M Theory Lesson 68
Matti Pitkanen now has a post about Farey sequences and the Riemann hypothesis in TGD. The idea that the hypothesis is not provable within standard mathematics appears to be gaining a foothold within physical constructions.
On the other hand, it is possible that the physical axioms could guide a concrete proof within a convenient model, such as the Jordan algebra M Theory, in which U duality is algebraically manifest. But the zeta function itself only enters here with the (operad) algebras associated to moduli integrals. So it is difficult to avoid the higher categorical framework in studying exact (eg. MHV) amplitudes, and this lands us back in the world of post ZF axioms.
After inhabiting this world for some time, it becomes difficult to look at zeta functions any other way. One simply can't help looking at the Selberg axioms and thinking of closure under products, or factorisation, as topos-like axioms, even though these are radically different things. Recall that the interplay of + and x here is thought of as a higher distributive law for monads. This suggests that the Euler relation for zeta functions is about equating invariants based on monads, or rather that the distributivity $+ \times \rightarrow \times +$ is an identity. That is, that the distributivity of complex arithmetic is somehow more responsible for Euler's product relation than the notion of primeness, which is used through the application of the fundamental theorem of arithmetic only after the product has been expanded.
This suggests that the higher dimensional versions of the Riemann zeta function should be thought of as non-commutative, non-associative and even non-distributive L-functions. Ah! So that's why Goncharov likes Shimura varieties. Note that such considerations are necessary for understanding even the values of the Riemann function, since its arguments extend throughout the heirarchy.
Update: Khalkhali has a new post on Determinants and Traces in which he notes: "... Bost and Connes in their paper Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), no. 3, 411--457, right in the beginning show that the above formula (5) gives the Euler product formula for the zeta function ... In fact their paper starts by quantizing the set of prime numbers ... Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality."
Formula (6) is $Tr_{s}(\Lambda A) Tr(SA) = 1$, a relation between trace and supertrace. Hmm. I would like to understand Koszul duality better because it applies to operads, and more generally PROPS.
On the other hand, it is possible that the physical axioms could guide a concrete proof within a convenient model, such as the Jordan algebra M Theory, in which U duality is algebraically manifest. But the zeta function itself only enters here with the (operad) algebras associated to moduli integrals. So it is difficult to avoid the higher categorical framework in studying exact (eg. MHV) amplitudes, and this lands us back in the world of post ZF axioms.
After inhabiting this world for some time, it becomes difficult to look at zeta functions any other way. One simply can't help looking at the Selberg axioms and thinking of closure under products, or factorisation, as topos-like axioms, even though these are radically different things. Recall that the interplay of + and x here is thought of as a higher distributive law for monads. This suggests that the Euler relation for zeta functions is about equating invariants based on monads, or rather that the distributivity $+ \times \rightarrow \times +$ is an identity. That is, that the distributivity of complex arithmetic is somehow more responsible for Euler's product relation than the notion of primeness, which is used through the application of the fundamental theorem of arithmetic only after the product has been expanded.
This suggests that the higher dimensional versions of the Riemann zeta function should be thought of as non-commutative, non-associative and even non-distributive L-functions. Ah! So that's why Goncharov likes Shimura varieties. Note that such considerations are necessary for understanding even the values of the Riemann function, since its arguments extend throughout the heirarchy.
Update: Khalkhali has a new post on Determinants and Traces in which he notes: "... Bost and Connes in their paper Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), no. 3, 411--457, right in the beginning show that the above formula (5) gives the Euler product formula for the zeta function ... In fact their paper starts by quantizing the set of prime numbers ... Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality."
Formula (6) is $Tr_{s}(\Lambda A) Tr(SA) = 1$, a relation between trace and supertrace. Hmm. I would like to understand Koszul duality better because it applies to operads, and more generally PROPS.
posted by Kea | 9:36 AM
7 Comments:
The fundamental variational principle of mathematics would be in spirit of Leibniz: Platonia would be the cognitively best possible world with "laws of Nature".
RH as "law of Nature" would optimize cognitive representations relying on discretization using rationals of Farey sequence on unit inverval and corresponding algebraic phases on the unit circle. These numbers correspond to the first lowest N levels of dark matter hierarchy.
This principle would allow to make educated guesses about the plausibility of unproven conjectures. Axioms could be identified as the collection of most plausible candidates for the "laws of Nature" in Platonia.
Dear Kea,
this is somewhat out of topic and relates more closely to yesterday's posting where you gave a link to very interesting article of Kauffman and Lambropoulou about rational 2-tangles having commutative sum and product allowing to map them to rationals. This raised the question how tangles could be realized in TGD Universe.
Wow Kea, lesson 68! By now you have enough material to write a thick book! Have you considered that ? I would advertise it!
Cheers,
T.
Great idea Tommaso. Maybe she can title it "Operads and Motives in Physics" or something.
By now you have enough material to write a thick book!
Thanks everybody. Yes, a book is a good idea, Tommaso, but I think I would need some kind of stable existence before I took on such a project.
1 - Apparently there is another new word in the Englich language:
blook
"A blook can refer to either an object manufactured to imitate a bound book, an online book published via a blog, or a printed book that contains or is based on content from a blog."
2 - Thanks for the reference in MT_67, Kauffman and Lambropoulou, 'Hard Unknots and Collapsing Tangles'.
I see that Carl Brannen has a blog entry on 'Why does DNA only use 4 nucleotides?' at "Mass". I will make some nucleic acid [NA] comments there.
3 - The reciprocals of the Farey sequence from [1,oo] are also interesting.
kneemo: Maybe she can title it "Operads and Motives in Physics" or something.
I would want to buy "Operads and Motives in Physics for Dummies" :-]
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