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Patrick Iglesias-Zemmour

Publications and source records attributed to Patrick Iglesias-Zemmour.

15 recordsLinked to original sources
arXiv2026

Geometric Quantization by Paths Part II: The General Case

In Part I, we constructed the prequantum groupoid for connected and simply connected parasymplectic spaces. This second part extends the construction to arbitrary connected parasymplectic diffeological spaces $(X, ω)$: the obstruction to the additivity of the action form on loops is measured by a surfacic cocycle on $π_1(X)$, and a choice of rectifying cochain $c$ yields a prequantum groupoid $T_c$ whose isotropy group is the torus of periods $\mathbf{R}/P_ω$ and which carries a canonical invariant $1$-form of curvature $\operatorname{trg}^*ω- \operatorname{src}^*ω$. The resulting family $\{T_c\}_c$ is an affine space directed by the character group $\mathrm{Hom}(π_1(X), \mathbf{R}/P_ω)$.

math.DG↗
arXiv2026

Does the motor cortex draw on a wire plane?

The two-thirds power law of human motor control ($v \propto κ^{-1/3}$) is geometrically equivalent to constant equi-affine speed. In classical differential geometry, however, the equi-affine metric is not a tensor: it depends on acceleration, which does not transform covariantly under arbitrary coordinate changes. To recover tensorial behavior, one must either restrict the symmetry group to the affine group or introduce an affine connection -- sacrificing full diffeomorphism covariance. This article proposes a different geometric setting. We equip the Euclidean plane with the "wire diffeology', the smooth structure generated by all smooth curves. In this diffeological space, the equi-affine metric becomes a true covariant $3$-tensor under the **full** diffeomorphism group -- no restriction of symmetries, no additional structure required. The construction is motivated by a simple fact: the motor cortex traces curves, not two-dimensional patches. Accordingly, curves are taken as primitive, echoing the motor control literature in which movements are built from a repertoire of elementary building blocks -- motor primitives. The wire plane offers a geometric formalization of this idea in which the two-thirds power law emerges as a fully covariant invariant.

q-bio.NC↗
arXiv2026

Bosonic and Fermionic Singularities in Diffeology

We explore the differential geometry of the quadrant $C_2 = [0,\infty[^2$, equipped with the subset diffeology of $\mathbf{R}^2$. We show a striking dichotomy between differential forms and symmetric tensors. While differential forms on $C_2$ are simply restrictions of smooth forms on $\mathbf{R}^2$ (a "Fermionic" behavior where singularities are hidden), symmetric tensors exhibit a "Bosonic" behavior where singularities accumulate. We prove a decomposition theorem identifying exactly the singular parts: they are purely axial. Surprisingly, the mixed interaction term is forced to be regular by the symmetries of the corner. Finally, we introduce the notion of \emph{singular capacity} to quantify the order of singularity a tensor can support.

math.DG↗
arXiv2026

Geometric Quantization by Paths, Part III: The Metaplectic Anomaly

In the previous parts of this work, we established the Prequantum Groupoid $\mathbf{T}_ω$ as the universal geometric container for quantum mechanics. This approach, which we call the "Geometric Quantization by Paths" (GQbP) framework, replaces the traditional construction of principal bundles with the distillation of the space of histories. In this third part, we cross the "Threshold of Analysis" by constructing the intrinsic observable algebra of the system. The harmonic oscillator is treated here as a validation case, demonstrating that the standard resolution via complex polarization and half-forms is naturally integrated into the GQbP framework. Starting from the complexified groupoid, we define the algebra using symplectic half-densities to ensure a canonical convolution product. We then show that the transition to a polarized representation forces a factorization of these densities. The action of the symmetry group on the polarized half-forms generates a divergence term, which we identify as the source of the zero-point energy of the harmonic oscillator, $E_0 = n\hbar/2$. This derivation resolves the "Metaplectic Anomaly" as a necessary geometric consequence of the intrinsic quantization process.

math-ph↗
arXiv2026

Diffeology and Arithmetic of Irrational Tori

The irrational torus, $\mathrm{T}_α$, originally introduced as a geometric model for quasicrystals, is a foundational object in the theory of diffeology. This paper, after recalling its main algebraic properties, provides a comprehensive analysis of a new geometric invariant for this singular space: the group of flows, $\mathbf{Fl}(\mathrm{T}_α, \mathbf{R})$. This invariant, which is trivial for all manifolds, arises as the core of the obstruction to the de Rham theorem in the diffeological setting. We provide a complete computation and geometric interpretation of this group, proving the isomorphism $\mathbf{Fl}(\mathrm{T}_α, \mathbf{R}) \simeq \mathbf{R} \times \mathrm{coker}(Δ_α)$.

math-ph↗
arXiv2025

Why Diffeology?

