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, and yet many mathematical problems, both major and minor, still remain unsolved.[1]

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[edit]

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

List Number of problems Proposed by Proposed in
Hilbert's problems[2] 23 David Hilbert 1900
Landau's problems[3] 4 Edmund Landau 1912
Taniyama's problems[4] 36 Yutaka Taniyama 1955
Thurston's 24 questions[5][6] 24 William Thurston 1982
Smale's problems 18 Stephen Smale 1998
Millennium Prize problems 7 Clay Mathematics Institute 2000
Simon problems 15 Barry Simon 2000
Unsolved Problems on Mathematics for the 21st Century[7] 22 Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges[8][9] 23 DARPA 2007

Millennium Prize Problems[edit]

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved, as of 2018:[10]

The seventh problem, the Poincaré conjecture, has been solved.[11] 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.[12]

Unsolved problems[edit]

Algebra[edit]

Algebraic geometry[edit]

Analysis[edit]

Combinatorics[edit]

Differential geometry[edit]

  • The filling area conjecture, that a hemisphere has the minimum area among shortcut-free surfaces in Euclidean space whose boundary forms a closed curve of given length[22]
  • The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds[23]

Discrete geometry[edit]

Euclidean geometry[edit]

Dynamical systems[edit]

Games and puzzles[edit]

Graph theory[edit]

Paths and cycles in graphs[edit]

Graph coloring and labeling[edit]

Graph drawing[edit]

Miscellaneous graph theory[edit]

Group theory[edit]

Model theory[edit]

  • 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.[82]
  • Determine the structure of Keisler's order[83][84]
  • 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?[85]
  • The Stable Forking Conjecture for simple theories[86]
  • 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?[87]
  • Shelah's eventual Categority conjecture: For every cardinal there exists a cardinal such that If an AEC K with LS(K)<= is categorical in a cardinal above then it is categorical in all cardinals above .[82][88]
  • Shelah's categoricity conjecture for : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[82]
  • Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[89]
  • If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?[90][91]
  • 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[92]
  • 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?[93]
  • The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[94]

Number theory[edit]

General[edit]

Additive number theory[edit]

Algebraic number theory[edit]

Combinatorial number theory[edit]

  • Singmaster's conjecture: Is there a finite upper bound on the number of times that a number other than 1 can appear in Pascal's triangle?

Computational number theory

Prime numbers[edit]

Partial differential equations[edit]

Ramsey theory[edit]

Set theory[edit]

Topology[edit]

Other[edit]

Problems solved since 1995[edit]

References[edit]

