Fujita conjecture
In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved as of 2017. It is named after Takao Fujita, who formulated it in 1985.
Erdős distinct distances problem
In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly-linear number of distinct distances. It was posed by Paul Erdős in 1946 and almost
Hopf conjecture
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
Space form
In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic
Goldberg–Seymour conjecture
In graph theory the Goldberg–Seymour conjecture states that where is the edge chromatic number of G and Note this above quantity is twice the arboricity of G. It is sometimes called the density of G.
De Branges's theorem
In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the comple
Four exponentials conjecture
In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transce
Ravenel's conjectures
In mathematics, the Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel at the end of a paper published in 1984. It was earlier ci
Szymanski's conjecture
In mathematics, Szymanski's conjecture, named after Ted H. Szymanski, states that every permutation on the n-dimensional doubly directed hypercube graph can be routed with edge-disjoint paths. That is
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's La
Pacman conjecture
The Pacman conjecture holds that durable-goods monopolists have complete market power and so can exercise perfect price discrimination, thus extracting the total surplus. This is in contrast to the Co
Szpiro's conjecture
In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named fo
Carmichael's totient function conjecture
In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states
Crouzeix's conjecture
Crouzeix's conjecture is an unsolved (as of 2018) problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it refines Crouzeix's theorem, which states: where the set is the field of
Six exponentials theorem
In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a
Yau's conjecture
In differential geometry, Yau's conjecture from 1982, is a mathematical conjecture which states that a closed Riemannian 3-manifold has an infinite number of smooth closed immersed minimal surfaces. I
Chern's conjecture for hypersurfaces in spheres
Chern's conjecture for hypersurfaces in spheres, unsolved as of 2018, is a conjecture proposed by Chern in the field of differential geometry. It originates from the Chern's unanswered question: Consi
Fröberg conjecture
In algebraic geometry, the Fröberg conjecture is a conjecture about the possible Hilbert functions of a set of forms. It is named after Ralf Fröberg, who introduced it in Fröberg . The Fröberg–Iarrobi
Kaplansky's conjectures
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kap
Hadwiger conjecture (combinatorial geometry)
In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2n or fewer smaller bodies homothetic with the original body, and that
Hilbert–Smith conjecture
In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithful
Fuglede's conjecture
Fuglede's conjecture is a closed problem in mathematics proposed by Bent Fuglede in 1974. It states that every domain of (i.e. subset of with positive finite Lebesgue measure) is a spectral set if and
Homological mirror symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists s
Hilbert–Pólya conjecture
In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riema
Grothendieck–Katz p-curvature conjecture
In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogou
N conjecture
In number theory the n conjecture is a conjecture stated by as a generalization of the abc conjecture to more than three integers.
Albertson conjecture
In combinatorial mathematics, the Albertson conjecture is an unproven relationship between the crossing number and the chromatic number of a graph. It is named after Michael O. Albertson, a professor
Vojta's conjecture
In mathematics, Vojta's conjecture is a conjecture introduced by Paul Vojta about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diopha
Kalai's 3^d conjecture
In geometry, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every d-dimensional centrally symmetric p
Entropy influence conjecture
In mathematics, the entropy influence conjecture is a statement about Boolean functions originally conjectured by Ehud Friedgut and Gil Kalai in 1996.
Chronology protection conjecture
The chronology protection conjecture is a hypothesis first proposed by Stephen Hawking that laws of physics beyond those of standard general relativity prevent time travel on all but microscopic scale
Edgeworth conjecture
In economics, the Edgeworth conjecture is the idea, named after Francis Ysidro Edgeworth, that the core of an economy shrinks to the set of Walrasian equilibria as the number of agents increases to in
Chang's conjecture
In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by , p. 309), states that every model of type (ω2,ω1) for a countable language has an elementary sub
Sierpiński number
In number theory, a Sierpiński number is an odd natural number k such that is composite for all natural numbers n. In 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which
Duffin–Schaeffer conjecture
The Duffin–Schaeffer conjecture was a conjecture (now a theorem) in mathematics, specifically, the Diophantine approximation proposed by R. J. Duffin and A. C. Schaeffer in 1941. It states that if is
Homological conjectures in commutative algebra
In mathematics, homological conjectures have been a focus of research activity in commutative algebra since the early 1960s. They concern a number of interrelated (sometimes surprisingly so) conjectur
André–Oort conjecture
In mathematics, the André–Oort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the Manin–Mumford conjecture, which is now a th
Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an o
Determinantal conjecture
In mathematics, the determinantal conjecture of Marcus and de Oliveira asks whether the determinant of a sum A + B of two n by n normal complex matrices A and B lies in the convex hull of the n! point
Dixmier conjecture
In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. in 2005, and independently and Kontsevich in 2007, sh
Carathéodory conjecture
In differential geometry, the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 192
Reconstruction conjecture
Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to Kelly and Ulam.
