We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence relation, and the observable space is obtained as a recognition quotient. Locality is introduced through a neighborhood system, without assuming any metric or topological structure. A finite local resolution axiom formalizes the fact that any observer can distinguish only finitely many outcomes within a local region. We prove that the induced observable map R¯:CR→E is injective, establishing that observable states are uniquely determined by measurement outcomes with no hidden structure. The framework connects deeply with existing approaches: C*-algebraic quantum theory, information geometry, categorical physics, causal set theory, noncommutative geometry, and topos-theoretic foundations all share the measurement-first philosophy, yet RG provides a unified axiomatic foundation synthesizing these perspectives. Comparative recognizers allow us to define order-type relations based on operational comparison. Under additional assumptions, quantitative notions of distinguishability can be introduced in the form of recognition distances, defined as pseudometrics. Several examples are provided, including threshold recognizers on Rn, discrete lattice models, quantum spin measurements, and an example motivated by Recognition Science. In the last part, we develop the composition of recognizers, proving that composite recognizers refine quotient structures and increase distinguishing power. We introduce symmetries and gauge equivalence, showing that gauge-equivalent configurations are necessarily observationally indistinguishable, though the converse does not hold in general. A significant part of the axiomatic framework and the main constructions are formalized in the Lean 4 proof assistant, providing an independent verification of logical consistency.

Using the framework of non-commutative geometry, specifically the concepts of symplectic structure and the Weyl product, unitary representations of the Galilei group are constructed with the operators [Formula: see text] and [Formula: see text]. This formulation allows the Schrödinger equation to be expressed in phase space, where the quark-antiquark interaction is modeled using the Hulthen potential plus a linear term. However, direct application of this potential to meson spectroscopy presents certain challenges in the context of Moyal quantization. To address these difficulties, a methodological approach based on the Levi-Civita transformation is introduced to derive the mass spectrum of bound states. The results of these calculations are then applied to the study of several heavy mesons, including charmonium [Formula: see text], bottomonium [Formula: see text], and the mixed-flavor meson [Formula: see text], as well as heavy-light mesons such as [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The analysis is restricted to [Formula: see text]-wave states, as our theoretical model is found to be applicable primarily to such configurations. A key insight of our approach is that the Levi-Civita transformation naturally leads to quark confinement within mesons. Furthermore, the calculated meson masses exhibit good agreement with experimental data, reinforcing the validity of the method. The Wigner functions of the studied mesons are also analyzed. It is observed that they exhibit negative values in the first excited state, whereas they remain strictly positive in the ground state.

To a symplectic Lefschetz pencil on a monotone symplectic manifold, we associate an algebraic structure, which is a pencil of categories in the sense of noncommutative geometry.

Charged Dirac fermions with anomalous magnetic moment in the presence of the chiral magnetic effect and of a noncommutative phase space

In this work, we analyze the dynamical evolution of locally rotationally symmetric anisotropic cosmological models of Bianchi type I (flat curvature) and Bianchi type III (open curvature) within a noncommutative phase space framework characterized by a deformation parameter γ. Using a Hamiltonian formulation based on Schutz’s formalism for a perfect radiation fluid, we introduce noncommutative Poisson brackets that allow for geometric corrections to commutative dynamics. The resulting equations are solved numerically, which allows for the study of the impact of γ and the energy density C on the expansion of the universe and the evolution of anisotropy. The results show that γ<0 improves expansion and favors isotropization, while γ>0 tends to slow expansion and preserve residual anisotropy, especially in the open curvature model. It is estimated that the influence of noncommutativity was significant during the early stages of the universe, decreasing toward the present time, suggesting that this approach could serve as an effective alternative to the cosmological constant in describing the evolution of the early universe.

We construct a non-commutative version of the Grassmann variety G(2,4) as a non-commutative moduli space of linear subspaces in a projective space.

