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, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, mathematical logic, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention.

This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance.

Notable lists For over a century, various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions, with the Millennium Prize Problems each containing a reward of one million dollars.

List Number ofproblems Number unsolved or incompletely solved Proposed by Proposedin Hilbert's problems 23 13 David Hilbert 1900 Landau's problems. 4 4 Edmund Landau 1912 Taniyama's problems 36 – Yutaka Taniyama 1955 Thurston's 24 questions 24 2 William Thurston 1982 Smale's problems 18 14 Stephen Smale 1998 Millennium Prize Problems 7 6 Clay Mathematics Institute 2000 Simon problems 15 12 Barry Simon 2000 DARPA's math challenges 23 – DARPA 2007 Erdős's problems ≥ 1221 635 Paul Erdős Over six decades of Erdős' career, from the 1930s to 1990s

Millennium Prize Problems Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:

Birch and Swinnerton-Dyer conjecture Hodge conjecture Navier–Stokes existence and smoothness P versus NP Riemann hypothesis Yang–Mills existence and mass gap

The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003.

Contrary to most of the Hilbert problems and many other problems described in this article which allow for varying interpretations, the Millennium Prize problems are precisely described to provide clear conditions to award winners. After Perelman's resolution of the Poincaré conjecture, the official descriptions of the remaining six problems were codified by six specialists and released publicly. While the problems can touch upon greater research topics, and can themselves be only special cases of more general open problems, the Clay Institute decided upon these particular cases of the problems: The Poincaré conjecture can be formulated in terms of the generalized Poincaré conjecture for different categories of manifolds. In the category of piecewise linear manifolds, the generalized conjecture is still an open problem for 4-dimensional manifolds. For differential manifolds, it is known to be false generally, but it is an open problem to determine in which dimensions it is true. In other words, it is not yet fully known on which spheres there exist no exotic smooth structures. The Millennium Prize was offered for the topological version of the conjecture, that remained open only for 3-dimensional manifolds during the publication of the list. The Birch and Swinnerton-Dyer conjecture can be formulated for elliptic curves over any number field, and generalizations for more general arithmetic objects have been proposed. The Millennium Prize is offered for resolving the conjecture for elliptic curves over rational numbers. The Navier–Stokes equations have a general form that can take into account many physical phenomena including the difference between volume viscosity and dynamic viscosity, compressible flow, change of viscosities in the flow, and an external force. The Millennium Prize is offered for proving any one of four statements, two of which deal with the Navier–Stokes equations with no external force while the other two deal with the Navier–Stokes equations with a smooth external force. The Riemann hypothesis has many generalizations for different types of L-functions that are expected to be true. The Millennium Prize problem is offered for showing the original Riemann hypothesis for the Riemann zeta function.

Notebooks The Kourovka Notebook () is a collection of unsolved problems in group theory, first published in 1965 and updated many times since. The Sverdlovsk Notebook () is a collection of unsolved problems in semigroup theory, first published in 1965 and updated every 2 to 4 years since. The Dniester Notebook () lists several hundred unsolved problems in algebra, particularly ring theory and modulus theory. The Erlagol Notebook () lists unsolved problems in algebra and model theory.

