ABSTRACTS
OF 
PART I: NUMBER THEORETICAL VISION 
PART II: TGD AND pADIC NUMBERS 
Does Riemann Zeta Code for Generic Coupling Constant Evolution? A general model for the coupling constant evolution is proposed. The analogy of Riemann zeta and fermionic zeta ζ_{F}(s)/ζ_{F}(2s) with complex square root of a partition function natural in Zero Energy Ontology suggests that the the poles of ζ_{F}(ks), k=1/2, correspond to complexified critical temperatures identifiable as inverse of Kähler coupling strength itself having interpretation as inverse of critical temperature. One can actually replace the argument s of ζ_{F} with Möbius transformed argument w= (as+b)/(cs+d) with a,b,c,d real numbers, rationals, or even integers. For α_{K} w= (s+b)/2 is proper choices and gives zeros of ζ(s) and s=2b as poles. The identification α_{K}= α_{U(1)} leads to a prediction for α_{em}, which deviates by .7 per cent from the experimental value at low energies (atomic scale) if the experimental value of the Weinberg angle is used. The conjecture generalizes also to weak, color and gravitational interactions when general Möbius transformation leaving upper halfplane invariant is allowed. One ends up with a general model predicting successfully the entire electroweak coupling constant evolution successfully from the values of fine structure constant at atomic or electron scale and in weak scale 
PART III: MISCELLANEOUS TOPICS 
The chapter represents a comparison of ultrapower fields (loosely surreals, hyperreals, long line) and number fields generated by infinite primes having a physical interpretation in Topological Geometrodynamics. Ultrapower fields are discussed in very physicist friendly manner in the articles of Elemer Rosinger and these articles are taken as a convenient starting point. The physical interpretations and principles proposed by Rosinger are considered against the background provided by TGD. The construction of ultrapower fields is associated with physics using the close analogies with gauge theories, gauge invariance, and with the singularities of classical fields. Nonstandard numbers are compared with the numbers generated by infinite primes and it is found that the construction of infinite primes, integers, and rationals has a close similarity with construction of the generalized scalars. The construction replaces at the lowest level the index set Λ=N of natural numbers with algebraic numbers A, Frechet filter of N with that of A, and R with unit circle S^{1} represented as complex numbers of unit magnitude. At higher levels of the hierarchy generalized possibly infinite and infinitesimal algebraic numbers emerge. This correspondence maps a given set in the dual of Frechet filter of A to a phase factor characterizing infinite rational algebraically so that correspondence is like representation of algebra. The basic difference between two approaches to infinite numbers is that the counterpart of infinitesimals is infinitude of real units with complex number theoretic anatomy: one might loosely say that these real units are exponentials of infinitesimals. 
In this chapter the goal is to find whether the general mathematical structures associated with twistor approach, superstring models and Mtheory could have a generalization or a modification in TGD framework. The contents of the chapter is an outcome of a rather spontaneous process, and represents rather unexpected new insights about TGD resulting as outcome of the comparisons. 1. Infinite primes, Galois groups, algebraic geometry, and TGD In algebraic geometry the notion of variety defined by algebraic equation is very general: all number fields are allowed. One of the challenges is to define the counterparts of homology and cohomology groups for them. The notion of cohomology giving rise also to homology if Poincare duality holds true is central. The number of various cohomology theories has inflated and one of the basic challenges to find a sufficiently general approach allowing to interpret various cohomology theories as variations of the same motive as Grothendieck, who is the pioneer of the field responsible for many of the basic notions and visions, expressed it. Cohomology requires a definition of integral for forms for all number fields. In padic context the lack of wellordering of padic numbers implies difficulties both in homology and cohomology since the notion of boundary does not exist in topological sense. The notion of definite integral is problematic for the same reason. This has led to a proposal of reducing integration to Fourier analysis working for symmetric spaces but requiring algebraic extensions of padic numbers and an appropriate definition of the padic symmetric space. The definition is not unique and the interpretation is in terms of the varying measurement resolution. The notion of infinite has gradually turned out to be more and more important for quantum TGD. Infinite primes, integers, and rationals form a hierarchy completely analogous to a hierarchy of second quantization for a supersymmetric arithmetic quantum field theory. The simplest infinite primes representing elementary particles at given level are in oneone correspondence with manyparticle states of the previous level. More complex infinite primes have interpretation in terms of bound states.
