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Renaissance, every century has seen the solution of more mathematical problems that the century before, and yet many mathematical problems, both major and minor, still elude solution. Most graduate students, in order to earn a Ph.D. in mathematics, are expected to produce new, original mathematics. That is, they are expected to solve problems that are not routine, and which cannot be solved by standard methods. [1 ]
An unsolved problem in mathematics does not refer to the kind of problem found as an exercise in a textbook, but rather to the answer to a major question or a general method that provides a solution to an entire class of problems. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems receive considerable attention. This article reiterates the
Millennium Prize list of unsolved problems in mathematics as of October 2014, and lists further unsolved problems in algebra, additive and algebraic number theories, analysis, combinatorics, algebraic, discrete, and Euclidean geometries, dynamical systems, partial differential equations, and graph, group, model, number, set and Ramsey theories, as well as miscellaneous unsolved problems. A list of problems solved since 1975 also appears, alongside some sources, general and specific, for the stated problems.
Lists of unsolved problems in mathematics [ edit ]
Over the course of time, several lists of unsolved mathematical problems have appeared. The following is a listing of those lists.
Millennium Prize Problems [ edit ]
Of the seven
Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved, as of October 2014, : [6 ]
The seventh problem, the
Poincaré conjecture, has been solved. The [7 ] smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved. [8 ]
Other still-unsolved problems [ edit ]
Pompeiu problem Are
(the Euler–Mascheroni constant), + π , e π − e, π e, π/ e, π , e π , √ 2 π , e π , π 2 ln π, 2 , e e , e Catalan's constant or Khinchin's constant rational, algebraic irrational, or transcendental? What is the irrationality measure of each of these numbers? [9 ] [10 ] [11 ] The
Khabibullin’s conjecture on integral inequalities
Vaught's conjecture The
Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in 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 -saturated models of a countable theory. [17 ] Determine the structure of Keisler's order
[18 ] [19 ] 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
decidable? of the field of polynomials over ? (BMTO) Is the Borel monadic theory of the real order
decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable? [20 ] The Stable Forking Conjecture for simple theories
[21 ] 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 does it have a model of cardinality continuum? [22 ] Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?
[23 ] If the class of atomic models of a complete first order theory is
categorical in the , is it categorical in every cardinal? [24 ] [25 ] Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
[26 ] 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?
[27 ] The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?
(Proof claimed in 2012, currently under review.) abc conjecture
Carmichael's totient function conjecture
Erdős–Straus conjecture Do any
odd perfect numbers exist? Are there infinitely many
perfect numbers? Do
quasiperfect numbers exist? Do any odd
weird numbers exist? Do any
Lychrel numbers exist? Is 10 a
solitary number? Do any
Taxicab(5, 2, n) exist for n>1?
Brocard's problem: existence of integers, n, m, such that n!+1= m 2 other than n=4,5,7
Distribution and upper bound of mimic numbers
Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
Lehmer's totient problem: if φ( n) divides n − 1, must n be prime? Are there infinitely many
amicable numbers? Are there any pairs of
relatively prime amicable numbers? The
Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?
Problems solved since 1975 [ edit ]
This article needs additional citations for . verification (December 2014)
Kadison–Singer problem ( Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)
Willmore conjecture ( Fernando Codá Marques and André Neves, 2012) [32 ]
Gromov's problem on distortion of knots ( John Pardon, 2011)
Circular law ( Terence Tao and Van H. Vu, 2010)
Hirsch conjecture ( Francisco Santos Leal, 2010) [33 ] [34 ]
Serre's modularity conjecture ( Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008) [35 ] [36 ]
Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2007)
Weinstein conjecture for closed 3-dimensional manifolds ( Clifford Taubes, 2007)
Road coloring conjecture ( Avraham Trahtman, 2007) The
Angel problem (Various independent proofs, 2006) The
Langlands–Shelstad fundamental lemma ( Ngô Bảo Châu and Gérard Laumon, 2004)
Stanley–Wilf conjecture ( Gábor Tardos and Adam Marcus, 2004)
Green–Tao theorem ( Ben J. Green and Terence Tao, 2004)
Cameron–Erdős conjecture ( Ben J. Green, 2003, Alexander Sapozhenko, 2003) [37 ]
Strong perfect graph conjecture ( Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)
Poincaré conjecture ( Grigori Perelman, 2002)
Catalan's conjecture ( Preda Mihăilescu, 2002)
Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, 2001) The
Langlands correspondence for function fields ( Laurent Lafforgue, 1999)
Taniyama–Shimura conjecture (Wiles, Breuil, Conrad, Diamond, and Taylor, 1999)
Kepler conjecture ( Thomas Hales, 1998)
Milnor conjecture ( Vladimir Voevodsky, 1996)
Fermat's Last Theorem ( Andrew Wiles and Richard Taylor, 1995)
Bieberbach conjecture ( Louis de Branges, 1985)
Princess and monster game ( Shmuel Gal, 1979) [38 ] [39 ] [40 ]
Four color theorem ( Appel and Haken, 1977)
References [ edit ]
^ Eves, An Introduction to the History of Mathematics 6th Edition, Thomson, 1990, ISBN 978-0-03-029558-4.