The impact of non-expansive entropies on thermodynamics of high order correction of Schwarzschild-AdS black hole in non-commutative geometry

Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces

This paper investigates the relationship between harmonic functions and several classes of ideals in polynomial rings. Focusing on prime, primary, maximal, and weakly primary ideals (see [M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra (Addison-Wesley, 1969); D. Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry (Springer-Verlag, New York, 1995); O. Zariski and P. Samuel, Commutative Algebra, Vol. II (Van Nostrand, Princeton,1960)]), we analyze how these algebraic structures reflect and influence the behavior of harmonic functions. Through theoretical analysis and illustrative examples, we examine the algebraic foundations of harmonics within the context of commutative algebra. Furthermore, we discuss the role of D-modules as a powerful tool for studying differential operators, particularly the Laplacian, in relation to harmonic functions (see [J.-E. Bj[Formula: see text]rk, Rings of Differential Operators, North-Holland Mathematical Library, Vol. 21 (North Holland, Amsterdam, 1979); S. C. Coutinho, A Primer of Algebraic D-Modules (Cambridge University Press, 1995); H. Johannes, P. Elizabeth, S. Anna-Laura and Z. Simone, D-modules and analytic representation of differential equations, preprint (2023); V. A. Lunts and A. Rosenberg, Differential operators and D-module structures in noncommutative geometry, Adv. Math. 415 (2023) 108901; M. Saito, K. Takeuchi and H. Terao, D-modules and Microlocal Geometry, Springer Monographs in Mathematics (Springer, 2022)]).

This study explores the thermodynamic and geometric properties of phantom BTZ black holes within a noncommutative spacetime framework, where noncommutativity is implemented through Lorentzian smearing of mass and charge distributions. The resulting metric exhibits significant modifications in curvature and horizon structure, particularly in the near-horizon regime. We perform a comparative thermodynamic analysis between phantom and Maxwell field cases, calculating quantities such as Hawking temperature, entropy, heat capacity, and Gibbs free energy. Our findings reveal that noncommutative corrections strongly affect phase transitions and stability conditions. Furthermore, we model the black hole as a heat engine and compute its efficiency, showing how noncommutative effects enhance or suppress energy extraction. This work underscores the interplay between spacetime fuzziness and exotic field dynamics in lower-dimensional gravity, offering new insights into quantum-modified black hole thermodynamics.

Mathematics provides the structural foundation through which the universe’s expansion becomes intelligible. This review synthesizes the differential-geometric, dynamical, and computational formalisms that define modern relativistic cosmology. Beginning from the Einstein–Hilbert action, spacetime is described as a pseudo-Riemannian manifold whose curvature and energy content obey the Einstein field equations. Under symmetry reductions, these equations yield the Friedmann relations that govern the temporal evolution of the scale factor. Analytical and numerical studies of these equations have refined the ΛCDM paradigm and motivated extensions such as f(R) gravity, scalar–tensor, and Gauss–Bonnet models. Relativistic perturbation theory links these global frameworks to the growth of cosmic structure, while advances in numerical relativity and symbolic computation have transformed Einstein’s equations into solvable systems across regimes from linear perturbations to nonlinear curvature dynamics. The review further explores mathematical frontiers—singularity analysis, alternative geometries, and quantum corrections—highlighting how new tools such as fractional calculus, noncommutative geometry, and category theory seek to reconcile continuous spacetime with quantum discreteness. By uniting geometry, dynamics, and computation, this study positions mathematical innovation as the key driver of cosmological progress. The expanding universe thus emerges not merely as a physical phenomenon but as an evolving mathematical construct whose form and evolution are written in the language of geometry and logic