This construction would realize the number theoretical, algebraic geometrical, and topological content in the construction of quantum states in TGD framework in accordance with TGD as almost TQFT philosophy, TGD as infiniteD geometry, and TGD as generalized number theory visions. 2. pAdic integration and cohomology This picture leads also to a proposal how padic integrals could be defined in TGD framework.
3. Floer homology, GromovWitten invariants, and TGD Floer homology defines a generalization of Morse theory allowing to deduce symplectic homology groups by studying Morse theory in loop space of the symplectic manifold. Since the symplectic transformations of the boundary of δ M^{4}_{+/}× CP_{2} define isometry group of WCW, it is very natural to expect that Kähler action defines a generalization of the Floer homology allowing to understand the symplectic aspects of quantum TGD. The hierarchy of Planck constants implied by the onetomany correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates leads naturally to singular coverings of the imbedding space and the resulting symplectic Morse theory could characterize the homology of these coverings. One ends up to a more precise definition of vacuum functional: Kähler action reduces ChernSimons terms (imaginary in Minkowskian regions and real in Euclidian regions) so that it has both phase and real exponent which makes the functional integral welldefined. Both the phase factor and its conjugate must be allowed and the resulting degeneracy of ground state could allow to understand qualitatively the delicacies of CP breaking and its sensitivity to the parameters of the system. The critical points with respect to zero modes correspond to those for Kähler function. The critical points with respect to complex coordinates associated with quantum fluctuating degrees of freedom are not allowed by the positive definiteness of Kähler metric of WCW. One can say that Kähler and Morse functions define the real and imaginary parts of the exponent of vacuum functional. The generalization of Floer homology inspires several new insights. In particular, spacetime surface as hyperquaternionic surface could define the 4D counterpart for pseudoholomorphic 2surfaces in Floer homology. Holomorphic partonic 2surfaces could in turn correspond to the extrema of Kähler function with respect to zero modes and holomorphy would be accompanied by supersymmetry. GromovWitten invariants appear in Floer homology and topological string theories and this inspires the attempt to build an overall view about their role in TGD. Generalization of topological string theories of type A and B to TGD framework is proposed. The TGD counterpart of the mirror symmetry would be the equivalence of formulations of TGD in H=M^{4}× CP_{2} and in CP_{3}× CP_{3} with spacetime surfaces replaced with 6D sphere bundles. 4. Ktheory, branes, and TGD Ktheory and its generalizations play a fundamental role in superstring models and Mtheory since they allow a topological classification of branes. After representing some physical objections against the notion of brane more technical problems of this approach are discussed briefly and it is proposed how TGD allows to overcome these problems. A more precise formulation of the weak form of electricmagnetic duality emerges: the original formulation was not quite correct for spacetime regions with Euclidian signature of the induced metric. The question about possible TGD counterparts of RR and NSNS fields and S, T, and U dualities is discussed. 5. pAdic spacetime sheets as correlates for Boolean cognition pAdic physics is interpreted as physical correlate for cognition. The so called Stone spaces are in oneone correspondence with Boolean algebras and have typically 2adic topologies. A generalization to padic case with the interpretation of p pinary digits as physically representable Boolean statements of a Boolean algebra with 2^{n}>p>p^{n1} statements is encouraged by padic length scale hypothesis. Stone spaces are synonymous with profinite spaces about which both finite and infinite Galois groups represent basic examples. This provides a strong support for the connection between Boolean cognition and padic spacetime physics. The Stone space character of Galois groups suggests also a deep connection between number theory and cognition and some arguments providing support for this vision are discussed. 