Eilenberg–Ganea conjecture
The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group G has cohom
Brouwer's conjecture
In the mathematical field of spectral graph theory, Brouwer's conjecture is a conjecture by Andries Brouwer on upper bounds for the intermediate sums of the eigenvalues of the Laplacian of a graph in
Farrell–Jones conjecture
In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms. The
Hilbert's twelfth problem
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any bas
New digraph reconstruction conjecture
The reconstruction conjecture of Stanisław Ulam is one of the best-known open problems in graph theory. Using the terminology of Frank Harary it can be stated as follows: If G and H are two graphs on
Lafforgue's theorem
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic
Harborth's conjecture
In mathematics, Harborth's conjecture states that every planar graph has a planar drawing in which every edge is a straight segment of integer length. This conjecture is named after Heiko Harborth, an
Montgomery's pair correlation conjecture
In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit
Tate conjecture
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois
Thrackle
A thrackle is an embedding of a graph in the plane, such that each edge is a Jordan arcand every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoi
Baum–Connes conjecture
In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the reduced C*-algebra of a group and the K-homology of the classifying space of p
Second neighborhood problem
In mathematics, the second neighborhood problem is an unsolved problem about oriented graphs posed by Paul Seymour. Intuitively, it suggests that in a social network described by such a graph, someone
Herzog–Schönheim conjecture
In mathematics, the Herzog–Schönheim conjecture is a combinatorial problem in the area of group theory, posed by Marcel Herzog and Jochanan Schönheim in 1974. Let be a group, and let be a finite syste
Arnold–Givental conjecture
The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of a Lagrangian subm
Erdős–Gyárfás conjecture
In graph theory, the unproven Erdős–Gyárfás conjecture, made in 1995 by the prolific mathematician Paul Erdős and his collaborator András Gyárfás, states that every graph with minimum degree 3 contain
Erdős conjecture on arithmetic progressions
Erdős' conjecture on arithmetic progressions, often referred to as the Erdős–Turán conjecture, is a conjecture in arithmetic combinatorics (not to be confused with the Erdős–Turán conjecture on additi
Weinstein conjecture
In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the conjecture claims that on a compact conta
Coase conjecture
The Coase conjecture, developed first by Ronald Coase, is an argument in monopoly theory. The conjecture sets up a situation in which a monopolist sells a durable good to a market where resale is impo
Casas-Alvero conjecture
In mathematics, the Casas-Alvero conjecture is an open problem about polynomials which have factors in common with their derivatives, proposed by Eduardo Casas-Alvero in 2001.
Demazure conjecture
In mathematics, the Demazure conjecture is a conjecture about representations of algebraic groups over the integers made by Demazure . The conjecture implies that many of the results of his paper can
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be
Serre's multiplicity conjectures
In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since Andr
Goodman's conjecture
Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.
Redshift conjecture
In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory has chromatic level one higher than that of a complex-oriented ring spe
Bombieri–Lang conjecture
In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variet
Ramanujan–Petersson conjecture
In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan , states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12 where , satisfies
Ehrhart's volume conjecture
In the geometry of numbers, Ehrhart's volume conjecture gives an upper bound on the volume of a convex body containing only one lattice point in its interior. It is a kind of converse to Minkowski's t
Surface subgroup conjecture
In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup
Falconer's conjecture
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact -dimensional spaces. Intui
Littlewood conjecture
In mathematics, the Littlewood conjecture is an open problem (as of May 2021) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α a
Virtually fibered conjecture
In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with i
Log-rank conjecture
In theoretical computer science, the log-rank conjecture states that the deterministic communication complexity of a two-party Boolean function is polynomially related to the logarithm of the rank of
Barnette's conjecture
Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after , a professor emeritus at the University of California
Abc conjecture
The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of
Final state conjecture
The final state conjecture is that the end state of the universe will consist of black holes and gravitational radiation.
