Best Possible World: Gateway to the Millennium and Eschaton

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Where Stands Functoriality Today? -- Robert Langlands (1997). This is a mildly pessimistic assessment of the field.

Endoscopy and beyond -- Robert Langlands (2000). Robert finds this endeavor more hopeful than functoriality.

[...] my goal is rather to persuade the younger, more vigorous members of the audience, that the path to a successful treatment of the number theoretical problems to whose solution functoriality promises to contribute may very well lie through the trace formula and that they had best be prepared to master it; it is not to discuss in any serious way the arguments that enter into the complete treatment of the formula. That would be a task not for a single lecture but for a year’s seminar.
Dimension of spaces of automorphic forms -- Robert Langlands (1963). His early work on the trace formula.
"Trace formula" & endoscopy (102 hits).
"Trace formula" (6100 hits).
Selberg trace formula and zeta functions -- Matthew Watkins:
"On a compact (i.e. closed and bounded) two-dimensional surface of negative Gaussian curvature the classical motion [of a point mass] takes place on the geodesics, and it is as chaotic and nonintegrable as possible (being Bernoullian). On this surface there exists also a well-defined quantum dynamics, where the Laplace-Beltrami operator (the invariant Laplacian) acts as the Hamiltonian in the Schrödinger equation. A limiting procedure, exactly parallel to the semiclassical tradition in ordinary quantum mechanics, takes the quantum theory into the classical one when the energy E becomes large, E-1/2 playing the role of Planck's constant... If in addition the curvature is constant, this semiclassical transition is even understood in a certain sense, exemplified by the Selberg trace formula. This formula, which was motivated by Riemann's zeta function, relates in an exact way the spectrum of the quantal motion on compact surfaces of negative curvature to the classical motion. The so-resulting mathematical literature has deep connections with manifold theory, automorphic functions, number theory, etc..."
-- A. Voros and N.L. Balasz, "Chaos on the pseudosphere", Physics Reports 143 no. 3, p. 112.
The resemblance between this formula [Selberg's trace] and the Riemann-Weil explicit formula is such that the N{P} correspond to the prime numbers, and the r(i) on the left-hand side (which are directly linked to the Laplacian spectrum of the surface) correspond to the nontrivial zeros of the Riemann zeta function. Consequently, the resemblance is a major source of support for the spectral interpretation of the Riemann zeta function. Put very simply, the spectral interpretation argues that "the nontrivial zeros of the Riemann zeta function are eigenvalues in some setting".
Selberg zeta function and trace formula for the BTZ black hole -- PETER A. PERRY & FLOYD L. WILLIAMS (2001)
A Selberg zeta function is attached to the three-dimensional BTZ black hole, and a trace formula is developed for a general class of test functions. The trace formula differs from those of more standard use in physics in that the black hole has a fundamental domain of infinite hyperbolic volume. Various thermodynamic quantities associated with the black hole are conveniently expressed in terms of the zeta function.
Peter A. Perry:
Research Statement: Inverse spectral geometry is the study of the geometric content of eigenvalues of the Laplacian on a compact surface or the scattering resonances of the Laplacian on a non-compact surface. These problems model more realistic problems such as target identification through the scattering resonances of the target. [...]
Note that inverse (scattering) problems generally have a holographic basis.

At this point I am stuck on the math, trying to find the thread that leads to organicity, so lets bring in the big gun, which in our case is the Big Six: AORSAM.

