arXiv subjects
Publications and source records attributed to Christian Pierre.
The modular forms are revisited from a geometric and an algebraic point of view leading to a geometric interpretation of the weak Maass forms connecting them to the Ramanujan Mock Theta functions and to the cusp forms generated from the Langlands global program.
The curious connection between the spacings of the eigenvalues of random matrices and the corresponding spacings of the non trivial zeros of the Riemann zeta function is analyzed on the basis of the geometric dynamical global program of Langlands whose fundamental structures are shifted quantized conjugacy class representatives of bilinear algebraic semigroups.The considered symmetry behind this phenomenology is the differential bilinear Galois semigroup shifting the product,right by left,of automorphism semigroups of cofunctions and functions on compact transcendental quanta.
The paper revisits concretely the algebraic K-theory in the light of the global program of Langlands by taking into account the new algebraic interpretation of homotopy viewed as deformation(s) of Galois representations given by compactified algebraic groups. More concretely, we introduce higher algebraic bilinear K-theories referring to homotopy and cohomotopy and related to the reducible bilinear global program of Langlands as well as mixed higher bilinear KK-theories related to dynamical geometric bilinear global program of Langlands.
An algebraic extended bilinear Hilbert semispace is proposed as being the natural representation space for the algebras of von Neumann.This bilinear Hilbert semispace has a well defined structure given by the representation space of an algebraic general bilinear semigroup over the product of sets of archimedean completions characterized by increasing degrees.This representation space,decomposing into subbisemimodules according to the pseudounramified or pseudoramified conjugacy classes,is in one-to-one correspondence with the corresponding cuspidal representation according to the Langlands global program.In this context,towers of von Neumann bisemialgebras on the graded bilinear Hilbert semispaces are constructed algebraically which allows to envisage the classification of the factors of von Neumann from an algebraic point of view.
the program of Langlands is studied here on the basis of: a)new concepts of global class field theory related to the explicit construction of global class fields and of reciprocity laws; b)the representations of the reductive algebraic groups GL(n) constituting the n-dimensional representations of the associated global Weil groups; c)a toroidal compactification of the conjugacy classes of these reductive algebraic groups whose analytic representations constitute the cuspidal representations of these groups GL(n) in the context of harmonic analysis. This leads us to build two types of n-dimensional global bilinear correspondences of Langlands by taking into account the irreducibility or the reducibility of the representations of the considered bilinear algebraic semigroups. The major outcome of this global approach is the generation of general algebraic symmetric structures, consisting of double symmetric towers of conjugacy class representatives of algebraic groups,so that the analytic toroidal representations of these conjugacy classes are the cuspidal conjugacy class representatives of these bilinear algebraic semigroups.
All kinds of global correspondences of Langlands are evaluated from the functional representation spaces of the algebraic bilinear semigroups GL2(.x.) with entries in products,right by left,of sets of archimedean increasing completions. Degenerate singularities on these functional representation spaces can give rise,by versal deformations and blowups of these,to one or two new covering functional representation spaces of GL2(.x.) according to the type of considered singularities. The discovered correspondences of Langlands are associated with singular and nonsingular universal GL(2)-structures.
In the basic general frame of the Langlands global program, a local p-adic elliptic semimodule corresponding to a local (left) cuspidal form is constructed from its global equivalent covered by p-th roots. In the same context, global and local bilinear deformations of Galois representations inducing the invariance of their respective residue fields are introduced as well as global and local bilinear quantum deformations leaving invariant the orders of the inertia subgroups. More particularly, the inverse quantum deformation of a closed curve responsible for its splitting directly leads to the Goldbach conjecture.
With the theory of special relativity, time has been linked with space into a four-dimensional space-time from which a basic question must be asked: can space be really transformed into time and vice-versa? The response is affirmative if time has the same structural topological structure as space at the subplanckian quantum level in such a way that a discrete structural quantum time constitutes the time part of the space-time internal vacuum of every elementary particle. It has thus been shown that a quantum time, quantized algebraically according to a lattice of time quanta, really exists and is emergent in the sense that time quanta can be transformed into space quanta and vice-versa. Furthermore,this quantum time, only relevant at the subplanckian scale, is proved to be in one-to-one correspondence with the absolute and relative clock times.
The seven non euclidean geometries of the Thurston's geometrization program are proved to originate naturally from singularization morphisms and versal deformations on euclidean 3-manifolds generated in the frame of the Langlands global program. The Poincare conjecture for a 3-manifold appears as a particular case of this new approach of the Thurston'program.
The Langlands functoriality conjecture envisaged in the bisemialgebra framework is proved to correspond to the nonorthogonal completely reducible cuspidal representations of the bilinear algebraic semigroups.
Based upon new global class field concepts leading to Langlands two-dimensional global correspondences,a modular representation of cusp forms is proposed in terms of global elliptic (bisemi)modules which are (truncated) Fourier series over R.As application,the conjectures of Shimura-Taniyama-Weil,Birch-Swinnerton-Dyer and Riemann are analyzed.
The two quantizations of QFT,as well as the attempt of unifying it with general relativity,lead us to consider that the internal structure of an elementary fermion must be twofold and composed of three embedded internal (bi)structures which are vacuum and mass (physical) bosonic fields decomposing into packets of pairs of strings behaving like harmonic oscillators characterized by integers mu corresponding to normal modes at mu (algebraic) quanta.
The interactions inside the (bisemi)particles are envisaged from two points of view: The first approach, based on the reducible representations of algebraic bilinear semigroups, allows to describe explicitly the interactions between (bisemi)particles by means of gravito-electro-magnetic fields of interaction while the second approach,based on the consideration of (bi)connections, leads to Maxwell extended equations involving the gravitational field.
New fundamental mathematical structures are introduced by the triples (left semistructure,right semistructure,bisemistructure) associated with the classical mathematical structures and such that the bisemistructures,resulting from the reciprocal actions of left semistructures on right semistructures,are composed of bielements which are either diagonal bielements or cross bielements.
A rather complete phenomenology of the singularities is developed according to a new algebraic point of view in the frame of Langlands global correspondences. That is to say,a process of: -singularizations and versal deformations of these, -singularizations and monodromies of these, is envisaged on all the sections of sheaves of differentiable (bi)functions on (bi)linear algebraic (semi)groups constituting the n-dimensional representations of the global Weil groups. To get the searched holomorphic and cuspidal representations,it is necessary to consider: -the resolutions of the singularities and the blowups of the versal deformations; -the resolutions of the singularities in the monodromy cases. Furthermore, the geometry of the versal deformations and of their blowups is studied, as well as the associated dynamics leading to the consideration of singular hyperbolic attractors and of singular strange attractors.
In the perspective of unifying quantum field theories with general relativity,the equations of the internal dynamics of the vacuum and mass structures of a set of interacting particles are proved to be in one-to-one correspondence with the equations of general relativity. This leads us to envisage a high value for the cosmological constant,as expected theoretically.
The main objective consists in endowing the elementary particles with an algebraic space-time structure in the perspective of unifying quantum field theory and general relativity: this is realized in the frame of the Langlands global program based on the infinite dimensional representations of algebraic groups over adele rings. In this context, algebraic quanta, strings and fields of particles are introduced.