Number theoretic Langlands program can be seen as an attempt to unify number theory on one hand and theory of representations of reductive Lie groups on the other hand. So called automorphic functions to which various zeta functions are closely related define the common denominator. Geometric Langlands program tries to achieve a similar conceptual unification in the case of function fields. This program has caught the interest of physicists during last years. TGD can be seen as an attempt to reduce physics to infinitedimensional Kähler geometry and spinor structure of the "world of classical worlds" (WCW). Since TGD ce be regarded also as a generalized number theory, it is difficult to escape the idea that the interaction of Langlands program with TGD could be fruitful. More concretely, TGD leads to a generalization of number concept based on the fusion of reals and various padic number fields and their extensions implying also generalization of manifold concept, which inspires the notion of number theoretic braid crucial for the formulation of quantum TGD. TGD leads also naturally to the notion of infinite primes and rationals. The identification of Clifford algebra of WCW as a hyperfinite factors of type II_{1} in turn inspires further generalization of the notion of imbedding space and the idea that quantum TGD as a whole emerges from number theory. The ensuing generalization of the notion of imbedding space predicts a hierarchy of macroscopic quantum phases characterized by finite subgroups of SU(2) and by quantized Planck constant. All these new elements serve as potential sources of fresh insights. 1. The Galois group for the algebraic closure of rationals as infinite symmetric group? The naive identification of the Galois groups for the algebraic closure of rationals would be as infinite symmetric group S_{∞} consisting of finite permutations of the roots of a polynomial of infinite degree having infinite number of roots. What puts bells ringing is that the corresponding group algebra is nothing but the hyperfinite factor of type II_{1} (HFF). One of the many avatars of this algebra is infinitedimensional Clifford algebra playing key role in Quantum TGD. The projective representations of this algebra can be interpreted as representations of braid algebra B_{∞} meaning a connection with the notion of number theoretical braid. 2. Representations of finite subgroups of S_{∞} as outer automorphisms of HFFs Finitedimensional representations of Gal(Qbar/Q) are crucial for Langlands program. Apart from onedimensional representations complex finitedimensional representations are not possible if S_{∞} identification is accepted (there might exist finitedimensional ladic representations). This suggests that the finitedimensional representations correspond to those for finite Galois groups and result through some kind of spontaneous breaking of S_{∞} symmetry.
3. Correspondence between finite groups and Lie groups The correspondence between finite and Lie group is a basic aspect of Langlands.
4. Could there exist a universal rational function giving rise to the algebraic closure of rationals? One could wonder whether there exists a universal generalized rational function having all units of the algebraic closure of rationals as roots so that S_{∞} would permute these roots. Most naturally it would be a ratio of infinitedegree polynomials. With motivations coming from physics I have proposed that zeros of zeta and also the factors of zeta in product expansion of zeta are algebraic numbers. Complete story might be that nontrivial zeros of Zeta define the closure of rationals. A good candidate for this function is given by (ξ(s)/ξ(1s))× (s1)/s), where ξ(s)= ξ(1s) is the symmetrized variant of ζ function having same zeros. It has zeros of zeta as its zeros and poles and product expansion in terms of ratios (ss_{n})/(1s+s_{n}) converges everywhere. Of course, this might be too simplistic and might give only the algebraic extension involving the roots of unity given by exp(iπ/n). Also products of these functions with shifts in real argument might be considered and one could consider some limiting procedure containing very many factors in the product of shifted ζ functions yielding the universal rational function giving the closure. 5. What does one mean with S_{∞}? There is also the question about the meaning of S_{∞}. The hierarchy of infinite primes suggests that there is entire infinity of infinities in number theoretical sense. Any group can be formally regarded as a permutation group. A possible interpretation would be in terms of algebraic closure of rationals and algebraic closures for an infinite hierarchy of polynomials to which infinite primes can be mapped. The question concerns the interpretation of these higher Galois groups and HFF:s. Could one regard these as local variants of S_{∞} and does this hierarchy give all algebraic groups, in particular algebraic subgroups of Lie groups, as Galois groups so that almost all of group theory would reduce to number theory even at this level? Be it as it may, the expressive power of HFF:s seem to be absolutely marvellous. Together with the notion of infinite rational and generalization of number concept they might unify both mathematics and physics!

