboration paper now expands on that 'quasi-classical' rationale (building from lst years essay) and I sincerely hope after the contest you'll be able to read, comment, advice and hopefully assist mathematically (I think I cited it in the essay).

    I also identify a simple 'new' mathematical formalism in my essay which I hope you'll review.

    Thank you, very well done, best wishes, and best of luck in the final judging.

    Peter

    Dear Peter,

    Thank you so much for your vote of strong confidence.

    I already had a look at your essay prior to this post. I hope to be able to understand what you are doing more easily that some other respectable essays with a strong philosophical taste. Recently I red "John Bell and the Nature of the Quantum World" by Bertlmann himself and this should help me. You can expect my feedback by the end of the contest that is of course very close.

    Best wishes,

    Michel

      Dear Alma,

      You essay and comments are insightful and you seem to be a charming person. I was also interested in Leifer's essay viewing the whole of knowledge as a scale-free network. Your idea of looking at possible phase transitions is developed in his Ref. [13], Sec. G, p. 63 where you can read that "the critical exponents of the phase transition equal the critical exponents of the infinite-dimensional percolation". On my side, in my Neuroquantology paper quant- ph/0403020, I wrote in the abstract "Time perception is shown to depend on the thermodynamics of a quantum algebra of number and phase operators already proposed for quantum computational tasks, and to evolve according to a Hamiltonian mimicking Fechner's law. The mathematics is Bost and Connes quantum model for prime numbers. The picture that emerges is a unique perception state above a critical temperature and plenty of them allowed below, which are parametrized by the symmetry group for the primitive roots of unity." We recently revisited the BC model in the context of Riemann hypothesis and quantum computation http://iopscience.iop.org/1751-8121/labtalk-article/45421. This is a good sign that a good mathematical theory may have many inequivalent applications.

      Today I have in mind to approach the subject of cognition with the tools I am advocating in my essay, it may take a while. I already mentioned that rivalry between the two cerebral hemisphres looks like a qubit.

      Thank you very much Alma for the stimulus you are giving me. My very best regards.

      Michel

      Dear Michel,

      Yours is certainly a very eclectic essay, as you cover a lot of mathematical ground, some of which was completely new to me! The relationship between the Monster group, the j-function and sting theory is quite something, isn't it? As I argue in my essay, I believe that the only way to construct an explanation of reality that is self-contained is to somehow link the whole of mathematics (an infinite ensemble that taken as a whole contains zero information, like Borges' library) and the whole of physical existence. But the ever-puzzling question, "Why is our universe so lawful and so simple", is hard to answer within such a broad hypothesis of universal existence. There's clearly something missing, some process that empowers certain mathematical possibilities (and not others) to become actualized as physical realities. Could the monstrous moonshine conjecture be a hint at some convergent universal properties that physical universes share with particularly rich and fundamental mathematical structures? What if the ultimate answer to Life, the Universe and Everything is not 42, but 24? ;)

      I really like the Frenkel quote, "Mathematics is not about studying boring and useless equations. It is about accessing a new way of thinking and understanding reality at a deeper level."

      Let's push forward into the unknown!

      Cheers,

      Marc

        Dear Michel,

        I found your dialogue to be a dizzyingly fast-paced voyage through many different areas of mathematics and physics which require a high sophistication to follow. It is good to be able to recognize novel connections between different areas of specialization, for then one is more likely to be able to approach a given problem from a new angle. Also the bigger one's toolbox, the greater the likelihood one will have just the right tool at hand when it is needed.

        I had found out about the connection of the monster group to number theory via string theory concepts very recently, and not knowing much about it, it does seem rather unexpected. So perhaps there are other unexpected connections lurking in the back, waiting to be discovered by scientists who can connect fields that seem otherwise to be widely apart.

        I wish you all the best in these endeavors,

        Armin

        PS. Since I saw that your essay had a major emphasis on quantum contextuality and non-locality, I am adding an additional response to your comment in my blog post regarding what I meant by "pseudo-nonlocality" (It is a very different concept from what most opponents of non-locality believe)

        Dear Mark,

        Thank you for your kind comments. I doubt that the moonshine conjecture is useful for approaching MUH or your Maxiverse. It is just an amazing sporadic anomaly of our mathematical universe. But it may be useful to describe parts of our universe because the characters (the Fourier transform of the group) have the same number theoretical structure than modular forms (fonctions defined on the Poincaré upper-half plane). Within these structures, some have a non-local and contextual flavour that I am investigating at the moment.

        Best,

        Michel

        Hi Michel,

        I finally got a chance to read your essay today. Wow! It covered quite a broad spectrum of ideas. I need to digest them a bit. In fact, your essay is definitely one that I will most likely come back to. I think my primary criticism of it is that I didn't feel as if I understood the "big picture" - what was the point you were trying to get across? Or was it really just lots of little points?

        Cheers,

        Ian

        Dear Ian,

        Thank you for going to my essay. I expect that my essay will not be red as a tree but as a surface with punctures, it is non local in some sense. I also hope a big picture is emerging, some points are ongoing research (as those pointed out in the abstract), some technical aspects may not be familiar to quantum physicists (e.g. modular forms and characters).

        If you go to reference [17] just appearing in QIP, you can see that I cite a work of yours on the "order theoretic quantification of contextuality", meanwhile I also found another measure of geometrical contextuality that I am currently working on. A mathematician would say the Langlands program but I stay closer to physics.

        Best,

        Michel