The connecting theme appears to be the cycle, and recall that it was the e&pi syzygy that triggered the present tangent in an effort to resolve the apparent contradiction between anthropics and the MG. We were then led to consider the cosmogonic role of the cycle.
For a cycle to be cosmogonic it must break its essential circular symmetry. It does so by becoming eccentric, i.e. elliptic. It is precisely the elliptic functions that are at the core of our 'exceptional beauty', or so we are told. I can barely make out the thread.
Before the elliptic functions came the circular functions. These are the good old sine and cosine functions that dominate much of math and physics, particularly on the engineering side. The next step in complexity beyond the circular functions are the elliptic ones, and these dominate the rest of math and physics. Between them, the circle and the ellipse have a firm grip on much of our scientific or material reality.
The specific connection between elliptical functions and exceptional beauty is to be found in the modular form:
Modular forms satisfy rather spectacular and special properties resulting from their surprising array of internal symmetries. Hecke discovered an amazing connection between each modular form and a corresponding Dirichlet L-series. A remarkable connection between rational elliptic curves and modular forms is given by the Taniyama-Shimura conjecture, which states that any rational elliptic curve is a modular form in disguise. This result was the one proved by Andrew Wiles in his celebrated proof of Fermat's last theorem.
The circular functions are periodic. The elliptic functions are doubly periodic. There can be no triply periodic functions, according to Jakob Jacobi. In some significant sense, the ellipse defines the limits of complexity. Let us see, therefore, how direct is the path from the 'simple' ellipse to the monster group. Need we look any further than the j-function:
It turns out that the j-function also is important in the classification theorem for finite simple groups, and that the factors of the orders of the sporadic groups, including the celebrated monster group, are also related.
For the explicit modularity of the j-function see Klein's absolute invariant, J, by which it is defined:
Every rational function of J is a modular function, and every modular function can be expressed as a rational function of J. The Fourier series of J, modulo a constant multiplicative factor, is called the j-function.
And recall that the j-function is also probably related to our syzygy via class field theory, the class numbers and the Heegner numbers:
The Heegner numbers have a number of fascinating connections with amazing results in prime number theory. In particular, the j-function provides stunning connections between e, pi, and the algebraic integers
The lowly ellipse has certainly managed to keep the mathematical pot boiling. Is there a Chef stirring this pot? What sort of seasoning has been added to come up with the recipe for the BPW? I continue to speculate that the answer may lie in our double syzygy: e^pi - pi = 19.999099979.... Here we have a transcendental manifesting as an almost rational to the second degree. In that numerical hiccup must lie a tale. Do I dare plug this into Integer-Relations? 19.999099979189475767266442984669 ==>>. I ask it to find the minimal polynomial and it returns two very large coefficients. I plug it into the Simon Plouffe - P&J Borwein inverse symbolic calculator, and sure enough it recovers the above formula. Then I hit the browse button.... Here are the names that come up: Thue-Morse and Lehmer. In the other direction appears Fibonacci and Bessel. Arbitrarily switching a digit, i.e. 1997909997918947, yields similar results. I see nothing yet that grabs my attention.
What does catch my attention is the browser itself. I had no idea! Imagine what Srinivasa could have done with this baby! I'll wager there's a Srini II out there right now, hot on the grail trail of the organic math. Me? I'll just trip through some tulips.
If I plug in plain pi, the nearest hit is ln (Parking + Madelung). When I try e, the nearest hits are bunch of sums of different integer sequences. I wonder what the story is there? With e^pi, I don't see anything striking, but shouldn't I have gotten another 'Parking' ticket, or is my slide rule busted? It may be due to a shifted decimal. This could be addictive, however. Does it do horoscopes?
There must be an innocent explanation for all those sequences near to e. It must be rounding errors of some kind. Otherwise someone is really missing something. OK, it seems to be hitting on all the nearly sequential sequences of integers, i.e. e = S (n/n!). Perfectly innocent! Well, another Fields Medal gets away. Somehow I think Srini wouldn't have fallen for that one! Just checking.
There ought to be a way to filter through these symbol sequences to pull out significant hits and misses. Has this not been done? Is this the best we can come up with? Surely we are missing something. I'm still wondering if Renyi ever met Madelung before they bumped into each other here at pi? Are they trying to stir up trouble, or are they just passing the time? Is there no cop on this beat?
One might choose to compare the elliptic generator with the Mandelbrot generator. Both are superficially simple and yet they both yield a nearly unbounded complexity, but is there any doubt as to which is more essential to math? The ellipse is in a class of its own. I don't think we fully understand its rationale. The area of an ellipse is pi*a*b. The perimeter is something else. One of its most elegant approximations is pi(a+b)(1+3h/(10+sqrt(4-3h))), due to Srinivasa. He calculated it on the toes of his left foot. Imagine pulling that out of the blue, and this is perhaps the simplest formula to appear is his notebooks.