Quantum Adeles
Quantum arithmetics provides a possible resolution of a longlasting challenge of finding a mathematical justification for the canonical identification mapping padics to reals playing a key role in TGD  in particular in padic mass calculations. pAdic numbers have padic pinary expansions ∑ a_{n}p^{n} satisfying a_{n}<p. of powers p^{n} to be products of primes p_{1}<p satisfying a_{n}<p for ordinary padic numbers. One could map this expansion to its quantum counterpart by replacing a_{n} with their counterpart and by canonical identification map p→ 1/p the expansion to real number. This definition might be criticized as being essentially equivalent with ordinary padic numbers since one can argue that the map of coefficients a_{n} to their quantum counterparts takes place only in the canonical identification map to reals. One could however modify this recipe. Represent integer n as a product of primes l and allow for l all expansions for which the coefficients a_{n} consist of primes p_{1}<p but give up the condition a_{n}<p. This would give 1tomany correspondence between ordinary padic numbers and their quantum counterparts. It took time to realize that l<p condition might be necessary in which case the quantization in this sense  if present at all  could be associated with the canonical identification map to reals. It would correspond only to the process taking into account finite measurement resolution rather than replacement of padic number field with something new, hopefully a field. At this step one might perhaps allow l>p so that one would obtain several real images under canonical identification. This did not however mean giving up the notion of the idea of generalizing number concept. One can replace integer n with ndimensional Hilbert space and sum + and product × with direct sum ⊕ and tensor product ⊗ and introduce their cooperations, the definition of which is highly nontrivial. This procedure yields also Hilbert space variants of rationals, algebraic numbers, padic number fields, and even complex, quaternionic and octonionic algebraics. Also adeles can be replaced with their Hilbert space counterparts. Even more, one can replace the points of Hilbert spaces with Hilbert spaces and repeat this process, which is very similar to the construction of infinite primes having interpretation in terms of repeated second quantization. This process could be the counterpart for construction of n^{th} order logics and one might speak of Hilbert or quantum mathematics. The construction would also generalize the notion of algebraic holography and provide selfreferential cognitive representation of mathematics. This vision emerged from the connections with generalized Feynman diagrams, braids, and with the hierarchy of Planck constants realized in terms of coverings of the imbedding space. Hilbert space generalization of number concept seems to be extremely well suited for the purposes of TGD. For instance, generalized Feynman diagrams could be identifiable as arithmetic Feynman diagrams describing sequences of arithmetic operations and their cooperations. One could interpret ×_{q} and +_{q} and their coalgebra operations as 3vertices for number theoretical Feynman diagrams describing algebraic identities X=Y having natural interpretation in zero energy ontology. The two vertices have direct counterparts as two kinds of basic topological vertices in quantum TGD (stringy vertices and vertices of Feynman diagrams). The definition of cooperations would characterize quantum dynamics. Physical states would correspond to the Hilbert space states assignable to numbers. One prediction is that all loops can be eliminated from generalized Feynman diagrams and diagrams are in projective sense invariant under permutations of incoming (outgoing legs).

About Absolute Galois Group Absolute Galois Group defined as Galois group of algebraic numbers regarded as extension of rationals is very difficult concept to define. The goal of classical Langlands program is to understand the Galois group of algebraic numbers as algebraic extension of rationals  Absolute Galois Group (AGG)  through its representations. Invertible adeles ideles  define Gl_{1} which can be shown to be isomorphic with the Galois group of maximal Abelian extension of rationals (MAGG) and the Langlands conjecture is that the representations for algebraic groups with matrix elements replaced with adeles provide information about AGG and algebraic geometry. I have asked already earlier whether AGG could act is symmetries of quantum TGD. The basis idea was that AGG could be identified as a permutation group for a braid having infinite number of strands. The notion of quantum adele leads to the interpretation of the analog of Galois group for quantum adeles in terms of permutation groups assignable to finite l braids. One can also assign to infinite primes braid structures and Galois groups have lift to braid groups. Objects known as dessins d'enfant provide a geometric representation for AGG in terms of action on algebraic Riemann surfaces allowing interpretation also as algebraic surfaces in finite fields. This representation would make sense for algebraic partonic 2surfaces, and could be important in the intersection of real and padic worlds assigned with living matter in TGD inspired quantum biology, and would allow to regard the quantum states of living matter as representations of AGG. Adeles would make these representations very concrete by bringing in cognition represented in terms of padics and there is also a generalization to Hilbert adeles.
