Open research questions in Algebraic and Geometric Analysis

What the literature leaves open

• To define and investigate the 3PGQic extension of the DGC Leonardo sequence of type-1. To derive Binet's formulas, generating functions, and fundamental identities for the 3PGQic extension of the DGC Leonardo sequence of type-1. To explore the applications of the 3PGQic DGC sequences of type-2 in fields such as signal processing and computer graphics.

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

• Ensuring bit-perfect global parity in distributed architectures. Validating the analytic index parity in SU(2) Yang-Mills theory. Resolving the vulnerability of 'Logic-Blur' by proving that topological charges remain invariant. Generalizing the framework to SU(N) for N > 2.

  Topological Charge Conservation in SU(2) Yang-Mills Theory: The Atiyah-Singer Index Handshake and Non-Local Braid-Lock Validation Framework · 2026 · DOI

• The nonlocal nature of fractional operators poses a challenge. The lack of a satisfactory fractional tangent space is a technical challenge. The construction of a controlled pointwise substitute is a domain-specific challenge.

  A fractional de Rham complex for coframe-attached Maxwell equations · 2026 · DOI

• The challenge of resolving divergences in quantum field theory without relying on analytic continuation. The challenge of preserving gauge invariance in the regularization method. The challenge of applying the method to Yang-Mills theory to establish a mass gap.

  DUAL ARCHITECTURE IN YANG-MILLS THEORY: ALGEBRAIC REGULARIZATION, WIGHTMAN AXIOMS, AND MASS GAP · 2026 · DOI

• Future research could focus on generalizing the authors' strategy to other areas of nonassociative geometry. Future research could focus on constructing spectral geometries for other types of nonassociative algebras. Future research could focus on applying the novel form of bimodule introduced by the authors to other fields.

  Spectral geometry with exceptional symmetry and charged Higgs fields · 2026 · DOI

• The gap is the lack of a general approach to nonassociative spectral geometry. The gap is the lack of understanding of nonassociative algebras. The gap is the challenge of constructing spectral geometries for nonassociative algebras.

  Spectral geometry with exceptional symmetry and charged Higgs fields · 2026 · DOI

• The lack of understanding of how Lorentz symmetry emerges from an almost-commutative twisted spectral triple. The need for a new perspective on the Lorentzian signature problem.

  Emergence of Lorentz symmetry from an almost-commutative twisted spectral triple · 2026 · DOI

• Future research can build on the results of this paper to further study the properties of E theory and Siegel theory. Future research can explore the implications of the results for our understanding of string theory and general relativity.

  Local symmetry and the dependence on extended spacetime · 2026 · DOI

• The paper identifies a gap in current understanding of the local symmetry of E theory. The paper highlights the need to study the local symmetry of E theory and its dependence on extended spacetime.

  Local symmetry and the dependence on extended spacetime · 2026 · DOI

• The paper suggests further research on the T-dual interpretation of the generalized complex structure on the twistor space. The paper leaves the total-space formulation to future work.

  Rank-3 Generalized Clifford Manifold and Its Twistor Space · 2026 · DOI

• The paper identifies a gap in the existing literature on generalized geometric structures. The paper addresses the need for a unified framework for generalized geometric structures.

  Rank-3 Generalized Clifford Manifold and Its Twistor Space · 2026 · DOI

• The development of qudit-based quantum computation is challenging due to the need for new mathematical frameworks and computational methods. The use of qudits requires a deeper understanding of their mathematical structure and potential applications. The paper notes that the study of qudits is still in its early stages and further research is needed.

  In Praise of Qudits: Why Higher-Dimensional Quantum Systems Deserve Centre Stage · 2026 · DOI

• The need for a proof of the Riemann Hypothesis. The lack of understanding of the fundamental geometric background of the physical universe.

  RENASCENT-Q Theory v.47: The Origin of the Resonant Negentropic Asymptotic Coherent Entangled with Quantum Field Fundamental Force · 2026 · DOI

• To further develop the quaternionic representation of minimal surfaces. To apply the method in computational geometry and geometric modeling.