Monomial conjecture
In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following: Let A be a Noetherian local ring of Krull dimension d and let x1, ..., xd be a system of
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by André Weil. They led to a successful multi-decade program to prove them, in which many leading researchers developed the frame
Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot. Paul A. Smith showed that a non-tr
Fontaine–Mazur conjecture
In mathematics, the Fontaine–Mazur conjectures are some conjectures introduced by Fontaine and Mazur about when p-adic representations of Galois groups of number fields can be constructed from represe
Brennan conjecture
The Brennan conjecture is a mathematical hypothesis (in complex analysis) for estimating (under specified conditions) the integral powers of the moduli of the derivatives of conformal maps into the op
Wormhole
A wormhole (Einstein-Rosen bridge) is a speculative structure connecting disparate points in spacetime, and is based on a special solution of the Einstein field equations. A wormhole can be visualized
Lander, Parkin, and Selfridge conjecture
The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theor
Feit–Thompson conjecture
In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by Walter Feit and John G. Thompson. The conjecture states that there are no distinct prime numbers p and q suc
Arthur's conjectures
In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local fields made
List of conjectures by Paul Erdős
The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects, and in many cases Erdős offered monetary rewards for solvi
Unique games conjecture
In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the a
Standard conjectures on algebraic cycles
In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications o
Grand Riemann hypothesis
In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic L-functions lie on
Gillies' conjecture
In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the disc
Lehmer's totient problem
In mathematics, Lehmer's totient problem asks whether there is any composite number n such that Euler's totient function φ(n) divides n − 1. This is an unsolved problem. It is known that φ(n) = n − 1
Gan–Gross–Prasad conjecture
In mathematics, the Gan–Gross–Prasad conjecture is a restriction problem in the representation theory of real or p-adic Lie groups posed by Gan Wee Teck, Benedict Gross, and Dipendra Prasad. The probl
List of unsolved problems in computer science
This article is a list of notable unsolved problems in computer science. A problem in computer science is considered unsolved when no solution is known, or when experts in the field disagree about pro
Chern's conjecture (affine geometry)
Chern's conjecture for affinely flat manifolds was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem. Chern's conjecture st
Novikov conjecture
The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965. The Novikov conjecture concerns the homot
P versus NP problem
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.
Jacobson's conjecture
In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring. It has only been proven for special typ
Mahler's 3/2 problem
In mathematics, Mahler's 3/2 problem concerns the existence of "Z-numbers". A Z-number is a real number x such that the fractional parts of are less than 1/2 for all positive integers n. Kurt Mahler c
Nagata–Biran conjecture
In mathematics, the Nagata–Biran conjecture, named after Masayoshi Nagata and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces.
Zilber-Pink conjecture
In mathematics, the Zilber–Pink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as André-Oort, Manin–Mumford, and Mordell-Lang. For algebraic to
Filling area conjecture
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientable surfaces that fill a closed curve of given length without introduci
Köthe conjecture
In mathematics, the Köthe conjecture is a problem in ring theory, open as of 2022. It is formulated in various ways. Suppose that R is a ring. One way to state the conjecture is that if R has no nil i
Stark conjectures
In number theory, the Stark conjectures, introduced by Stark and later expanded by Tate, give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-fu
1/3–2/3 conjecture
In order theory, a branch of mathematics, the 1/3–2/3 conjecture states that, if one is comparison sorting a set of items then, no matter what comparisons may have already been performed, it is always
Sendov's conjecture
In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex varia
Blattner's conjecture
In mathematics, Blattner's conjecture or Blattner's formula is a description of the discrete series representations of a general semisimple group G in terms of their restricted representations to a ma
Hirsch conjecture
In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more
Vaught conjecture
The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order comp
Dinitz conjecture
In combinatorics, the Dinitz theorem (formerly known as Dinitz conjecture) is a statement about the extension of arrays to partial Latin squares, proposed in 1979 by Jeff Dinitz, and proved in 1994 by
Erdős–Faber–Lovász conjecture
In graph theory, the Erdős–Faber–Lovász conjecture is a problem about graph coloring, named after Paul Erdős, Vance Faber, and László Lovász, who formulated it in 1972. It says: If k complete graphs,