Diffeology extends differential geometry to spaces beyond smooth manifolds. This paper explores diffeology's key features and illustrates its utility with examples including singular and quotient spaces, and applications in symplectic geometry and prequantization. Diffeology provides a natural and effective framework for handling complexities from singularities and infinite-dimensional settings.

math.DG↗
arXiv2025

A Non-Trivial $(\mathbf{R},+)$ Principal Bundle over a Contractible Base?

We investigate the properties of a specific quotient space construction, the "warped projection'" $π: W_α\to D_α$, over a smoothly contractible base. In a previous version of this work, it was claimed that this structure constituted a non-trivial principal bundle. We revisit this claim and observe that, due to the existence of flat functions in the smooth category, the projection fails indeed to satisfy the strict condition of local triviality along the plots, required for diffeological bundles. However, the structure remains rich: it possesses a smooth, free, and fiber-transitive group action. Drawing on the concept of vector pseudo-bundles introduced by Christensen and Wu, we propose that this object is best understood as a non-trivial principal pseudo-bundle. This example thus serves to clarify the boundary between strict bundles and generalized pseudo-bundles in the context of singular base spaces.

math.DG↗
arXiv2025

A Conformal Co-Symplectic Structure on the Space of Pseudo-Riemannian Geodesics

The classical construction of the symplectic structure on the space of geodesic trajectories via Hamiltonian reduction fails in the pseudo-Riemannian setting due to a dimensional mismatch created by the null geodesics. This paper proposes a new, unified approach. We first construct the space of all geodesic trajectories $\mathcal{G}_\text{traj}$ directly as the quotient of the space of geodesics curves $\mathcal{G}_\text{curv}$ by the affine reparametrization group. The analysis of the orbits of this group action reveals a key geometric distribution. To describe this distribution globally, we introduce a canonical object, the "conformal co-symplectic structure" $σ$, defined by pushing forward the conformal class of the inverse $ω^{-1}$ of the original symplectic form $ω$. We prove that the image of this structure coincides with the geometric distribution identified previously. On the subspace of time-like and space-like geodesics, this structure is non-degenerate and defines a conformal class of symplectic forms. On the null subspace, its image is a codimension-$1$ distribution that we prove is the canonical contact structure on the space of light rays.

math.DG↗
arXiv2025

Groupoids in Diffeology

This expository paper recounts the development and application of the concept of the diffeological groupoid, from its introduction in 1985 to its use in current research. We demonstrate how this single concept has served as a powerful and unifying tool for defining fundamental structures, analyzing the stratification of complex spaces like orbifolds, building a bridge to noncommutative geometry, and, most recently, forging new approaches to geometric quantization. The paper aims to provide a cohesive narrative of this journey, making explicit certain concepts like the "Klein groupoid" and showcasing the enduring vitality of the diffeological groupoid in modern geometry and physics.

math.DG↗
arXiv2025

On Diffeology of Orbit Spaces

We investigate the correspondence between the geometry of a smooth compact Lie group action on a manifold $\mathrm{M}$ and the intrinsic smooth structure of the orbit space $\mathrm{M}/\mathrm{G}$. While the action on $\mathrm{M}$ is classically organized by the orbit-type stratification, we show this structure fails to predict the intrinsic $Klein\ stratif\!ication$ of the quotient, which partitions the space into the orbits of local diffeomorphisms, thereby classifying the space by its intrinsic singularity types. The correct correspondence, we prove, is governed by a finer partition on $\mathrm{M}$: the $isostabilizer\ decomposition$. We establish a surjective map from the components of this partition to the Klein strata of $\mathrm{M}/\mathrm{G}$. As a corollary, we obtain by pullback a new canonical stratification on $\mathrm{M}$, the $Inverse\ Klein\ Stratif\!ication$, and clarify its relationship with classical structures.