  1. ^ Eves, An Introduction to the History of Mathematics 6th Edition, Thomson, 1990, ISBN 978-0-03-029558-4.
  2. ^ Thiele, Rüdiger (2005), "On Hilbert and his twenty-four problems", in Van Brummelen, Glen, Mathematics and the historian's craft. The Kenneth O. May Lectures, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 21, pp. 243–295, ISBN 0-387-25284-3
  3. ^ Guy, Richard (1994), Unsolved Problems in Number Theory (2nd ed.), Springer, p. vii, ISBN 978-1-4899-3585-4.
  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.
  5. ^ http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/dmv_091514.pdf
  6. ^ "THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY" (PDF).
  7. ^ Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 9051994907.
  8. ^ "DARPA invests in math". CNN. 2008-10-14. Archived from the original on 2009-03-04. Retrieved 2013-01-14.
  9. ^ "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.
  10. ^ "Millennium Problems".
  11. ^ "Poincaré Conjecture". Clay Mathematics Institute. Archived from the original on 2013-12-15.
  12. ^ "Smooth 4-dimensional Poincare conjecture".
  13. ^ For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi ([1]), e ([2]), Khinchin's Constant ([3]), irrational numbers ([4]), transcendental numbers ([5]), and irrationality measures ([6]) at Wolfram MathWorld, all articles accessed 15 December 2014.
  14. ^ 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 [7], accessed 15 December 2014.
  15. ^ 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 [8], accessed 15 December 2014.
  16. ^ Bruhn, Henning; Schaudt, Oliver (2015), "The journey of the union-closed sets conjecture" (PDF), Graphs and Combinatorics, 31 (6): 2043–2074, arXiv:1309.3297, doi:10.1007/s00373-014-1515-0, MR 3417215
  17. ^ Tao, Terence (2017), "Some remarks on the lonely runner conjecture", arXiv:1701.02048
  18. ^ Singmaster, D. (1971), "Research Problems: How often does an integer occur as a binomial coefficient?", American Mathematical Monthly, 78 (4): 385–386, doi:10.2307/2316907, JSTOR 2316907, MR 1536288.
  19. ^ Liśkiewicz, Maciej; Ogihara, Mitsunori; Toda, Seinosuke (2003-07-28). "The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes". Theoretical Computer Science. 304 (1): 129–156. doi:10.1016/S0304-3975(03)00080-X.
  20. ^ Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, doi:10.1007/BF01110378, MR 1368815.
  21. ^ Murnaghan, F. D. (1938), "The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups", American Journal of Mathematics, 60 (1): 44–65, doi:10.2307/2371542, MR 1507301, PMC 1076971, PMID 16577800
  22. ^ Katz, Mikhail G. (2007), Systolic geometry and topology, Mathematical Surveys and Monographs, 137, American Mathematical Society, Providence, RI, p. 57, doi:10.1090/surv/137, ISBN 978-0-8218-4177-8, MR 2292367
  23. ^ Rosenberg, Steven (1997), The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds, London Mathematical Society Student Texts, 31, Cambridge: Cambridge University Press, pp. 62–63, doi:10.1017/CBO9780511623783, ISBN 0-521-46300-9, MR 1462892
  24. ^ Morris, Walter D.; Soltan, Valeriu (2000), "The Erdős-Szekeres problem on points in convex position—a survey", Bull. Amer. Math. Soc., 37 (4): 437–458, doi:10.1090/S0273-0979-00-00877-6, MR 1779413; Suk, Andrew (2016), "On the Erdős–Szekeres convex polygon problem", J. Amer. Math. Soc., 30: 1047–1053, arXiv:1604.08657, doi:10.1090/jams/869
  25. ^ Dey, Tamal K. (1998), "Improved bounds for planar k-sets and related problems", Discrete Comput. Geom., 19: 373–382, doi:10.1007/PL00009354, MR 1608878; Tóth, Gábor (2001), "Point sets with many k-sets", Discrete Comput. Geom., 26 (2): 187–194, doi:10.1007/s004540010022, MR 1843435.
  26. ^ Boltjansky, V.; Gohberg, I. (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
  27. ^ Weisstein, Eric W. "Kobon Triangle". MathWorld.
  28. ^ Matoušek, Jiří (2002), Lectures on discrete geometry, Graduate Texts in Mathematics, 212, Springer-Verlag, New York, p. 206, doi:10.1007/978-1-4613-0039-7, ISBN 0-387-95373-6, MR 1899299
  29. ^ Gardner, Martin (1995), New Mathematical Diversions (Revised Edition), Washington: Mathematical Association of America, p. 251
  30. ^ Conway, John H.; Neil J.A. Sloane (1999), Sphere Packings, Lattices and Groups (3rd ed.), New York: Springer-Verlag, pp. 21–22, ISBN 0-387-98585-9
  31. ^ Brass, Peter; Moser, William; Pach, János (2005), "5.1 The Maximum Number of Unit Distances in the Plane", Research problems in discrete geometry, Springer, New York, pp. 183–190, ISBN 0-387-23815-8, MR 2163782