Goormaghtigh conjecture
In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. The conjecture is that the only non-trivial integer solutions of the
Nakayama's conjecture
In mathematics, Nakayama's conjecture is a conjecture about Artinian rings, introduced by Nakayama. The generalized Nakayama conjecture is an extension to more general rings, introduced by Auslander a
Nakai conjecture
In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.It states that if V is a complex algeb
Calogero conjecture
The Calogero conjecture is a minority interpretation of quantum mechanics. It is a quantization explanation involving quantum mechanics, originally stipulated in 1997 and further republished in 2004 b
Sidorenko's conjecture
Sidorenko's conjecture is a conjecture in the field of graph theory, posed by in 1986. Roughly speaking, the conjecture states that for any bipartite graph and graph on vertices with average degree ,
Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integ
Erdős sumset conjecture
In additive combinatorics, the Erdős sumset conjecture is a conjecture which states that if a subset of the natural numbers has a positive upper density then there are two infinite subsets and of such
N! conjecture
In mathematics, the n! conjecture is the conjecture that the dimension of a certain module of is n!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald's pos
Union-closed sets conjecture
In combinatorics, the union-closed sets conjecture is a problem, posed by Péter Frankl in 1979 and is still open. A family of sets is said to be union-closed if the union of any two sets from the fami
Cartan–Hadamard conjecture
In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized
Kummer–Vandiver conjecture
In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime p does not divide the class number hK of the maximal real subfield of the p-th cyclotomic field. The conject
Novikov self-consistency principle
The Novikov self-consistency principle, also known as the Novikov self-consistency conjecture and Larry Niven's law of conservation of history, is a principle developed by Russian physicist Igor Dmitr
Uniform boundedness conjecture for rational points
In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field and a positive integer that there exists a number depending only on and such that f
ER = EPR
ER = EPR is a conjecture in physics stating that two entangled particles (a so-called Einstein–Podolsky–Rosen or EPR pair) are connected by a wormhole (or Einstein–Rosen bridge) and is thought by some
Fatou conjecture
In mathematics, the Fatou conjecture, named after Pierre Fatou, states that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters.
Generalized Poincaré conjecture
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere is a sphere. More precisely, one fixes a category of manifolds: topo
Main conjecture of Iwasawa theory
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying th
Mirror symmetry conjecture
In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curve
Singmaster's conjecture
Singmaster's conjecture is a conjecture in combinatorial number theory, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on th
Sumner's conjecture
Sumner's conjecture (also called Sumner's universal tournament conjecture) states that every orientation of every -vertex tree is a subgraph of every -vertex tournament. David Sumner, a graph theorist
Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be desc
Halperin conjecture
In rational homotopy theory, the Halperin conjecture concerns the Serre spectral sequence of certain fibrations. It is named after the Canadian mathematician Stephen Halperin.
Thomas–Yau conjecture
In mathematics, and especially symplectic geometry, the Thomas–Yau conjecture asks for the existence of a stability condition, similar to those which appear in algebraic geometry, which guarantees the
Pompeiu problem
In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929, as follows. Suppose f is a nonzero continuous function defined on a
Sato–Tate conjecture
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all pr
Virasoro conjecture
In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro
Gyárfás–Sumner conjecture
In graph theory, the Gyárfás–Sumner conjecture asks whether, for every tree and complete graph , the graphs with neither nor as induced subgraphs can be properly colored using only a constant number o
Hadwiger conjecture (graph theory)
In graph theory, the Hadwiger conjecture states that if is loopless and has no minor then its chromatic number satisfies . It is known to be true for . The conjecture is a generalization of the four-c
Cereceda's conjecture
In the mathematics of graph coloring, Cereceda’s conjecture is an unsolved problem on the distance between pairs of colorings of sparse graphs. It states that, for two different colorings of a graph o
Dissection into orthoschemes
In geometry, it is an unsolved conjecture of Hugo Hadwiger that every simplex can be dissected into orthoschemes, using a number of orthoschemes bounded by a function of the dimension of the simplex.
Langlands program
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Rober
Kahn–Kalai conjecture
The Kahn–Kalai conjecture, also known as the expectation threshold conjecture, is a conjecture in the field of graph theory and statistical mechanics, proposed by Jeff Kahn and Gil Kalai in 2006.