math.DG↗
arXiv2025

The Geometry of Motions, Vol. I: Mechanics as Geometries

This work presents a group-theoretic interpretation of the historical evolution of mechanics, proposing that each fundamental theory of motion corresponds to a distinct geometry in the sense of Felix Klein. The character of each geometry is uniquely determined by its Inertia Group-the group of spacetime transformations that preserves its privileged class of inertial motions. We trace the three major epistemological ruptures in the history of mechanics by translating foundational physical principles into the unambiguous language of group theory. The analysis begins with Aristotelian mechanics, where absolute Space and Time are shown to be homogeneous spaces of the Group of Aristotle, defined by its preservation of rest. The second rupture, driven by the relativity of motion (Bruno, Galileo), leads to the abandonment of absolute Space and the construction of the Galilean Group, which preserves the class of uniform rectilinear motions. The final rupture, precipitated by the crisis in electromagnetism, results in the dissolution of absolute Time and the emergence of the Poincaré Group, preserving affine lines and the Minkowski metric. Central results of this approach are theorems demonstrating, with mathematical certainty, the non-existence of a Galilean-invariant "Space" and a Poincaré-invariant "Time," where invariance is defined with respect to the action of the corresponding inertia group. This geometric framework provides a unified perspective on the transition from classical to modern physics and allows for a rigorous distinction between 'primary' epistemological ruptures, which alter the Inertia Group itself, and 'secondary' ruptures, which introduce new formalisms for the management of dynamics within a stable geometry.

math.HO↗
arXiv2025

Geometric Quantization by Paths -- Part I: The Simply Connected Case

For any connected and simply connected parasymplectic space $(\mathrm{X},ω)$ with group of periods $\mathrm{P}_ω\subsetneq \mathbf{R}$, we construct a prequantum groupoid $\pmb{\mathrm{T}}_ω$ as a diffeological quotient of the space $\mathrm{Paths}(\mathrm{X})$ of paths in $\mathrm{X}$. This object, built from the geometry of the classical system, serves as a unified structure for prequantization. The groupoid $\pmb{\mathrm{T}}_ω$ has $\mathrm{X}$ as its objects, and its space of morphisms $\mathcal{Y}$ carries a canonical left-right invariant $1$-form $\pmbλ$ whose curvature encodes $ω$. A key property is that the isotropy group $\pmb{\mathrm{T}}_{ω,x}$ at any point $x$, naturally arising as a quotient of the space of loops, is isomorphic to the torus of periods $\mathrm{T}_ω= \mathbf{R}/\mathrm{P}_ω$. Furthermore, the entire symmetry group $\mathrm{Diff}(\mathrm{X}, ω)$ acts as faithful automorphisms of $(\pmb{\mathrm{T}}_ω, \pmbλ)$ without central extensions at this level. Built within the framework of diffeology, this construction generalizes classical prequantization by applying to broad classes of spaces, including those with singularities or infinite-dimensional aspects, and by accommodating generalized (e.g., irrational) tori of periods. This paper focuses on the simply connected case; the construction will be extended to general diffeological spaces in a subsequent publication.

math-ph↗
arXiv2020

Quasifolds, Diffeology and Noncommutative Geometry

After embedding the objects quasifolds into the category {Diffeology}, we associate a C*-agebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry (beginning with the today classical example of the irrational torus) which associates a Morita class of C*-algebras with a diffeomorphic class of quasifolds.

math.DG↗
arXiv2013

Primary Spaces, Mackey's Obstruction, and the Generalized Barycentric Decomposition

We call a hamiltonian N-space \emph{primary} if its moment map is onto a single coadjoint orbit. The question has long been open whether such spaces always split as (homogeneous) x (trivial), as an analogy with representation theory might suggest. For instance, Souriau's \emph{barycentric decomposition theorem} asserts just this when N is a Heisenberg group. For general N, we give explicit examples which do not split, and show instead that primary spaces are always flat bundles over the coadjoint orbit. This provides the missing piece for a full "Mackey theory" of hamiltonian G-spaces, where G is an overgroup in which N is normal.

math.SG↗