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Open research questions in advanced mathematical theories

83 gap statements mined from advanced mathematical theories papers in our 4.5M-paper local library, which holds 314 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

  • Standard adelic theory is limited to algebraic number fields. The adelic approach in number theory does not provide a unified framework for treating all completions of a number field.

    ULTRAMETRIC PHYSICS: Module 4: Adelic Theory for General Number Fields · 2026 · DOI

  • The conventional p-adic framework relies on integer primes, which may not be natural for fundamental physics. There is a need for a generalized valuation theory that accommodates irrational scaling bases.

    ULTRAMETRIC PHYSICS: Module 1: Generalized Valuation Theory for Irrational Scaling Bases · 2026 · DOI

  • These open issues are not fatal; they are research frontiers.

    Quantum Laws of Form: A Syntactic Foundation for Physics: From The Calculus of Distinction to Ultrametric Cosmology · 2026 · DOI

  • The current research on the subject has limitations. There is a gap in the understanding of the distribution of critical points of random Fourier series. The paper aims to fill this gap.

    A law of large numbers concerning the distribution of critical points of random Fourier series · 2026 · DOI

  • The paper identifies a need for new formulas and theorems for the p-adic valuations of Legendre polynomials.

    On the p-adic valuations of values of Legendre polynomials · 2026 · DOI

  • The lack of a formula for connected bosonic N-point functions for the BKP hierarchy. The need for a new method for computing connected bosonic N-point functions for the BKP hierarchy.

    A formula for connected bosonic N-point functions for BKP hierarchy · 2026 · DOI

  • The paper does not provide a comprehensive survey of all related work. The study is limited to the specific context of the quantitative identity of the formalization remainder.

    The Quantitative Identity of the Remainder: From ρ≠∅ to Euler's Formula / 余项的定量身份:从ρ≠∅到欧拉公式 · 2026 · DOI

  • The gap is the lack of understanding of the relationship between remainders and the exponential map. The study identifies a need for further research on self-referential generation.

    The Quantitative Identity of the Remainder: From ρ≠∅ to Euler's Formula / 余项的定量身份:从ρ≠∅到欧拉公式 · 2026 · DOI

  • The paper identifies a gap in the development of a calculus that can read the unevenness of the universe. The calculus is built to be portable, but it requires collaboration to extend it. The paper does not claim to have found a complete or definitive solution, but rather invites further development and collaboration.

    A Coupling Calculus · v2 · The Classical Anchoring · 2026 · DOI

  • The document gives a path by which spectral geometry could imply BSD, but it also makes clear exactly what remains to be proved. Proving equality only up to an unspecified scalar is insufficient unless that scalar is absorbed into the real-period normalization and shown not to alter the BSD formula.

    A Spectral-Geometric Program of Birch and Swinnerton-Dyer Conjecture · 2026 · DOI

  • The lack of effective lower bounds for the distance between powers of 2 and powers of 3 is identified. The gap in previous results, including the lack of estimates for the greatest prime divisor of the distance, is identified.

    On the difference between perfect powers andintegral S-units · 2026 · DOI

  • The Logvinenko-Sereda theorem has not been generalized for lacunary spectra. Uniqueness sets for functions with lacunary spectra have not been characterized.

    A Logvinenko-Sereda Theorem for Lacunary Spectra · 2026 · DOI

  • The paper identifies a gap in the prior work, noting that a fully rigorous derivation of the quantum master equation requires careful treatment of the substrate's mode-decomposition under the entangled state and of the time-dependence of the c n amplitudes.

    APF Paper 24 — The Recruitment-Radius Extension — Foundations (Technical Supplement) · 2026 · DOI

  • The Fyodorov-Hiary-Keating conjecture was only known in the leading order for θ ∈ (−1, 0). There was a lack of precise upper bounds for the maximum of the Riemann zeta function on a typical short interval of the critical line.

    The Fyodorov–Hiary–Keating conjecture on mesoscopic intervals · 2026 · DOI

  • The need to generalize the results to apply to all Archimedean vector lattices.

