# List of unsolved problems in mathematics

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

## 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

### Combinatorics

• Number of magic squares (sequence A006052 in OEIS)
• Finding a formula for the probability that two elements chosen at random generate the symmetric group $S_n$
• Frankl's union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
• The Lonely runner conjecture: if $k+1$ runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance $1/(k+1)$ from each other runner) at some time?
• Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?
• The 1/3–2/3 conjecture: does every finite partially ordered set 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?

### Dynamical systems

• Furstenberg conjecture – Is every invariant and ergodic measure for the $\times 2,\times 3$ action on the circle either Lebesgue or atomic?
• Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups
• MLC conjecture - Is the Mandelbrot set locally connected ?

### 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.[14]
• Determine the structure of Keisler's order[15][16]
• 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?[17]
• The Stable Forking Conjecture for simple theories[18]
• 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?[19]
• Is there a logic satisfying the interpolation theorem which is compact?[20]
• If the class of atomic models of a complete first order theory is categorical in the $\aleph_n$, is it categorical in every cardinal?[21][22]
• Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
• Kueker's conjecture[23]
• 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?[24]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[25]

## References

1. ^
2. ^
3. ^
4. ^
5. ^
6. ^
7. ^ An introduction to irrationality and transcendence methods
8. ^ Some unsolved problems in number theory
9. ^ Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer 34 (1): 18–28, arXiv:1009.1419, doi:10.1007/s00283-011-9255-y, MR 2902144.
10. ^ Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society 61 (4): 346–253.
11. ^ Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete and Computational Geometry 7 (2): 153–162, doi:10.1007/BF02187832, MR 1139077.
12. ^ Wagner, Neal R. (1976), "The Sofa Problem", The American Mathematical Monthly 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022
13. ^ Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338.
14. ^ Shelah S, Classification Theory, North-Holland, 1990
15. ^ Keisler, HJ, “Ultraproducts which are not saturated.” J. Symb Logic 32 (1967) 23—46.
16. ^ Malliaris M, Shelah S, "A dividing line in simple unstable theories." http://arxiv.org/abs/1208.2140
17. ^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479-506.
18. ^ Peretz, Assaf, “Geometry of forking in simple theories.” J. Symbolic Logic Volume 71, Issue 1 (2006), 347-359.
19. ^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae 159 (1): 1–50. arXiv:9802134.
20. ^ Makowsky J, “Compactness, embeddings and definability,” in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645-715.
21. ^ Baldwin, John T. (July 24, 2009). Categoricity. American Mathematical Society. ISBN 978-0821848937. Retrieved February 20, 2014.
22. ^
23. ^ Hrushovski, Ehud, “Kueker's conjecture for stable theories.” Journal of Symbolic Logic Vol. 54, No. 1 (Mar., 1989), pp. 207-220.
24. ^ Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B 97 (3): 293–333. arXiv:0512218.
25. ^ Džamonja, Mirna, “Club guessing and the universal models.” On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
26. ^ Ribenboim, P. (2006). Die Welt der Primzahlen (in German) (2 ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
27. ^ Dobson, J. B. (June 2012) [2011], On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907
28. ^ Malliaris, M.; Shelah, S. (2012), Cofinality spectrum theorems in model theory, set theory and general topology, arXiv:1208.5424
29. ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society 125: 1503–1509, JSTOR 2162098.
30. ^ Franciscos Santos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics (Princeton University and Institute for Advanced Study) 176 (1): 383–412. doi:10.4007/annals.2012.176.1.7.
31. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (I)", Inventiones Mathematicae 178 (3): 485–504, doi:10.1007/s00222-009-0205-7 and Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (II)", Inventiones Mathematicae 178 (3): 505–586, doi:10.1007/s00222-009-0206-6.
32. ^ Green, Ben (2004), "The Cameron–Erdős conjecture", The Bulletin of the London Mathematical Society 36 (6): 769–778, arXiv:math.NT/0304058, doi:10.1112/S0024609304003650, MR 2083752.

### Books discussing unsolved problems

• Fan Chung; Ron Graham (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 1-56881-111-X.
• Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3.
• Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer. ISBN 0-387-20860-7.
• Victor Klee; Stan Wagon (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0-88385-315-9.
• Marcus Du Sautoy (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0-06-093558-8.
• John Derbyshire (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0-309-08549-7.
• Keith Devlin (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8.
• Vincent D. Blondel, Alexandre Megrestski (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0-691-11748-9.