  Minimal Surfaces via Complex Quaternions · 2026 · DOI

• The lack of a quaternionic representation of minimal surfaces. The need for a new approach to generating minimal surfaces.

  Minimal Surfaces via Complex Quaternions · 2026 · DOI

• This reformulation unifies standard modular and annulus constraints in two dimensions with the anchored neural approach for CFT correlators, providing a new way to reconstruct full partition functions from sparse data with remarkable accuracy.

  Neural Spectral Bias and Conformal Correlators II: Modular and Annulus Bootstrap · 2026

• 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 identifies a gap in the study of Witt type realizations of 2-D Cayley-Klein algebras with non-zero curvatures. 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

• 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

• The paper identifies a gap in the understanding of the 2-Cauchy-Fueter equation. 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 current state-of-the-art methods for computing Cayley's first hyperdeterminant have exponential time complexity. There is a need for a faster method for computing the hyperdeterminant of symmetric tensors.

  New identity for Cayley's first hyperdeterminant with applications to symmetric tensors and entanglement · 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 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

Most-cited papers in Algebraic and Geometric Analysis

• Isolated systems and their symmetries, part II: Local and global symmetries of field theories · Studies in History and Philosophy of Science Part A · 2022 · 11 citations
• An algebraic approach to physical fields · Studies in History and Philosophy of Science Part A · 2021 · 9 citations
• On gauge symmetries, indiscernibilities, and groupoid-theoretical equalities · Studies in History and Philosophy of Science Part A · 2022 · 7 citations
• Anomalous Dimensions at Large Charge in \(d=4\ \mathrm {O}(N)\) Theory · Acta Physica Polonica B · 2021 · 7 citations
• Angular momentum without rotation: Turbocharging relationalism · Studies in History and Philosophy of Science Part A · 2021 · 5 citations
• Interpreting the Wigner–Eckart Theorem · Studies in History and Philosophy of Science Part A · 2021 · 5 citations
• The Generalization of Gaussians and Leonardo’s Octonions · Annales Mathematicae Silesianae · 2023 · 5 citations
• Family of quadratic differential systems with invariant parabolas: a complete classification in the space <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msup> <a:mrow class="MJX-TeXAtom-ORD"> <a:mrow class="MJX-TeXAtom-ORD"> <a:mi mathvariant="double-struck">R</a:mi> </a:mrow> </a:mrow> <a:mrow class="MJX-TeXAtom-ORD"> <a:mn>12</a:mn> </a:mrow> </a:msup></a:math> · Electronic journal of qualitative theory of differential equations · 2024 · 3 citations
• Why Hamilton Couldn’t Multiply Triples · College Mathematics Journal · 2021 · 3 citations
• A Novel Generalisation of Supersymmetry: Quantum \(\mathbb {Z}_2^2\)-Oscillators and Their ‘Superisation’ · Acta Physica Polonica B · 2024 · 2 citations

Most recent work

• Carrollian physics and holography · Physics Reports · 2026
• Symbols from bi-projections · Journal of High Energy Physics · 2026
• 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) · Zenodo (CERN European Organization for Nuclear Research) · 2026
• The Recursive Harmonic Codex: And the unified field theory proofs · Zenodo (CERN European Organization for Nuclear Research) · 2026
• On polynomial equations over split-octonions: the arbitrary field case · Communications in Mathematics · 2026
• A Monster-Symmetric Admissibility Formulation of the SEXA Unified Field Theory: Operator-Glyph Closure, Σ₆₀ Exciternion Logic, and Falsifiable Reduction to General Relativity, Quantum Field Theory, and Yukawa Interaction Regimes · JOURNAL OF ADVANCES IN MATHEMATICS · 2026
• 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. · Zenodo (CERN European Organization for Nuclear Research) · 2026
• The 731 Frog Calculus, Part 1: Three-Dimensional Spin Foams, Magmoidal Category Theory, and Non-Associative Topology · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Relativité Générale Hypercomplexe RGH-Oλ Écriture symplectique d'une incursion octonionique activée · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Witt type Realizations of 2-D Cayley-Klein Algebras with non-zero curvatures · Physica Scripta · 2026

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