GNRS conjecture
In theoretical computer science and metric geometry, the GNRS conjecture connects the theory of graph minors, the stretch factor of embeddings, and the approximation ratio of multi-commodity flow prob
Nagata's conjecture on curves
In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with
Kneser–Tits conjecture
In mathematics, the Kneser–Tits problem, introduced by Tits based on a suggestion by Martin Kneser, asks whether the Whitehead group W(G,K) of a semisimple simply connected isotropic algebraic group G
Atiyah conjecture
In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.
Graceful labeling
In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with some subset of the integers from 0 to m inclusive, such that no two vertices share a label, and each edg
Andrews–Curtis conjecture
In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on th
Naimark's problem
Naimark's problem is a question in functional analysis asked by Naimark. It asks whether every C*-algebra that has only one irreducible -representation up to unitary equivalence is isomorphic to the -
Erdős–Hajnal conjecture
In graph theory, a branch of mathematics, the Erdős–Hajnal conjecture states that families of graphs defined by forbidden induced subgraphs have either large cliques or large independent sets. It is n
Rota's basis conjecture
In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if X is either a vector space o
Ryu–Takayanagi conjecture
The Ryu–Takayanagi conjecture is a conjecture within holography that posits a quantitative relationship between the entanglement entropy of a conformal field theory and the geometry of an associated a
Catalan's conjecture
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderbor
Brumer–Stark conjecture
The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's
Jacobian conjecture
In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself has Jaco
Lindelöf hypothesis
In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line. This hypothes
Lovász conjecture
In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally Lás
SYZ conjecture
The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and
Hall's conjecture
In mathematics, Hall's conjecture is an open question, as of 2015, on the differences between perfect squares and perfect cubes. It asserts that a perfect square y2 and a perfect cube x3 that are not
Beal conjecture
The Beal conjecture is the following conjecture in number theory: Unsolved problem in mathematics: If where A, B, C, x, y, z are positive integers and x, y, z are ≥ 3, do A, B, and C have a common pri
Fermat–Catalan conjecture
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many
NLTS Conjecture
In quantum information theory, the No Low-Energy Trivial State (NLTS) conjecture is a precursor to a and posits the existence of families of Hamiltonians with all low energy states of non-trivial comp
Schanuel's conjecture
In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions o
Birch–Tate conjecture
The Birch–Tate conjecture is a conjecture in mathematics (more specifically in algebraic K-theory) proposed by both Bryan John Birch and John Tate.
Turán's brick factory problem
Unsolved problem in mathematics: Can any complete bipartite graph be drawn with fewer crossings than the number given by Zarankiewicz? (more unsolved problems in mathematics) In the mathematics of gra
Vizing's conjecture
In graph theory, Vizing's conjecture concerns a relation between the domination number and the cartesian product of graphs. This conjecture was first stated by Vadim G. Vizing, and states that, if γ(G
Elementary function arithmetic
In proof theory, a branch of mathematical logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual
Parshin's conjecture
In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson–Parshin conjecture) states that for any smooth projective variety X defined over a fini
Lonely runner conjecture
In number theory, specifically the study of Diophantine approximation, the lonely runner conjecture is a conjecture about the long-term behavior of runners on a circular track. It states that runners
Weak gravity conjecture
The weak gravity conjecture (WGC) is a conjecture regarding the strength gravity can have in a theory of quantum gravity relative to the gauge forces in that theory. It roughly states that gravity sho
Dittert conjecture
The Dittert conjecture, or Dittert–Hajek conjecture, is a mathematical hypothesis (in combinatorics) concerning the maximum achieved by a particular function of matrices with real, nonnegative entries
Greenberg's conjectures
Greenberg's conjecture is either of two conjectures in algebraic number theory proposed by Ralph Greenberg. Both are still unsolved as of 2021.
Atiyah conjecture on configurations
In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by Atiyah stating that a certain n by n matrix depending on n points in R3 is always non-singular.
