Wikipedia

Linas

Joined 30 December 2004
This is an old revision of this page, as edited by Pdn~enwiki (talk | contribs) at 14:34, 21 June 2005 (Thanks but probably "no"). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Linas is a common given name among people of Lithuanian descent. Lithuanian children are traditionally given one pagan name and one Christian name; thus Linas is the given pagan name, in honor of Linai, the name of the flax or linseed plant, from which linen cloth is woven. Traditional Lithuanian folk clothing is made primarily of woven linen cloth. The Indo-European root lin- also appears in the words linseed oil and linoleum.

User:Linas runs a personal web site at http://www.linas.org/

User:Linas is currently employed by IBM as a hacker working on the Linux kernel for PowerPC-based mainframes. The Linux on the PowerPC wiki is a good place to find out more about IBM Linux mainframes and systems.

What am I doing?

I'm dabbling in quantum chaos, trying to understand, well, really basic things about quantum mechanics that should be obvious, but are not. Things like wave function collapse, and all that jazz. At the moment, I'm trying to figure out what a wave function is. Well, OK, I was dealt a setback on that point. I'm trying to figure out what a function is. Actually, turns out I'm deeply confused about what a number actually is. Specifically, a real number. As in, "the real numbers". I vaguely understand what a rational number is. Sometimes. On a good day. Problem is, I was brainwashed in first grade into thinking that I knew what an integer was; only recently have I made strides in resolving the deep misconceptions about integers that were planted in my young, pliable mind.


It has been a big, backwards journey in my eductation; once upon a time, I studied quantum field theory and supersymmetry in grad school, and actually wrote a thesis on the confinment of quarks in the nucleon. But that was before I found out that I didn't know what numbers were.

Falling out of my chair

Integers hold many deep, dark secrets. Dare I say, maybe even the secrets of the universe? I read a paper on string theory the other day that equated the strong coupling coefficient to 3=22-1 (plus corrections), the weak coupling coefficient to 7=23-1 (plus corrections), the fine structure constant to 127=27-1 (plus corrections on the order of ten percent) and planck's constant to 2127-1. The careful reader will note these are a series of Mersenne primes. Well, I fell out of my chair at that point. Of course, its just numerology, but it rattles the brain-pan, nonetheless.


Another article I read recently pointed out that the charge of many simple Lie algebras, when used as generators of free groups, generate the disjoint cosets that underly the Banach-Tarski paradox. In particular, SU(3) is a simple Lie algebra; its charges are fractional, and thus the problem of quark confinement may be a manifestation of Banach-Tarski paradox. (Oh, I didn't understand the article, and it may well have been not even wrong, but ...) That should make anybody fall out of their chair.

My KAM conjecture

Recently, I have noticed that dozens of different kinds of fractals all have the modular group symmetry; I have a web page, http://www.linas.org/math/sl2z.html exploring this relation. As I've been studying the modular group, and modular forms, I am suddenly struck by the fact that the KAM torus also probably has the modular group symmetry. How can this be? Well we have this suspicious link: the Jacobian elliptic functions and elliptic integrals occur naturally in the classical equations of motion of the pendulum. But the elliptic functions also have a close relationship to modular forms, which have modular group symmetry. But the KAM torus is nothing more (and nothing less) than the study of the chaotic dynamics of the perturbed pendulum (well, planetary orbits are elliptical too). So this is my very own fall-out-of-chair conjecture: that the modular group symmetry extends not only to fractals, but also to dynamical systems, starting first and foremost with the KAM torus.

If that didn't make things go bump, think of it this way: the modular group is usually defined in terms ot the upper half-plane, which, ohh, by the way, has this hyperbolic poincare metric on it. That means that geodesics will have positive Lyapunov exponents. But the upper half-plane can also be thought of as a Riemann surface, which, oh, by the way is a Kahler manifold, which is a symplectic manifold. But, duhh, this is what Hamiltonian dynamics is all about: its the expression of the symplectic group. So why, exactly, are we surprised that dyanmical systems are chaotic? More precisely, why is it that we don't see the modular group at every turn?

The Linas Riemann conjecture

I've also been working on a conjecture that generalizes the Riemann hypothesis for dynamical systems. The conjecture starts by noticing that almost all of the discrete eigenvalues of the transfer operators of iterated functions are less than one. This essentially means that almost all iterated function systems have dynamics that consist mostly of decaying eigenstates. This is a fancy variant of the Lefschetz fixed-point theorem. Now we make the leap: the other place where we see decaying eigenstates is in quantum mechanics. Here, one establishes a simple model for some physical system, and then perturbs the system to get closer to the 'true physics'. The perturbations mean that the eigenstates of the simple system are not the true eigenstates of "real" system: the "true" eigenstates are mixtures, and so it appears that states "decay". But note also that the "true" system is invariably chaotic: for example, the "true" planetary orbits, the KAM torus, billiards, etc.

