Open research questions in Mathematics and Applications
149 gap statements mined from Mathematics and Applications papers in our 4.5M-paper local library, which holds 1,219 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
There was a gap in proving a conjecture of Aluffi. The relationship between the adjoint polynomial and the covolume polynomial was not fully understood.
The lack of a minimal commutative foundation for paired complex numbers. The need for a symmetric system of two planes, two angles, and two angle differences.
The periodic table's natural shape and topology are not well understood. The relationships between the table's elements are not fully captured by traditional 2D representations.
The Helical Periodic Table: The Pre-Mendeleev 3D Origin, Rubber-Sheet Pliability, and the C₆₀ as One Admissible Host Among Many · 2026 · DOIThe lack of a unified solution to the Millennium Prize Problems. The need for a new propulsion system that can access a propulsive force far exceeding any chemical or nuclear reaction.
The following study addresses the issue of calculating with precision the volume of a truncated right circular cone while provided with limited information on the dimensions of the object itself (being given the relation between the radii, the vertical heights, or the slanted heights).
Future research can focus on the applications of the experiment in biology and chemistry. Further study can be done on the combinatorial geometric blueprints based on Hamiltonian paths.
Further research can be done to study properties of lattice polygons. Further research can be done to apply the results to other areas of geometry and computer science.
The question of extending the t-Feynman's configurations to tetrahedra appears naturally. There is a lack of understanding of the properties of t-Feynman's tetrahedra.
Heronian quadrilaterals have not been investigated thoroughly. The finiteness of amicable Heronian triangles and rectangles seems to end with this case.
Whether such an approach yields additional arithmetic information remains an open question. Whether the present resolution framework can be related to operatortheoretic or spectral-geometric constructions remains an open problem and is left for future investigation.
The Pell Spine as an Optimal Rational Separator: Extremal Resolution and Smooth Transport in Primitive Pythagorean Triples · 2026 · DOIThe paper identifies a gap in the understanding of the incidence results that involve points of tangency between the circles in the construction. The paper identifies a gap in the study of Steiner chains and their relation to conic structures.
The need for a deterministic parameterization of a classical spatial envelope. The lack of a compact representation for reconstruction of the Matrix boundary geometry, planar bounding envelope, and three-dimensional volume envelope.
There is a need for a unified approach to studying the cohomology theories of various operators on Leibniz algebras. Existing cohomologies of operators on Leibniz algebras are not unified.
The proof assumes certain conditions on the field F, which may not be met in all cases.- The proof does not provide a general method for finding the roots of a polynomial.- The proof is limited to the realm of field theory and does not explore other areas of mathematics.
The existing literature lacks a systematic construction of solutions of simplex equations from polygon equations. The paper identifies a gap in the understanding of the relationship between polygon equations and simplex equations.
The lack of a characterization of the relations ∼i in terms of the poset of subsets of antiflags. The need to explore the implications of such a characterization for the automorphisms of the graph and the structure of the point-hyperplane anti-flags.
The individual divisibility result is a known consequence of standard digit-sum congruences, but the systematic organization of the full family of inversion differences is new.
The Pythagorean theorem does not explain why a rope of fixed length pinned at three points must form a specific triangle. The theorem does not provide a constructional framework for deriving the right triangle family.
Extension of the helical form to predicted elements and physics past Z = 118. Application of the topological framework to other fields, such as materials science.
The Helical Periodic Table: The Pre-Mendeleev 3D Origin, Rubber-Sheet Pliability, and the C₆₀ as One Admissible Host Among Many · 2026 · DOIFurther development of the Yett Paradigm and its applications. Experimental verification of the Geodesic Pothole Mechanism. Study of the implications of the Yett Paradigm for the development of new propulsion systems.
The paper identifies a gap in the understanding of the mathematics behind the morphogenesis of icosahedral shapes in nature. The study aims to fill this gap by presenting a novel mathematical video experiment.
The paper identifies a gap in the study of convex lattice pentagons with three pairs of parallel sides and diagonals. The paper aims to characterize all obtained convex lattice pentagons of minimal area.
To study the number of almost ordinary triangles in a set of points in higher dimensions. To develop more efficient algorithms for counting all c-ordinary triangles. To apply the technique for distinguishing between two cases to other geometric problems.
Exploring the implications of the proof for the study of other areas of mathematics.- Developing a general method for finding the roots of a polynomial.- Investigating the properties of algebraically closed fields.
Further study of the automorphisms of the graph of point-hyperplane anti-flags. Exploration of the implications of the paper's characterization of the relations for other areas of mathematics.
Most-cited papers in Mathematics and Applications
- There is More than One Way to Frame a Curve · American Mathematical Monthly · 1975 · 1,908 citations
- The Determinants of Agglomeration · Journal of Urban Economics · 2001 · 747 citations
- The Cosine-Haversine Formula · American Mathematical Monthly · 1957 · 539 citations
- The Surveyor's Area Formula · College Mathematics Journal · 1986 · 347 citations
- Simplified Calculation of Principal Components · Psychometrika · 1936 · 188 citations
- The Longest Run of Heads · College Mathematics Journal · 1990 · 161 citations
- Components of geometric analogy solution · Cognitive Psychology · 1980 · 156 citations
- Ice-Ray: A Note on the Generation of Chinese Lattice Designs · Environment and Planning B Planning and Design · 1977 · 155 citations
- Ten compactness properties of circles: measuring shape in geography · Canadian Geographies / Géographies canadiennes · 2010 · 150 citations
- The Hyperbolic Number Plane · College Mathematics Journal · 1995 · 147 citations
Most recent work
- The Helical Periodic Table: The Pre-Mendeleev 3D Origin, Rubber-Sheet Pliability, and the C₆₀ as One Admissible Host Among Many · Zenodo (CERN European Organization for Nuclear Research) · 2026
- Clean up your mesh! Part 1: plane and simplex · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · 2026
- Carroll geometry meets de Sitter space via holography · Physics Letters B · 2026
- A proof of Brouwer's conjecture for k = 3 · Linear Algebra and its Applications · 2026
- Constructing Poncelet Polygons · Discrete & Computational Geometry · 2026
- Géométries combinatoires · Journées mathématiques X-UPS · 2026
- Menelaus’ and Ceva’s Theorems for Translation Triangles in Thurston Geometries · Results in Mathematics · 2026
- Curvilinear coordinate systems for 3D design: building cyclidic nets with conformal geometric algebra · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · 2026
- On the Tambara affine line · Advances in Mathematics · 2026
- Algebraic surfaces as Hadamard products of curves · Journal of Algebra · 2026
Find a gap in your own Mathematics and Applications sub-topic
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