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Unsolved problems in mathematics: Wikis

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From Wikipedia, the free encyclopedia

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

Contents

Millennium Prize Problems

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

Only 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

Additive number theory

Number theory: prime numbers

General number theory

Algebraic number theory

Discrete geometry

Ramsey theory

General algebra

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 Sn
  • 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 more than 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?

Graph theory

Analysis

Partial differential equations

  • Regularity of solutions of Vlasov-Maxwell equations
  • Regularity of solutions of Euler equations

Group theory

Set theory

Other

Problems solved recently

See also

References

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 0060935588. 
  • Keith Devlin (2006). The Millennium Problems - The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 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. 

Books discussing recently solved problems

  • Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1841157910. 
  • Donal O'Shea (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-846-14012-9. 
  • George G. Szpiro (2003). Kepler's Conjecture. Wiley. ISBN 0-471-08601-0. 
  • Mark Ronan (2006). Symmetry and the Monster. Oxford. ISBN 0-19-280722-6. 

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