No-three-in-line problem
The no-three-in-line problem in discrete geometry asks how many points can be placed in the grid so that no three points lie on the same line. This number is at most , because points in a grid would i
Geometrization conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue
Ulam's packing conjecture
Ulam's packing conjecture, named for Stanislaw Ulam, is a conjecture about the highest possible packing density of identical convex solids in three-dimensional Euclidean space. The conjecture says tha
Scholz conjecture
In mathematics, the Scholz conjecture is a conjecture on the length of certain addition chains.It is sometimes also called the Scholz–Brauer conjecture or the Brauer–Scholz conjecture, after Arnold Sc
Rudin's conjecture
Rudin's conjecture is a mathematical hypothesis (in additive combinatorics and elementary number theory) concerning an upper bound for the number of squares in finite arithmetic progressions. The conj
Aanderaa–Karp–Rosenberg conjecture
In theoretical computer science, the Aanderaa–Karp–Rosenberg conjecture (also known as the Aanderaa–Rosenberg conjecture or the evasiveness conjecture) is a group of related conjectures about the numb
Selberg's 1/4 conjecture
In mathematics, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by Selberg , states that the eigenvalues of the Laplace operator on Maass wave forms of congruence subg
Torsion conjecture
In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian vari
Cycle double cover
In graph-theoretic mathematics, a cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph
Erdős–Turán conjecture on additive bases
The Erdős–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erdős conjecture on arithmetic progressions) posed by Paul Erdős and Pál Turán in 1941. The que
Whitehead conjecture
The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcompl
Zeeman conjecture
In mathematics, the Zeeman conjecture or Zeeman's collapsibility conjecture asks whether given a finite contractible 2-dimensional CW complex , the space is collapsible. The conjecture, due to Christo
Volume conjecture
In the branch of mathematics called knot theory, the volume conjecture is the following open problem that relates quantum invariants of knots to the hyperbolic geometry of knot complements. Let O deno
Yau's conjecture on the first eigenvalue
In mathematics, Yau's conjecture on the first eigenvalue is, as of 2018, an unsolved conjecture proposed by Shing-Tung Yau in 1982. It asks: Is it true that the first eigenvalue for the Laplace–Beltra
Manin conjecture
In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his
Khabibullin's conjecture on integral inequalities
In mathematics, Khabibullin's conjecture, named after , is related to Paley's problem for plurisubharmonic functions and to various extremal problems in the theory of entire functions of several varia
Pierce–Birkhoff conjecture
In abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first sta
Berman–Hartmanis conjecture
In structural complexity theory, the Berman–Hartmanis conjecture is an unsolved conjecture named after Leonard C. Berman and Juris Hartmanis that states that all NP-complete languages look alike, in t
Pollock's conjectures
Pollock's conjectures are two closely related unproven conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also
Lehmer's conjecture
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant such tha
Hardy–Littlewood zeta-function conjectures
In mathematics, the Hardy–Littlewood zeta-function conjectures, named after Godfrey Harold Hardy and John Edensor Littlewood, are two conjectures concerning the distances between zeros and the density
Leopoldt's conjecture
In algebraic number theory, Leopoldt's conjecture, introduced by H.-W. Leopoldt , states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual r
Bing–Borsuk conjecture
In mathematics, the Bing–Borsuk conjecture states that every -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and
Local Langlands conjectures
In mathematics, the local Langlands conjectures, introduced by Robert Langlands , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive a
Frankel conjecture
In the mathematical fields of differential geometry and algebraic geometry, the Frankel conjecture was a problem posed by Theodore Frankel in 1961. It was resolved in 1979 by Shigefumi Mori, and by Yu
Mean value problem
In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks: For a given complex polynomial of degree and a complex numbe
Borel conjecture
In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. I
Dade's conjecture
In finite group theory, Dade's conjecture is a conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups, introduced by Everet
Homotopy hypothesis
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geo
Kaplan–Yorke conjecture
In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. By arranging the Lyapunov exponents in order from largest to smallest , let j be t
List of unsolved problems in mathematics
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, alg
Littlewood polynomial
In mathematics, a Littlewood polynomial is a polynomial all of whose coefficients are +1 or −1.Littlewood's problem asks how large the values of such a polynomial must be on the unit circle in the com
Hodge conjecture
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subva
Erdős–Straus conjecture
Unsolved problem in mathematics: Does have a positive integer solution for every integer ? (more unsolved problems in mathematics) The Erdős–Straus conjecture is an unproven statement in number theory
Weil's conjecture on Tamagawa numbers
In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number of a simply connected simple algebraic group defined over a number field is 1. In this case, simply co