Some Chitchat

The biology blogger Dcase complimented me recently on my knowledge of biomathematics. Now, whether talking about the biology or the fancy String mathematics, either way my knowledge is actually very poor. But the point is that we both recognise a direction here, which I allude to in many of my posts. The application of trees, networks or categories to genetics, linguistics, computer science, physics, physiology or whatever else is not merely a coincidental appearance of a new type of calculus. Certainly this is one way to see things, because this combinatorics does open vast new vistas, mathematically speaking. But the biologists are not just talking about modelling systems. They are talking about a unified theory for understanding systems; something they have never had before.

Physicists are quite used to the idea of unifying laws of nature. Ever since the ancient Greeks they have worked with the unreasonable effectiveness of mathematics (to quote Weyl). Most physicists are therefore convinced that a theory of Quantum Gravity (a loose term for something that unifies QFT and GR) exists. Moreover, this theory must be predictive. The idea of a Landscape is outrageous and, since we already have better ideas anyway, one wonders why people persist with such investigations.

A theory of Quantum Gravity will say some radical things. Many physicists are now happy with the idea that spacetime disappears and is in some sense generated by the matter degrees of freedom. Einstein could not, in the end, incorporate a Machian inertia into GR, but we expect Quantum Gravity to be able to achieve this. After all, it only fell over with Einstein’s commitment to a classical differential geometry. However, to believe that any old background independent description of quantum covariance which yields roughly the standard cosmology would be radical enough is, perhaps, to underestimate the meaning of the word radical.

For starters, some of us are now fully convinced that Quantum Gravity will do for biology what QM did for chemistry. Of course, this is an arrogant physicist’s point of view. A biologist might say that the unified theory of biology happens to provide Quantum Gravity as well. Whatever. It’s the same theory.

A little while back we were talking about the number of generations in the Standard Model. It was pointed out that this follows from the orbifold Euler characteristic of the moduli of the six punctured sphere. Actually, the higher n-operads of Batanin highlight the fact that something special happens when one considers the statistics on six objects. Tamarkin showed that for n > 1 the polytopes that are usually considered cannot stabilise moduli. But Batanin’s can! The simplest Tamarkin example is for six points as branches of a two level nine edged tree. Since in this setting 2-operads are used to study points in the real plane, this enters into the correct combinatorics for the six punctured sphere. Look out for a paper on this soon!

5 Responses so far »

  1. 1

    Mahndisa S. Rigmaiden said,

    10 10 06

    I agree with you on the point that a unified framework to describe biology,physics, chemistry, or whatever else resides in categorial theory and other wondrous mathematical toolery! Braid theory seems to pop up just about everywhere I look and lately padic physics seems to pop up as well… I can see things tying together here…And about punctured six spheres, interesting pants that come out of them eh?:)

  2. 2

    Kea said,

    I can see things tying together here…

    It’s hard to know what to say, isn’t it? WOW just doesn’t quite do it.

    And about punctured six spheres, interesting pants that come out of them eh?

    Yeah!! Sort of like a full mountaineering suit: holes for neck, limbs and relieving oneself.

  3. 3

    CarlBrannen said,

    Kea, I uploaded most of chapter 2 in my book on density operators. The subject is primitive idempotents and the reason I’m bringing it up here is because it includes graphs of the idempotent structures of 2×2 and 3×3 matrices:

    I don’t know what these things have to do with the math you’re working on, or mountaineering suits, but thought you might see something. Two idempotents are connected by a vertical path if the lower one is equal to the product of the two.

    The corresponding physics is that the upper state can be treated as a composite state containing the thing below. Another way of saying this is that the projection operator for the upper state can be split into two projection operators, one of which is the lower projection operator.

    Meanwhile, I’m on the tropical island of Curacao, just off of Venezuela. I’m trying to get some laser telecommunications gear running.

  4. 4

    Kea said,

    Carl, great! Thanks. It looks like you’re drawing lattices of idempotents. These appear in the QM of Piron et al. A lattice has a 0 and a 1, an order (<) represented by lines, and operations such as AND and OR. Classical lattices underly classical topos theory: an example is the lattice of open sets of a topological space, where 0=empty and 1=space.

  5. 5

    CarlBrannen said,

    Thanks for the comment. I’ll look up Piron and lattice and see if I can tie the book more closely to the literature.

    It turns out that Curacao is very good for writing boring LaTex. I’ve got another section or two done: Right after the analysis of the idempotent lattice of 3×3 diagonal matrices, I do the idempotent lattice of 3×3 circulant matrices. Of course they are the same. And it is the ciruclant matrices that fit in with the Koide formula.

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