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Open research questions in Algebraic and Geometric Analysis

118 gap statements mined from Algebraic and Geometric Analysis papers in our 4.5M-paper local library, which holds 534 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

What the literature leaves open

  • The lack of a canonical form for the constant term c makes it challenging to solve the equations.

    On polynomial equations over split-octonions: the arbitrary field case · 2026 · DOI

  • String theory — a more ambitious dynamical framework in principle, but the landscape problem means specific predictions for SM parameters remain elusive.

    Watt ↔ Singh ↔ Boyle ↔ Furey ↔ Woit, A five-way complementary bridge joined through the composition-algebra geometry of (ℂ⊗𝕆)P² and its subspaces, with F₄ as candidate internal parent and twistor ℂP³ as candidate spacetime boundary. · 2026 · DOI

  • The lack of a direct method for constructing the symbol. The need for a systematic framework for analyzing the analytic properties of finite Feynman integrals.

    Symbols from bi-projections · 2026 · DOI

  • The paper suggests further research on the application of Cayley-Klein algebras in various fields. The paper proposes the study of other types of algebras and their relations with geometric spaces.

    Witt type Realizations of 2-D Cayley-Klein Algebras with non-zero curvatures · 2026 · DOI

  • The paper notes that prior work has not fully explored the use of Jacobi elliptic functions and modular transformations in this context.

    Witt type Realizations of 2-D Cayley-Klein Algebras with non-zero curvatures · 2026 · DOI

  • The paper addresses the need for a complete topological characterization for the solvability of the equation.

    A Pair of Multiplication-Type Operators in Quaternionic Analysis and the 2-Cauchy-Fueter Equation · 2026 · DOI

  • The problem of finding examples of prepositive cones that are not maximal remains open. The existence of prepositive cones that are not maximal remains elusive.

    Maximal prepositive cones on quaternion algebras with involution · 2026 · DOI

  • The need for a new perspective on quaternionic sequences and their applications. The lack of investigation on the 3PGQic DGC sequences of type-2.

    The 3-parameter generalized quaternionic sequences with $\mathcal{DGC}$ Leonardo numbers via doubling · 2026 · DOI

  • Future research directions include the extension of this work to entangled photon states. Another direction is the study of topological quantum materials. Further insights are needed on the physical, mathematical, and conceptual nature of geometrical phases.

    Geometrical derivation of Wigner's angle for arbitrary Lorentz transformations of massless particles · 2026

  • The gap is related to the non-uniqueness of the Wigner angle and its dependence on the path in momentum space.

    Geometrical derivation of Wigner's angle for arbitrary Lorentz transformations of massless particles · 2026

  • The method is limited to the study of $Jar{J}$-deformations. The calculation of the deformation of partition functions is limited to the case of higher genus Riemann surfaces.

    $J\bar{J}$-deformation as a Riemann bilinear dressing · 2026

  • The classification is limited to homogeneous definite-grade right multipliers K. The study does not consider arbitrary first-order linear systems or mixed-grade square roots.

    The Homogeneous First-Order Roots of the Klein-Gordon Operator · 2026 · DOI

  • The need for a unifying framework for geometric phenomena. The lack of a clear connection between classical rotation representation and quantum computing.

    Álgebra Geométrica como Marco Unificador: De las Rotaciones Clásicas a los Operadores Cuánticos Version Final · 2026 · DOI

  • The trigonometry of hyperspheres in higher dimensions has not been studied as extensively as spherical trigonometry. There is a need for a multidimensional generalization of the Euler top.

    Hyperspherical Trigonometry and Corresponding Elliptic Functions · 2026 · DOI

  • The paper does not address the implications of the results for physical systems. The study is limited to 2D GCA.

    Momentum space correlation functions in 2D Galilean conformal algebra · 2026 · DOI

  • The Riemann Hypothesis has resisted proof for 165 years. Prior work has not connected the four fields: noncommutative geometry, quantum information theory, econometrics, and analytic number theory.