Comparison of Levi-Civita connections in noncommutative geometry

Abstract We present an elementary construction of a (highly degenerate) Hopf pairing between the universal enveloping algebra U ⁢ ( 𝔤 ) {U(\mathfrak{g})} of a finite-dimensional Lie algebra 𝔤 {\mathfrak{g}} over arbitrary field 𝒌 {{\boldsymbol{k}}} , and the Hopf algebra 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} of regular functions on the automorphism group of 𝔤 {\mathfrak{g}} . This pairing induces a Hopf action of 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} on U ⁢ ( 𝔤 ) {U(\mathfrak{g})} , which, together with an explicitly given coaction, makes U ⁢ ( 𝔤 ) {U(\mathfrak{g})} into a braided commutative Yetter–Drinfeld 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} -module algebra. From these data one constructs a Hopf algebroid structure on the smash product algebra ♯ ⁡ 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) U ⁢ ( 𝔤 ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))\mathbin{\sharp}U(\mathfrak{g})} , retaining essential features from earlier constructions of a Hopf algebroid structure on infinite-dimensional versions of the Heisenberg double of U ⁢ ( 𝔤 ) {U(\mathfrak{g})} , including a noncommutative phase space of Lie algebra type, while avoiding the need of completed tensor products. We prove a slightly more general result, where the algebra 𝒪 ⁢ ( Aut ⁡ ( 𝔤 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{g}))} is replaced by 𝒪 ⁢ ( Aut ⁡ ( 𝔥 ) ) {\mathcal{O}(\operatorname{Aut}(\mathfrak{h}))} and where 𝔥 {\mathfrak{h}} is any finite-dimensional Leibniz algebra having 𝔤 {\mathfrak{g}} as its maximal Lie algebra quotient.

We investigate Snyder spacetime and its generalizations, including Yang and Snyder–de Sitter spaces, which constitute manifestly Lorentz-invariant noncommutative geometries. This work initiates a systematic study of gauge theory on such spaces in the semi-classical regime, formulated as Poisson gauge theory. As a first step, we construct the symplectic realizations of the relevant noncommutative spaces, a prerequisite for defining Poisson gauge transformations and field strengths. We present a general method for representing the Snyder algebra and its extensions in terms of canonical phase-space variables, enabling both the reproduction of known representations and the derivation of novel ones. These canonical constructions are employed to obtain explicit symplectic realizations for the Snyder–de Sitter space and to construct the deformed partial derivative which differentiates the underlying Poisson structure. Furthermore, we analyze the motion of freely falling particles in these backgrounds and comment on the geometry of the associated spaces.

Noncommutative differential geometry on infinitesimal spaces

Modular vector fields in non-commutative geometry

Wigner function for charged particles with spin 0 in non-commutative space

Charged noncommutative geometry inspired stringy black hole

Abstract In this work, we have studied the effects of noncommutative geometry on the properties of p-wave holographic superconductors with massive vector condensates in the probe limit. We have applied the St"{u}rm-Liouville eigenvalue approach to analyze the model. In this model, we have calculated the critical temperature and the value of the condensation operator for two different values of $m^2$. We have also shown how the influence of noncommutative geometry modifies these quantities. Finally, by applying a linearized gauge field perturbation along the boundary direction, we calculated the holographic superconductor's AC conductivity using a self-consistent approach and then carrying out a more rigorous analysis. The noncommutative effects are also found to be present in the result of AC conductivity. We have also found that just like the commutative case, here the DC conductivity diverges due to the presence of a first order pole in the frequency regime.

Abstract In this work, we investigate the effects of phase space noncommutativity on the dynamics of a Bianchi I (BI) cosmological model coupled to a reduced relativistic gas (RRG). The BI model provides a homogeneous but anisotropic framework suitable for exploring the transition from an early anisotropic Universe to the current isotropic stage. The RRG fluid interpolates between radiation and matter regimes, enabling a consistent treatment of the transition from the radiation-dominated to the matter-dominated era. In order to incorporate noncommutativity into the classical equations of motion, we employ the generalized symplectic formalism developed by Faddeev–Jackiw and extended by Barcelos–Wotzasek, which allows the introduction of noncommutative (NC) parameters via deformations in the symplectic structure. Within this approach, we derive a modified Hamiltonian expressed in terms of commutative variables that incorporate all NC effects. We then solve the resulting equations numerically and analyze the behavior of the scale factor and the anisotropic functions under variations of the NC parameters, as well as other physical and initial parameters of the model. Our results show that negative values of the NC parameters increase the expansion rate and reduce the isotropization timescale, partially mimicking the effect of a positive cosmological constant. We also estimate values for the NC parameters by numerically solving the integral expression for the age of the Universe, requiring consistency with observational data from the Planck 2018 mission. These findings support the possibility that noncommutativity may provide a geometric mechanism capable of accounting for the late-time acceleration and isotropization of the Universe, without requiring additional exotic energy components.