### M Theory Lesson 12

One of Devadoss's many papers on the real points of the moduli of punctured spheres is Cellular Structures determined by Polygons and Trees. In this paper he considers how the real case may be extended to the compactification of the complex moduli of $n$ points.

We saw that labels on trees were not necessary for the real case, which is described by associahedra. Devadoss points out that by labelling the leaves of trees with the numbers $1,2, \cdots n$ one can describe the complex moduli. Points on $\mathbb{CP}^1$ determine non-planar trees, because diffeomorphisms of the sphere can permute any two points without collision (recall that collisions were the basis for understanding the real case). This cyclicity is allowed on branch labellings, because the leaves should really be considered identical.

Now consider that we really want 2-dimensional operads to describe complex moduli. As it happens, 2-ordinal trees are exactly like 1-level trees with an integral number of branches on each leaf of the base 1-tree. However, 2-ordinals may have any collection of numbers $a_1, a_2, a_3, \cdots a_m$ of branches. So the simple labelled trees needed for the complex moduli are only a subset of the 2-ordinals. It remains to understand, therefore, in what sense the complex moduli is really a substructure of something, in this higher dimensional setting. Recall that in topos theory thinking, number fields are never to be considered fundamental in themselves.

We saw that labels on trees were not necessary for the real case, which is described by associahedra. Devadoss points out that by labelling the leaves of trees with the numbers $1,2, \cdots n$ one can describe the complex moduli. Points on $\mathbb{CP}^1$ determine non-planar trees, because diffeomorphisms of the sphere can permute any two points without collision (recall that collisions were the basis for understanding the real case). This cyclicity is allowed on branch labellings, because the leaves should really be considered identical.

Now consider that we really want 2-dimensional operads to describe complex moduli. As it happens, 2-ordinal trees are exactly like 1-level trees with an integral number of branches on each leaf of the base 1-tree. However, 2-ordinals may have any collection of numbers $a_1, a_2, a_3, \cdots a_m$ of branches. So the simple labelled trees needed for the complex moduli are only a subset of the 2-ordinals. It remains to understand, therefore, in what sense the complex moduli is really a substructure of something, in this higher dimensional setting. Recall that in topos theory thinking, number fields are never to be considered fundamental in themselves.

## 3 Comments:

Have you heard about www.indyscienceblogs.com? It might be a good opportunity. Your lessons on M-theory deserve a wide audience.

Thanks, Louise, but I'm quite happy to let Nature take its course! I am busy preparing a conference talk for next week.

Hi Kea

I skimmed through the Devadoss' paper and was intrigued by part 4 concerning trees.

In game theory trees are referred to as extensive form games.

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

I have been reading more Game Theory [GT] literature, especially dynamic noncooperative with static, discete and continuous time; noting GT semantics in a prestigious journal.

NATURE - Current issue: Volume 445 Number 7126 pp339-458

1 - Fish can infer social rank by observation alone p429

Logan Grosenick, Tricia S. Clement and Russell D. Fernald

doi:10.1038/nature05511

2 - 'Infotaxis' as a strategy for searching without gradients p406

Massimo Vergassola, Emmanuel Villermaux and Boris I Shraiman

doi:10.1038/nature05464

3 - Comparison of the Hanbury Brownâ€“Twiss effect [HBTE] for bosons and fermions p402

T Jeltes, ..., CI Westbrook, et al

doi:10.1038/nature05513

Please read editor's summary first, then article, if desred.

Fish observations and "infotaxis" [robotics] appear to be consistent with pursuit-evasion [P-E] games.

This is related to biophysics.

HBTE discusses the social life of atoms: HE-3 fermions and HE-4 bosons display bunching and anti-bunching [or attractor and dissipator] behavior.

Perhaps this type of high enegy physics [HEP] may be analyzed via P-E games.

Perhaps before there can be a grand unified theory of physics [both in mechanics and nature], there might be required a grand unified theory of mathematics.

GT appears to possibly encompass all branches of mathematics.

P-E may result in escape, equilibria or capture.

In nuclear physics, P-E may help explain:

escape: radioactice half-life

equilibria: stability of various electron shells about various nuclei

capture: k-capture.

A Tamer Basar slide lecture 'Different Shades of Robustness' on dynamic noncooperative game theory demonstrates how much this mathematics is like that of HEP.

Slide 33 /83 demonstrates lines of singular surfaces that may allow analysis of singularities such as dipoles, sinks and sources,

http://decision.csl.uiuc.edu/~tbasar/Delft05-slides.pdf

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