    Square closed pointed vector lattices · 2026 · DOI

  • To explore the applications of the framework in systems with exponential growth/decay, thermodynamics, and information theory. To investigate the properties of the tree construction with irrational branching characteristics. To develop new methods for generating effective continuous symmetries from strange loops in tree quotients.

    ULTRAMETRIC PHYSICS: Module 1: Generalized Valuation Theory for Irrational Scaling Bases · 2026 · DOI

  • Note: ζ(3) is odd and cannot arise from the Laurent-Bernoulli expansion. The mechanism is genuinely open. P2 (Central charge). Derive c = 6 from (1.1)–(1.2). Three motivated paths; none complete. P3 (Boundary value). Derive 20φ⁴ from a single argument. Three contributing factors, not unified. P4 (Coupling). Derive λ = 2 from R/r = 4. P5 (3+1 dimensions).

    Golden-Ratio Torus Holography: Cantor Spectrum, AdS₂ Geometry, and a Two-Term Formula for the Inverse Fine-Structure Constant · 2026 · DOI

  • The calculus can be applied to various fields, including biology and physics. The paper suggests that the development of the calculus can be done by running an anchor, breaking a seam, or tailoring it to a field that's stuck outside its own construct.

    A Coupling Calculus · v2 · The Classical Anchoring · 2026 · DOI

  • Experimental validation of the theory. Applications of the theory in quantum gravity and consciousness. Further development of the tree approximation.

    ULTRAMETRIC PHYSICS: Module 3: Vladimirov Operator and Ratio-Based Calculus · 2026 · DOI

  • The paper identifies a gap in the understanding of the Vladimirov operator for arbitrary scaling ratios q. The paper aims to fill this gap by developing a comprehensive theory.

    ULTRAMETRIC PHYSICS: Module 3: Vladimirov Operator and Ratio-Based Calculus · 2026 · DOI

  • To extend the results to other types of random functions. To apply the results to other areas of research. To further analyze the behavior of the random measures.

    A law of large numbers concerning the distribution of critical points of random Fourier series · 2026 · DOI

  • The paper identifies a gap in the literature on the derivation of interface evolution laws from microscopic models. The paper aims to connect mechanisms at a microscopic level to macroscopic models of image processing.

    Horizontal Mean Curvature Flow as a Scaling Limit of a Mean Field Equation in the Heisenberg Group · 2026 · DOI

  • To apply the result to other reductions of the KP hierarchy. To use the result to compute connected bosonic N-point functions for the BKP hierarchy. To study the Witten-Kontsevich tau-function and the Witten Conjecture/Kontsevich Theorem.

    A formula for connected bosonic N-point functions for BKP hierarchy · 2026 · DOI

  • The construction is limited to dimension 2 and codimension 3 - The result does not achieve full C¹¹ flexibility - The construction uses a specific iteration…

    The Monge–Ampère system in dimension two and codimension three · 2026 · DOI

  • Further study of the Monge-Ampère system's flexibility in higher dimensions - Development of new constructions to achieve higher Hölder exponents - Application of the result to other mathematical…

    The Monge–Ampère system in dimension two and codimension three · 2026 · DOI

Most-cited papers in advanced mathematical theories

Most recent work

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Open questions raised by papers listed on this page include: (1) There is a need for a generalized valuation theory that accommodates irrational scaling bases (“ULTRAMETRIC PHYSICS: Module 1: Generalized Valuation Theory for Irrational…”, 2026); (2) The lack of a formula for connected bosonic N-point functions for the BKP hierarchy (“A formula for connected bosonic N-point functions for BKP hierarchy”, 2026); (3) The gap is the lack of understanding of the relationship between remainders and the exponential map (“The Quantitative Identity of the Remainder: From ρ≠∅ to Euler's Formula /…”, 2026); (4) The document gives a path by which spectral geometry could imply BSD, but it also makes clear exactly what remains to be proved (“A Spectral-Geometric Program of Birch and Swinnerton-Dyer Conjecture”, 2026); (5) The lack of effective lower bounds for the distance between powers of 2 and powers of 3 is identified (“On the difference between perfect powers andintegral S-units”, 2026). 83 gap statements have been mined from advanced mathematical theories papers in our 4.5M-paper local library, which holds 314 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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