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# List of unsolved problems in mathematics

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 Title: List of unsolved problems in mathematics Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### List of unsolved problems in mathematics

This article lists some unsolved problems in mathematics. See individual articles for details and sources.

## Contents

• Millennium Prize Problems 1
• Other still-unsolved problems 2
• Algebra 2.2
• Algebraic geometry 2.3
• Algebraic number theory 2.4
• Analysis 2.5
• Combinatorics 2.6
• Discrete geometry 2.7
• Euclidean geometry 2.8
• Dynamical system 2.9
• Graph theory 2.10
• Group theory 2.11
• Model theory 2.12
• Number theory (general) 2.13
• Number theory (prime numbers) 2.14
• Partial differential equations 2.15
• Ramsey theory 2.16
• Set theory 2.17
• Other 2.18
• Problems solved recently 3
• References 5
• Books discussing unsolved problems 5.1
• Books discussing recently solved problems 5.2

## Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved:

The seventh problem, the Poincaré conjecture, has been solved. The smooth four-dimensional Poincaré conjecture is still unsolved. That is, can a four-dimensional topological sphere have two or more inequivalent smooth structures?

## Other still-unsolved problems

### Model theory

• Vaught's conjecture
• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in \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 \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 \mathbb{Z}_p decidable? of the field of polynomials over \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 \aleph_{\omega_1} does it have a model of cardinality continuum?
• 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 \aleph_n, is it categorical in every cardinal?
• Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
• 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?