^ Shimura, G. (1989). "Yutaka Taniyama and his time". Bulletin of the London Mathematical Society 21 (2): 186–196. doi: 10.1112/blms/21.2.186.
^ Abe, Jair Minoro; Tanaka, Shotaro (2001). . IOS Press. Unsolved Problems on Mathematics for the 21st Century ISBN 9051994907.
^ "DARPA invests in math". CNN. 2008-10-14. Archived from the original on 2009-03-04 . Retrieved . 2013-01-14
^ "Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO)". DARPA. 2007-09-10. Archived from the original on 2012-10-01 . Retrieved . 2013-06-25
^ "Poincaré Conjecture". Clay Mathematics Institute.
^ For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi ( ), e ( ), Khinchin's Constant ( ), irrational numbers ( ), transcendental numbers ( ), and irrationality measures ( ) at Wolfram MathWorld, all articles accessed 15 December 2014.
^ Michel Waldschmidt, 2008, "An introduction to irrationality and transcendence methods," at The University of Arizona The Southwest Center for Arithmetic Geometry 2008 Arizona Winter School, March 15–19, 2008 (Special Functions and Transcendence), see , accessed 15 December 2014.
^ John Albert, posting date unknown, "Some unsolved problems in number theory" [from Victor Klee & Stan Wagon, "Old and New Unsolved Problems in Plane Geometry and Number Theory"], in University of Oklahoma Math 4513 course materials, see , accessed 15 December 2014.
^ 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 .
^ Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society 61 (4): 346–253, doi: 10.1090/noti1100 .
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^ Shelah S, Classification Theory, North-Holland, 1990
^ Keisler, HJ, “Ultraproducts which are not saturated.” J. Symb Logic 32 (1967) 23—46.
^ Malliaris M, Shelah S, "A dividing line in simple unstable theories." http://arxiv.org/abs/1208.2140
^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
^ Peretz, Assaf, “Geometry of forking in simple theories.” J. Symbolic Logic Volume 71, Issue 1 (2006), 347–359.
^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae 159 (1): 1–50. arXiv: math/9802134.
^ Makowsky J, “Compactness, embeddings and definability,” in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
^ Baldwin, John T. (July 24, 2009). Categoricity (PDF). American Mathematical Society. ISBN 978-0821848937 . Retrieved . February 20, 2014
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^ Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B 97 (3): 293–333. arXiv: math/0512218. doi: 10.1016/j.jctb.2006.05.008.
^ Džamonja, Mirna, “Club guessing and the universal models.” On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
^ Ribenboim, P. (2006). (in German) (2nd ed.). Springer. pp. 242–243. Die Welt der Primzahlen doi: 10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
^ Dobson, J. B. (June 2012) , On Lerch's formula for the Fermat quotient, p. 15, arXiv: 1103.3907
^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society 125: 1503–1509, JSTOR 2162098 .
^ Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. arXiv: 1202.6036.
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^ Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica. Extra Volume "Optimization Stories": 75–85.
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^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (II)", Inventiones Mathematicae 178 (3): 505–586, doi: 10.1007/s00222-009-0206-6
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Further reading [ edit ]
Books discussing unsolved problems [ edit ]
Fan Chung; Graham, Ron (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 1-56881-111-X.
Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3.
Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN 0-387-20860-7.
Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0-88385-315-9.
Du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0-06-093558-8.
Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0-309-08549-7.
Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8.
Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0-691-11748-9.
Books discussing recently solved problems [ edit ]
Other works [ edit ]
External links [ edit ]