    The Riemann Hypothesis as a Corollary of the Granger Representation Theorem: ARIMA(35,1,∞) from the Standard Model Algebra A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ) · 2026 · DOI

  • To explore the applications of Geometric Algebra in various fields. To further develop the algebraic connection between SO(3), quaternions, and Geometric Algebra.

    Álgebra Geométrica como Marco Unificador: De las Rotaciones Clásicas a los Operadores Cuánticos Version Final · 2026 · DOI

  • The paper does not address the promotion of the global internal symmetry to a local Yang-Mills symmetry. The analysis is restricted to the Dirac-type sector of the homogeneous first-order factorisations of the Klein-Gordon operator.

    A Clifford-Algebraic Origin of a Doublet Structure and SU(2)L x U(1) Symmetry from First-Order Klein-Gordon Factorisation · 2026 · DOI

  • To explore the promotion of the global internal symmetry to a local Yang-Mills symmetry. To investigate the implications of the analysis for the development of new gauge theories or the interpretation of existing ones.

    A Clifford-Algebraic Origin of a Doublet Structure and SU(2)L x U(1) Symmetry from First-Order Klein-Gordon Factorisation · 2026 · DOI

  • This work has developed Twin-Fixed Kaiso as a minimal syntactic framework for representing hierarchical structures using only two complex variables. The central result is the closure theorem, which shows that the hierarchy remains stable across all regular levels: the twin structure is preserved, the parameters are propagated consistently, and the twist remains invariant.

    Tsuihenkaku Kaiso · 2026 · DOI

  • The fine structure constant has never been derived from a mathematical structure. Previous attempts have not produced agreement beyond a few significant figures.

    A Fixed Point of a Self-Coupled Clifford Algebra: The Fine Structure Constant From Algebraic Structure · 2026 · DOI

  • Further study of the connections between noncommutative geometry, quantum information theory, econometrics, and analytic number theory. Application of the Granger Representation Theorem and ARIMA representation to other problems in number theory and cryptography.

    The Riemann Hypothesis as a Corollary of the Granger Representation Theorem: ARIMA(35,1,∞) from the Standard Model Algebra A_F = ℂ ⊕ ℍ ⊕ M₃(ℂ) · 2026 · DOI

  • Further study of the 2-Cauchy-Fueter equation and its applications. Exploration of the implications of the results for theoretical physics and quaternionic analysis.

    A Pair of Multiplication-Type Operators in Quaternionic Analysis and the 2-Cauchy-Fueter Equation · 2026 · DOI

  • To apply the method to the study of string theory and the AdS/CFT correspondence. To generalize the method to other types of deformations. To study the properties of conformal field theories and their deformations using the new method.

    $J\bar{J}$-deformation as a Riemann bilinear dressing · 2026

  • Further study of the properties and applications of the classified roots. Investigation of the implications for relativistic quantum theory and the development of new quantum theories or models.

    The Homogeneous First-Order Roots of the Klein-Gordon Operator · 2026 · DOI

Most-cited papers in Algebraic and Geometric Analysis

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Open questions raised by papers listed on this page include: (1) The lack of a canonical form for the constant term c makes it challenging to solve the equations (“On polynomial equations over split-octonions: the arbitrary field case”, 2026); (2) String theory — a more ambitious dynamical framework in principle, but the landscape problem means specific predictions for SM parameters remain elusive (“Watt ↔ Singh ↔ Boyle ↔ Furey ↔ Woit, A five-way complementary bridge joined…”, 2026); (3) The paper notes that prior work has not fully explored the use of Jacobi elliptic functions and modular transformations in this context (“Witt type Realizations of 2-D Cayley-Klein Algebras with non-zero curvatures”, 2026); (4) The existence of prepositive cones that are not maximal remains elusive (“Maximal prepositive cones on quaternion algebras with involution”, 2026); (5) The lack of investigation on the 3PGQic DGC sequences of type-2 (“The 3-parameter generalized quaternionic sequences with $\mathcal{DGC}$…”, 2026). 118 gap statements have been mined from Algebraic and Geometric Analysis papers in our 4.5M-paper local library, which holds 534 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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