List of unsolved problems in mathematics

Many mathematical problems have not been solved yet. These unsolved problems occur in multiple domains, including theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, and partial differential equations. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.

The Riemann zeta function, subject of the celebrated and influential unsolved problem known as the Riemann hypothesis

This article is a composite of notable unsolved problems derived from many sources, including but not limited to lists considered authoritative. It does not claim to be comprehensive, it may not always be quite up to date, and it includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole.

Lists of unsolved problems in mathematicsEdit

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.

List Number of problems Number unresolved
or incompletely resolved
Proposed by Proposed in
Hilbert's problems[1] 23 15 David Hilbert 1900
Landau's problems[2] 4 4 Edmund Landau 1912
Taniyama's problems[3] 36 - Yutaka Taniyama 1955
Thurston's 24 questions[4][5] 24 - William Thurston 1982
Smale's problems 18 14 Stephen Smale 1998
Millennium Prize problems 7 6[6] Clay Mathematics Institute 2000
Simon problems 15 <12[7][8] Barry Simon 2000
Unsolved Problems on Mathematics for the 21st Century[9] 22 - Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges[10][11] 23 - DARPA 2007

Millennium Prize ProblemsEdit

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved as of May, 2021:[6]

The seventh problem, the Poincaré conjecture, has been solved;[12] however, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.[13]

Unsolved problemsEdit

AlgebraEdit

 
In the Bloch sphere representation of a qubit, a SIC-POVM forms a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.

Notebook problemsEdit

  • The Dneister Notebook (Dnestrovskaya Tetrad) collects several hundred unresolved problems in algebra, particularly ring theory and modulus theory.[14]
  • The Erlagol Notebook (Erlagolskaya Tetrad) collects unresolved problems in algebra and model theory.[15]

Conjectures and problemsEdit

AnalysisEdit

 
The area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.

Conjectures and problemsEdit

Open questionsEdit

OtherEdit

CombinatoricsEdit

Conjectures and problemsEdit

OtherEdit

Dynamical systemsEdit

 
A detail of the Mandelbrot set. It is not known whether the Mandelbrot set is locally connected or not.

Conjectures and problemsEdit

Open questionsEdit

Games and puzzlesEdit

Combinatorial gamesEdit

Games with imperfect informationEdit

GeometryEdit

Algebraic geometryEdit

ConjecturesEdit
OtherEdit

Covering and packingEdit

Conjectures and problemsEdit

Differential geometryEdit

Conjectures and problemsEdit

Discrete geometryEdit

 
In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.
Conjectures and problemsEdit
Open questionsEdit
OtherEdit

Euclidean geometryEdit

Conjectures and problemsEdit
Open questionsEdit
OtherEdit

Graph theoryEdit

Graph coloring and labelingEdit

 
An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.
Conjectures and problemsEdit

Graph drawingEdit

Conjectures and problemsEdit
OtherEdit

Paths and cycles in graphsEdit

Conjectures and problemsEdit

Word-representation of graphsEdit

Miscellaneous graph theoryEdit

Conjectures and problemsEdit
Open questionsEdit

Group theoryEdit

 
The free Burnside group   is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups   are finite remains open.

Notebook problemsEdit

  • The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.[120]

Conjectures and problemsEdit

Open questionsEdit

Model theory and formal languagesEdit

Conjectures and problemsEdit

  • The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in   is a simple algebraic group over an algebraically closed field.
  • Generalized star height problem
  • For which number fields does Hilbert's tenth problem hold?
  • Kueker's conjecture[122]
  • The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for  -saturated models of a countable theory.[123]
  • Shelah's categoricity conjecture for  : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[123]
  • Shelah's eventual categoricity conjecture: For every cardinal   there exists a cardinal   such that if an AEC K with LS(K)<=   is categorical in a cardinal above   then it is categorical in all cardinals above  .[123][124]
  • The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
  • The Stable Forking Conjecture for simple theories[125]
  • Tarski's exponential function problem
  • The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?[126]
  • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[127]
  • Vaught's conjecture

Open questionsEdit

  • Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality   does it have a model of cardinality continuum?[128]
  • Do the Henson graphs have the finite model property?
  • Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
  • Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
  • If the class of atomic models of a complete first order theory is categorical in the  , is it categorical in every cardinal?[129][130]
  • Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
  • (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[131]
  • Is the theory of the field of Laurent series over   decidable? of the field of polynomials over  ?
  • Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[132]

OtherEdit

  • Determine the structure of Keisler's order[133][134]

Number theoryEdit

GeneralEdit

 
6 is a perfect number because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them are odd.
Conjectures, problems and hypothesesEdit
Open questionsEdit
OtherEdit

Additive number theoryEdit

Conjectures and problemsEdit
Open questionsEdit
OtherEdit

Algebraic number theoryEdit

Conjectures and problemsEdit
OtherEdit
  • Characterize all algebraic number fields that have some power basis.

Computational number theoryEdit

Prime numbersEdit

 
Goldbach's conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.
Conjectures, problems and hypothesesEdit
Open questionsEdit

Set theoryEdit

Conjectures, problems, and hypothesesEdit

Open questionsEdit

TopologyEdit

 
The unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.

Conjectures and problemsEdit

Problems solved since 1995Edit

 
Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.

AlgebraEdit

AnalysisEdit

CombinatoricsEdit

Game theoryEdit

GeometryEdit

21st centuryEdit

20th centuryEdit

Graph theoryEdit

Group theoryEdit

Number theoryEdit

21st centuryEdit

20th centuryEdit

Ramsey theoryEdit

Theoretical computer scienceEdit

TopologyEdit

UncategorisedEdit

21st centuryEdit

2010sEdit
2000sEdit

20th centuryEdit

See alsoEdit

ReferencesEdit

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Further readingEdit

Books discussing problems solved since 1995Edit

Books discussing unsolved problemsEdit

External linksEdit

  1. ^ The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1979
  2. ^ The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1989
  3. ^ Fuks 1974, p. 47, 88, 116, 134, 158, 159, 186, 210, 242, 243, 292, 318.
  4. ^ Boltiansky 1965, p. 83.
  5. ^ Grunbaum 1971, p. 6.
  6. ^ V. G. Vizing Some unresolved problems for Graph theory // Russian Mathematical Surveys, 23:6(144) (1968), 117–134; Russian Math. Surveys, 23:6 (1968), 125–141
  7. ^ Sprinjuk 1967, p. 150—154.