Unsolved problems

Algebra

Birch–Tate conjecture on the relation between the order of the center of the Steinberg group of the ring of integers of a number field to the field's Dedekind zeta function. Casas-Alvero conjecture: if a polynomial of degree defined over a field of characteristic has a factor in common with its first through -th derivative, then must be the -th power of a linear polynomial? Connes embedding problem in Von Neumann algebra theory Determinantal conjecture on the determinant of the sum of two normal matrices. Eilenberg–Ganea conjecture: a group with cohomological dimension 2 also has a 2-dimensional Eilenberg–MacLane space . Farrell–Jones conjecture on whether certain assembly maps are isomorphisms. Bost conjecture: a specific case of the Farrell–Jones conjecture Finite lattice representation problem: is every finite lattice isomorphic to the congruence lattice of some finite algebra? Goncharov conjecture on the cohomology of certain motivic complexes. Green's conjecture: the Clifford index of a non-hyperelliptic curve is determined by the extent to which it, as a canonical curve, has linear syzygies. Grothendieck–Katz p-curvature conjecture: a conjectured local–global principle for linear ordinary differential equations. Hadamard conjecture: for every positive integer , a Hadamard matrix of order exists. Williamson conjecture: the problem of finding Williamson matrices, which can be used to construct Hadamard matrices. Lander's conjecture: if an abelian group of order contains a difference set of order , and a prime divides both and , must the Sylow -subgroup be non-cyclic? This implies Ryser's cyclic difference-set conjecture, and hence the circulant Hadamard conjecture. Ryser's conjecture on circulant Hadamard matrices: no real circulant Hadamard matrix has order greater than 4. Barker sequence conjecture: no Barker sequence has length greater than 13; equivalently, none has even length greater than 4. Hadamard's maximal determinant problem: what is the largest determinant of a matrix with entries all equal to 1 or −1? Hilbert's fifteenth problem: put Schubert calculus on a rigorous foundation. Hilbert's sixteenth problem: what are the possible configurations of the connected components of M-curves? Homological conjectures in commutative algebra Jacobson's conjecture: the intersection of all powers of the Jacobson radical of a left-and-right Noetherian ring is precisely 0. Kaplansky's conjectures Köthe conjecture: if a ring has no nil ideal other than , then it has no nil one-sided ideal other than . Monomial conjecture on Noetherian local rings Existence of perfect cuboids and associated cuboid conjectures Pierce–Birkhoff conjecture: every piecewise-polynomial is the maximum of a finite set of minimums of finite collections of polynomials. Rota's basis conjecture: for matroids of rank with disjoint bases , it is possible to create an matrix whose rows are and whose columns are also bases. Serre's conjecture II: if is a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most , then the Galois cohomology set is zero. Serre's positivity conjecture that if is a commutative regular local ring, and are prime ideals of , then implies . Uniform boundedness conjecture for rational points: do algebraic curves of genus over number fields have at most some bounded number of -rational points? Wild problems: problems involving classification of pairs of matrices under simultaneous conjugation. Zariski–Lipman conjecture: for a complex algebraic variety with coordinate ring , if the derivations of are a free module over , then is smooth. Zauner's conjecture: do SIC-POVMs exist in all dimensions?

Group theory

Andrews–Curtis conjecture: every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on relators and conjugations of relators Bounded Burnside problem: for which positive integers m, n is the free Burnside group finite? In particular, is finite? Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems Herzog–Schönheim conjecture: if a finite system of left cosets of subgroups of a group form a partition of , then the finite indices of said subgroups cannot be distinct. The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals? Isomorphism problem of Coxeter groups Are there an infinite number of Leinster groups? Does generalized moonshine exist? Is every finitely presented periodic group finite? Is every group surjunctive? Problems in loop theory and quasigroup theory consider generalizations of groups

Representation theory Arthur's conjectures Dade's conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups. Demazure conjecture on representations of algebraic groups over the integers. Kazhdan–Lusztig conjectures relating the values of the Kazhdan–Lusztig polynomials at 1 with representations of complex semisimple Lie groups and Lie algebras. McKay conjecture: in a group , the number of irreducible complex characters of degree not divisible by a prime number is equal to the number of irreducible complex characters of the normalizer of any Sylow -subgroup within .

Analysis

The Brennan conjecture: estimating the integral powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of Fuglede's conjecture on whether nonconvex sets in and are spectral if and only if they tile by translation. Goodman's conjecture on the coefficients of multivalued functions Invariant subspace problem – does every bounded operator on a complex Banach space send some non-trivial closed subspace to itself? Kung–Traub conjecture on the optimal order of a multipoint iteration without memory Lehmer's conjecture on the Mahler measure of non-cyclotomic polynomials The mean value problem: given a complex polynomial of degree and a complex number , is there a critical point of such that ? The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy Vitushkin's conjecture on compact subsets of with analytic capacity What is the exact value of Landau's constants, including Bloch's constant? Regularity of solutions of Euler equations Convergence of Flint Hills series Regularity of solutions of Vlasov–Maxwell equations Are there infinitely many Lehmer pairs?

Combinatorics

The 1/3–2/3 conjecture – does every finite partially ordered set that is not totally ordered contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?. The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition