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List of unsolved problems in mathematics

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Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved.

Unsolved problems remain in multiple domains, including physics, computer science, algebra, additive and algebraic number theories, analysis, combinatorics, algebraic, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and miscellaneous unsolved problems. 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.

Lists of unsolved problems in mathematics

Over the course of time, several lists of unsolved mathematical problems have appeared.

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

Millennium Prize Problems

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved, as of 2019:

The seventh problem, the Poincaré conjecture, has been solved. 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.

Unsolved problems

Algebra

Algebraic geometry

Analysis

Combinatorics

Differential geometry

Discrete geometry

Euclidean geometry

Dynamical systems

Games and puzzles

Graph theory

Paths and cycles in graphs

Graph coloring and labeling

Graph drawing

Miscellaneous graph theory

Group theory

Model theory

  • Vaught's conjecture
  • The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in 0 {\displaystyle \aleph _{0}} is a simple algebraic group over an algebraically closed field.
  • The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for 1 {\displaystyle \aleph _{1}} -saturated models of a countable theory.
  • Determine the structure of Keisler's order
  • The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
  • Is the theory of the field of Laurent series over Z p {\displaystyle \mathbb {Z} _{p}} decidable? of the field of polynomials over C {\displaystyle \mathbb {C} } ?
  • (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?
  • The Stable Forking Conjecture for simple theories
  • For which number fields does Hilbert's tenth problem hold?
  • Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ω 1 {\displaystyle \aleph _{\omega _{1}}} does it have a model of cardinality continuum?
  • Shelah's eventual Categority conjecture: For every cardinal λ {\displaystyle \lambda } there exists a cardinal μ ( λ ) {\displaystyle \mu (\lambda )} such that If an AEC K with LS(K)<= λ {\displaystyle \lambda } is categorical in a cardinal above μ ( λ ) {\displaystyle \mu (\lambda )} then it is categorical in all cardinals above μ ( λ ) {\displaystyle \mu (\lambda )} .
  • Shelah's categoricity conjecture for L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.
  • Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?
  • If the class of atomic models of a complete first order theory is categorical in the n {\displaystyle \aleph _{n}} , is it categorical in every cardinal?
  • 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.)
  • Kueker's conjecture
  • Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
  • Lachlan's decision problem
  • Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
  • Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
  • 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?
  • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?

Number theory

General

Additive number theory

See also: Problems involving arithmetic progressions

Algebraic number theory

Computational number theory

Prime numbers

Prime number conjectures

Partial differential equations

Ramsey theory

Set theory

Topology

Other

See also: List of conjectures

Problems solved since 1995

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