This has to do with symmetry. A circle has too much, a blob too little. The ellipse rules symmetry, certainly in two dimensions. The symmetry at higher dimensions is a shadow of what happens down here on the flat. Those extra dimensions can only buy so much. Beyond twenty-six, the action seems to fall off. If there is a higher dimensional version of the Leech, it is unable to cast any longer shadow, certainly not onto anything arithmetical, and arithmetic is the ultimate arbiter when it comes to numbers.

There is a similar thing going on with the octonions. These eight dimensional matrices define the limits of matroid complexity and symmetry. What exactly are the lines between elliptic, octonion and Leech symmetry and complexity is not within my present purview, but who is to doubt that those lines have been drawn?
Numbers are what we have been talking about these last few months, and it is Riemann who still owns them, as much as anyone else. The ellipse, octonions and the Leech, if they are to give up their secrets to anyone, it will be to the RH. Will there ever be another RH? Perhaps not. After the RH, math will be puzzle solving. It will be picking up the pieces. What will we know then? I'm suspecting we will know something more Godelian than Godel. We will begin to know the secret lives of numbers, and how we fit into their picture and they into ours.
Can we not proceed on this not so minor assumption? Do we have a choice? Besides, I'll be on a short break here, and need to tidy up around the ranch.
Along with the RH will go the quantum gravity enigma. Both are pushing complexity to what appears to be its logical limits. Both will have to reckon with the Monster Group, and both will have to confront our lowly syzygy. That latter item might even be the toughest challenge.
This intellectual chore of tying up our numero-logical loose ends is a principle prerequisite to the Millennium. Yes, the Omega does cast its shadow upon the Monster, and not the other way around. The numero-logical key to the Omega and the Monster is almost within the present the purview of the Symbol Calculator. We just need to know where to start looking. The syzygy is trying to tell us where and how. I just don't have the musical ear for it. Not here. Not now. Will Srini II have to be up to speed on the Omega? Yes, there will have to be some such eschatological motivation and insight to clear this final hurdle, to put it all into perspective.
It occurs to me that there must be something about Heegner that I'm not getting. It is this insight that will help to resolve our problem of the numerical coincidences. I have to remind myself that the same elliptical j-function that generates the almost integers also generates the Monster. e and pi cast their shadow on the Monster via j. e^i*pi is only the circle. Remove the i and we get the j or the ellipse, more or less. Someone is trying to tell us something here, but they'll have to speak up.

It is clear that e and pi are the most overdetermined numbers in mathematics. The are definitely overworked. What is their provenance? What is their progeny? What is their prodigy? We'll have to find out.

One of their progeny is i. It has been noted that e^i*pi = -1 is the most remarkable formula in mathematics. It is hard to imagine how sqrt(-1) could have existed before e and pi. It is this foundling that has done more to tame mathematics than any other numeral. It figures hugely in Riemann's strategy to tame the primes. Clearly it keeps e and pi on a short leash. Take away that iota and we get the Monster. Their iota-less syzygy at 20 is the shadow of the one with i.
Mixed up in all of this is the j. According to John McKay, Charles Hermite (1822-1901) invented the j in his transcendental solution to the quintic. The pentagon, by the way, yields up the golden ratio as nature's premier numeral. You gotta love Charles' general solution to the quintic in terms of Jacobi's elliptic exponential or theta functions.
Thus do we see the elliptic j figuring in the differential cubic and the algebraic quintic. Nothing beyond the three and the five seem to figure in mathematical complexity. Note that the thetas are 'quasi-doubly periodic'. Just what does that mean? What should the 'quasi' signify in this context? I see the formula, but it doesn't tell me anything.
The Jacobi theta functions satisfy an almost bewilderingly large number of identities involving the four functions, their derivatives, multiples of their arguments, and sums of their arguments.
What are all these quasi-symmetries? From whence do they come; whither do they go? They have to do with the limitation of complexity. Complexity is bounded by the j. Why did Srinivasa have to invent a mock theta? I notice that his name appears in 249 entries at Math World. Gauss: 277. Riemann: 308. Euler: 547. Jacobi: 225. Fermat: 249. Newton: 225. (Not counting Lie, Abel/Abelian, etc.) Who is missing from the >200 club?
The optimization of mathematical complexity must be related the the observer/anthropic problem. If there were a bigger Monster, might we not have been eaten alive? We could not have tamed it. Do not the primes define and contain all complexity? A bigger or lesser Monster might have rendered mathematical physics less amenable or less fertile, respectively. There may be limits to diversity. We may be pushing that envelope. There may be moral considerations.

Modular functions:

Here is a excellent exposition of the j-function, emphasizing its uniqueness:
On the Modular Function and Its Importance for Arithmetic -- Paula Cohen (2000). She refers to a proof by Chudnovsky of the algebraic independence of pi and e^pi. What does this imply about our syzygy? Might we infer that there are other folks out there beating around this same numerical coincidence? They are being mildly stealthy about it.
In the same volume of lecture notes is another interesting paper, Algebraic Dynamics and Transcendental Numbers -- M. Waldschmidt.

week125: John speaks of the '12-ness' of elliptic curves.

Harald Cramer and the distribution of prime numbers -- Andrew Granville (1995).
Unexpected irregularities in the distribution of prime numbers -- Andrew Granville (1994): primes are not purely probabilistic, e.g. mind the gaps. Taming the primes is no simple matter. RH not true? Are the prime numbers independent of each other? Their peculiar gappiness indicates otherwise.
The distribution of the primes seems deeply implicated in the core structure of mathematics. They are part of its alleged organicity. This is how they are related. This relatedness should then be manifested by the failure of the RH. At present my optimal Y2C scenario would be for a theoretical disproof of the RH to be quickly followed by heightened awareness of math's organicity. Via mathematical physics and the Anthropic principle, immaterialism and the BPWH would finally become established, along with their eschatological implications. I can imagine no more subtle or benign way to introduce the Omega. I am suggesting an incompatibility between a fully self-reflexive, organic math and the RH. This might not be necessary, but involving the RH in this historically consummate turn-around would be my concession to a minimalist drama.
A failure of the RH would, at first, strike many mathematicians as a blow against mathematical coherence. I submit that the studying of the manner of that failure, however, will lead them toward a much deeper sense of the coherence between math and the BPW. The breakdown of the RH would be part of the vital loophole that allows the abstractions of math to be 'cognizant' of, or commodious to, life and mind.
The 'unreasonable effectiveness of mathematics' is part of an ontogenetic bootstrap. It is a bootstrap in which the cosmic mind must participate, and we as well, through that mind.
It is indisputable that conjecture is playing an increasingly vital role is the advance of mathematics, particularly in number theory. This may be as much a symptom as a cause of the organicity of math. This reliance on the conjectural approach does render mathematics more unstable and more amenable to changes of philosophy. It is as if mathematics were approaching a critical point antecedent to a phase change in its structure. We just don't realize yet how we will be fitting into this larger picture.

A question that has struck me as significant is the relation between the prime numbers and the Monster Group, or, more succinctly, the connection between the zeta function and the MG. The shortest path between zeta and the MG is probably the 'moonlit' one, i.e. the modular elliptic j-function. Is there an elliptic aspect of the zeta? In 1937 Erich Hecke showed (also see the above) that if the Fourier coefficients of a modular cusp form are plugged into a Dirichlet L-series, then that series can be uniquely factored into an Euler product, i.e. a generalized zeta function. It was mainly Taniyama who connected the elliptic and modular functions, and his result was used by Wiles to prove Fermat. The moral of this moonlit excursion is that mathematical complexity has an ultimate source: be it the primes, the Monster, the j, or something beyond all of them. That source is vital to our 'physical' existence via anthropics.

The primes are not random, as we have seen above with Ulam, Odlyzko, de Bruijn, the 'conspiracy', etc, but presently I am rereading Conrey. Much of the zeta problem comes down to finding elliptic curves with many rational points (see p. 352 in Brian's article).