So my hypothesis is this: the true eigenstates of chaotic quantum systems are associated with the zeros of some zeta-type function for that dynamical system, and specifically, those zeros are organized in a straight line up the imaginary axis. If one tries to study the Hamiltonian dynamics of this system by considering discrete time intervals, one gets an iterated function system with decaying eigenstates. By focusing on the iteration, one is essentially focusing all attention on the "perturbations" and thus, will find only decaying eigenstates for the iterated map. However, these decaying eigenstates are really only the "off-diagonal" entries of the true Hamiltonian. The reason that number-theoretic things like Dirichlet characters and L-functions enter the picture is because iterated functions have periodic or almost-periodic orbits, and periodicity is naturally described in terms of the cyclic group. The Riemann zeta itself is associated, I think, with the (inverted ??) harmonic oscillator, which I think connects throught the Gauss-Kuzmin-Wirsing operator (GKW operator). The line of reasoning I'm currently pursuing is is: harmonic-oscillator implies the pendulum implies the elliptic functions implies modular forms implies modular group implies de Rham curve which is the dyadic iteration of two contraction mappings and is continuous, non-differentiable, and a generalization of both the Koch snowflake curve and the Minkowski question mark function, which both have modular group symmetry and tie to GKW, which is a transfer operator that has direct ties to the Riemann zeta. Oh, and Farey numbers show up in there. As does Pell's equation, which has, dohh, a deep relationship to the Riemann hypothesis as well, and whose solutions form a semigroup of the modular group. What a coincidence... wherever you look, the modular group is always a step or two away from the Riemann zeta. Why is that? I don't think its a coincidence.

Based on the Artin conjecture I'm wildly guessing that there are symplectic manifolds whose geodesics have a self-similarity that is best described by a non-abelian symmetry. That symplectic dynamics is essentially hyperbolic dynamics is essentially chaotic dynamics is essentially why the Riemann zeta looks so chaotic.

Or let me rephrase that in another way: a tweak of the Riemann zeta function will yield a modular form, but precisly how is opaque to me right now. However, once this established, it will link the Riemann zeta to Kahler manifolds, and thus to the Hamiltonian that Sir Michael Berry conjectures for the zeros.

Quantizing Riemannian manifolds

The equations of motion for the geodesics of a Riemannian manifold are given by solving the Hamiltonian for the cotangent bundle of the Riemannian manifold. What I'd like to know: how does one quantize this classical system? What is the quantum Hamiltonian corresponding to any given Riemann manifold? What is the spectrum of a given Riemann surface? Do Riemann surfaces correspond one-to-one with Dirichlet L-functions, or is that my imagination? If they don't, then why not?

Quantum madness

Surely the fact that the 3+1 Minkowski spacetime that we live in happens to be hyperbolic is somehow deeply at the root of quantum mechanics. Why would I think this? Well, Hamiltonian dynamics on a hyperbolic manifold is chaotic, as a rule. The hint of the connection comes from the exactly-solvable models in two dimensions. Here, we see fuchsian groups involved in fractals, we see modular forms that have semi-fractal shapes, and we have theta functions that represent the Heisenberg group. Yet the Heisenberg group is in a certain way all about quantum mechanics. And theta functions also play yet another role in the Hurwitz zeta function. Sooo....

The Heisenberg group is an example of a sub-Riemannian manifold. These occur during the investigation of geodesic bundles. But its also the Hamiltonian flow of the classical harmonic oscillator. Hmm...

Theory of Everything

Problem with current TOE's is that they have quantum mechanics built in as an assumption, rather than as an outcome. Yet the failure to fully understand the Riemann hypothesis seems to be give lie to such an approach. There is some basic quantization that ties together low-dimensional manifolds, simple groups, symmetric spaces and the like; all passing through the portal of Dirichlet L-functions; it seems to me that the solution to the Riemann hypothesis will be the Rosetta stone that will enable a true TOE to emerge.

Big Boom Number Theory

Given all the above, I fully expect a big boom in number theory in the next few decades, which will unify the treatment of previously disparate work. The stuff going on with polylogarithm ladders can't possibly be a lone accident, as compared with what's going on with partition function (number theory). There is just too much in common between the various "coincidences" there, and other crazy things like Indra's pearls. This "boom" will give a new understanding of Riemann surfaces and the relationship to geometric quantization. Or you can choose to believe I'm crazy, because I have no evidence for this.

Maybe I am a crank. Who knows. I find it remarkable that simple things like the taking of a derivative of some plain-old boring function leads to structures like the Faà di Bruno's formula which has absolutely remarkable number theory lurking below its surface. In high-school calculus, they don't tell you that number theory and Taylor series can be joined in this way. Maybe my teachers didn'tknow. After all, the fractals in Newton's method were only recently observed (in the last few decades).

More Ah ha! moments

I note that the study of umbral calculus and in general sheffer sequences and binomial type is reminiscent of the study irreducible polynomials over finite fields and thus has a certain resemblance ot the goings -on of elliptic curve cryptography.

ToDo List

Need articles for: quantum pointer state einselection

Resolve inaccuracy in: Fuchsian model

Finish edits before I forget/loose interest: Lagrangian foliation- moyal product - geometric quantization

Clarify per talk page: Vandermonde determinant

Read these articles: Analysis of flows - Field of sets - BRST - Complex multiplication - Abelian variety - Prym variety

Misunderstanding things

I've been trying to document my misunderstanding of numbers and math in general at an external website, titled The Modular Group Symmetries of Fractals. During this dabbling, I've started or made major revisions to the following articles (in historical order):

Purdy pictures

I'm visual in my thinking; I find 'pure algebra' hard. So I make pictures and post them on the net at my Art gallery. I've posted some of these pictures here too, with the help of the Wikipedia:picture tutorial.

File:Q-euler.jpeg
Euler pentagon number phi
 
invariant g_3 real part
 
invarient g_3 image part
 
Modular discriminant
 
Klein's J-invariant, real part
 
Klein's J-invariant, phase
 
Klein's J-invariant, modulus
 
Klein's J-invariant, on the Poincare disk


Category sweeps

I've recategorized maybe 500 articles on WP, and created more than a dozen categories. Major category work and cleanup includes:

Non-math entries

Thanks

How nice of you to invite me into the physics project. I already contributed a spelling correction (Porject->Project) and then found another on the KAM page (quisi-quasi). I am honored, and I do love physics and astrophysics, but I am about to move to Colorado, so I'll be pretty inactive except at odd hours or when very bored. Also at 70 yrs, I need to try to fix up a few physics/astrophysics ideas I've piddled with for 2 decades as I did spacecraft engineering. Finally, I am very disappointed with the way the creationists/"intelligent design" crowd is taking over so much of those topics, as well as religion itself, and more and more peripheral things. For example, in my opinion the article on "liquefaction" was written originally (or largely written) by a person Ungtss, who is an implacable follower of creationism, in order to cross link it to some other places and bring in biblical disasters (and maybe better miracles). Thus Ungtss, being countered by a couple of seemingly capable geophysicists, finally retreated turtle-like by reworking his home page (user page) to look less fanatical, but if you go to its history and an earlier version, you will see the true Ungtss. I picked it up as a random link and noticed it omitted gas liquefaction, slighting a bit Olzwski, Wroblewski, Faraday (liquified Chlorine) and poor old Heike Kamerlingh Onnes. When I edited to put in gas liquefaction, a storm arose (not over my making) to some link to fundamentalist items, and eventually the article was split. I think it is important to fight fundamentalist anti-scientific propaganda but I have a feeling that the better parts of Wikipedia are attracting intelligent viewers, some of whom (especially the younger ones) will fall into traps set by the creationists, much as Bilbo went, willy-nilly, down gullies to the Withywindle. I feel it is almost hopeless to fight the battle within Wikipedia, as sensible people (e.g. Joshuaschroeder, Aaarrrgghh, and so on) (and I) are so badly outnumbered. Letters to the editor, press releases, presentations to state, county and congressional committees may offer more hope. I have written to the entire Kansas school board and got 3 favorable responses, about this problem with teaching "intelligent design", but they say they are badly outnumbered. I believe that the way I got into this unending set of communications with Cleon_Teunissen was originally due to a creationist item on a page such as Big Bang or General Relativity, but I don't remember. Teunissen seems to be a reasonable chap and it was not his work that put in creationism, of course. I just checked out some links and ran into a citation to a badly bloated on-line "tutorial" by Kevin Brown, which started the dialogue. In view of my limited time, my age, my fear that the better the physics gets in the Wikipedia, the more it tends to legitimize the absurdities of the creationists (much as the AAAS let in parapsychology), I am trying to resist the temptation to put much time into it. Thanks again. Pdn

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