This is the last gasp before leaving for Ireland this pm. Our connecting flight to Newark was cancelled yesterday due to thunderstorms, which are expected again today.

Brian's article is the best I have seen as an overview of the RH. His main thesis is to demonstrate how convoluted the effort to prove the hypothesis has become. I am still struggling to absorb his cogent comments.
Here are some leads from Brian:
* Don Zagier: I am not finding a list of his publications; however: see his Periods (2001).

* An occasional collaborator is Fernando Rodriguez Villegas: Constructions of plane curves with many points, with Zagier (2000).

* Henryk Iwaniec -- no publication list. Four of his books are listed on Amazon.
On p. 351, note the connection between the Dirichlet and zeta zeros. And on p. 353:
The conjecture of Birch and Swinnerton-Dyer asserts that the multiplicity of the zero of the L-function associated with a given elliptic curve is equal to the rank of the group of rational points on the elliptic curve.
That's about it for now. Sorry to leave in the middle of this unresolved conspiracy.

I would like to attempt a better resolution of the above items, but first I must recall some old business. This is from way back when I first got into 'numerology', from 3-84 to 2-86.

The only specific items that might presently be salvaged from that excursion involve the metric-English conversion. According to my notes it was not until 1959 that the U.S. defined the inch as 2.54 centimeters. I do not know the situation in Britain. [But see 8/12 on the next page.] Once again we are confronted with a possible 'conspiracy' when considering the following additional points:
1. The Measure of All Things: The Seven-Year Odyssey and Hidden Error that Transformed the World -- Ken Alder, 2002.

2. 100/2*1.27 = 39.370... inches/meter. (3937 = 31*127)

3. 2*31*127^2 = 999998

4. c = 299792458 meters/sec.

One has to wonder if the French also had influence from across the channel, and to what end? And for the record, from the same notes:
1. e/phi = 1.67999005...

2. pi/phi^2 = 1.199982...

3. pi*phi^1/2 = 1.9990...^2

4. gamma^2 = 0.9996.../3

and from before
* e^pi - pi = 19.999099979...
If mathematicians are not becoming a bit paranoid, perhaps they are not paying attention. Someone somewhere has been working overtime to square the circle. Was a member of the French Academy privy to this rationale? If there were such a rationale it would explain much of the ancient wisdom as captured in the sacred geometry of the megalithic kind. The megalithic masons were operating on an intuition not unrelated to that of Srinivasa.
The key to these coincidences is still missing. If there were a convenient way to search for similar coincidences on the Internet, we might be close to putting 2 and 2 together, but see numerical coincidences (358 hits):
1. fine structure constant alpha = [31 x (PI)^6 ] / (21^5) (2 parts/million)

2. Cosmic Numerology -- Ivars Peterson

3. Stonehenge- Key to the Ancient World -- Richard Heath

4. Re- Path-Integrating over Differentiable Structures... -- John Baez

5. Numerical Coincidences -- Bill Debuque, et al.
Let's pause here. Bill points out that the e^pi coincidence is related to the fact that (pi + 20)^i ~= 1. I think this was also pointed out on Math World, but the formulas have become illegible (conspiracy?). I just checked this page again and the formulas are now more legible. [9/1 -- It is apparently only my browser that is failing. Finally, I fixed this problem by deleting my Internet cache.] The latter source now yields:
* 163*(pi - e) = 68.9996644...

* 163/ln163 = 1.999999984...^5

The presence of 163 in these coincidences may point to Heegner.

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My diversion into 'numerology' on the previous page was extensive; the most extensive since my two year numerical engagement. Laying on the floor beside my chair are several hundred pages of handwritten notes from that period. I have mixed feelings about that effort and how it relates to the present one. On the one hand there was an almost obsessive desperation to find the 'key', looking for relations between sacred geometry and astronomical data. Yet, I am certainly not the only one to have been bitten by the Pythagorean bug. The line between 'monstrous moonshine' (and see here) and outright numerology is a thin one. Professional mathematicians and physicists are very wary of this occupational hazard. They are not at all comfortable with the Ramanujan phenomenon.
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