  32. ^ Finch, S. R.; Wetzel, J. E. (2004), "Lost in a forest", American Mathematical Monthly, 11: 645–654, doi:10.2307/4145038, MR 2091541
  33. ^ Brandts, Jan; Korotov, Sergey; Křížek, Michal; Šolc, Jakub (2009), "On nonobtuse simplicial partitions", SIAM Review, 51 (2): 317–335, doi:10.1137/060669073, MR 2505583. See in particular Conjecture 23, p. 327.
  34. ^ 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
  35. ^ Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, MR 1252928
  36. ^ Arutyunyants, G.; Iosevich, A. (2004), "Falconer conjecture, spherical averages and discrete analogs", in Pach, János, Towards a Theory of Geometric Graphs, Contemp. Math., 342, Amer. Math. Soc., Providence, RI, pp. 15–24, doi:10.1090/conm/342/06127, MR 2065249
  37. ^ Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100
  38. ^ Katz, Nets; Tao, Terence (2002), "Recent progress on the Kakeya conjecture", Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), Publicacions Matemàtiques (Vol. Extra): 161–179, doi:10.5565/PUBLMAT_Esco02_07, MR 1964819
  39. ^ Weaire, Denis, ed. (1997), The Kelvin Problem, CRC Press, p. 1, ISBN 9780748406326
  40. ^ Brass, Peter; Moser, William; Pach, János (2005), Research problems in discrete geometry, New York: Springer, p. 457, MR 2163782
  41. ^ Stanley, Richard P. (1994), "A survey of Eulerian posets", in Bisztriczky, T.; McMullen, P.; Schneider, R.; Weiss, A. Ivić, Polytopes: abstract, convex and computational (Scarborough, ON, 1993), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 440, Dordrecht: Kluwer Academic Publishers, pp. 301–333, MR 1322068. See in particular p. 316.
  42. ^ 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
  43. ^ Wagner, Neal R. (1976), "The Sofa Problem" (PDF), The American Mathematical Monthly, 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022
  44. ^ 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
  45. ^ Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly, 59: 606–611, doi:10.2307/2306764, MR 0050303
  46. ^ ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, retrieved 2016-10-04.
  47. ^ Bereg, Sergey; Dumitrescu, Adrian; Jiang, Minghui (2010), "On covering problems of Rado", Algorithmica, 57 (3): 538–561, doi:10.1007/s00453-009-9298-z, MR 2609053
  48. ^ Kari, Jarkko (2009), "Structure of reversible cellular automata", Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7ÔÇô11, 2009, Proceedings, Lecture Notes in Computer Science, 5715, Springer, p. 6, Bibcode:2009LNCS.5715....6K, doi:10.1007/978-3-642-03745-0_5
  49. ^ a b c http://english.log-it-ex.com Ten open questions about Sudoku (2012-01-21).
  50. ^ "Higher-Dimensional Tic-Tac-Toe". PBS Infinite Series. YouTube. 2017-09-21. Retrieved 2018-07-29.
  51. ^ Florek, Jan (2010), "On Barnette's conjecture", Discrete Mathematics, 310 (10–11): 1531–1535, doi:10.1016/j.disc.2010.01.018, MR 2601261.
  52. ^ Broersma, Hajo; Patel, Viresh; Pyatkin, Artem (2014), "On toughness and Hamiltonicity of $2K_2$-free graphs", Journal of Graph Theory, 75 (3): 244–255, doi:10.1002/jgt.21734, MR 3153119
  53. ^ Jaeger, F. (1985), "A survey of the cycle double cover conjecture", Annals of Discrete Mathematics 27 – Cycles in Graphs, North-Holland Mathematics Studies, 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1.
  54. ^ Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7.
  55. ^ Akiyama, Jin; Exoo, Geoffrey; Harary, Frank (1981), "Covering and packing in graphs. IV. Linear arboricity", Networks, 11 (1): 69–72, doi:10.1002/net.3230110108, MR 0608921.
  56. ^ L. Babai, Automorphism groups, isomorphism, reconstruction, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447–1540.
  57. ^ Lenz, Hanfried; Ringel, Gerhard (1991), "A brief review on Egmont Köhler's mathematical work", Discrete Mathematics, 97 (1–3): 3–16, doi:10.1016/0012-365X(91)90416-Y, MR 1140782
  58. ^ Chung, Fan; Graham, Ron (1998), Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, pp. 97–99.
  59. ^ Chudnovsky, Maria; Seymour, Paul (2014), "Extending the Gyárfás-Sumner conjecture", Journal of Combinatorial Theory, Series B, 105: 11–16, doi:10.1016/j.jctb.2013.11.002, MR 3171779
  60. ^ Toft, Bjarne (1996), "A survey of Hadwiger's conjecture", Congressus Numerantium, 115: 249–283, MR 1411244.
  61. ^ Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991), Unsolved Problems in Geometry, Springer-Verlag, Problem G10.
  62. ^ Sauer, N. (2001), "Hedetniemi's conjecture: a survey", Discrete Mathematics, 229 (1–3): 261–292, doi:10.1016/S0012-365X(00)00213-2, MR 1815610.
  63. ^ Hägglund, Jonas; Steffen, Eckhard (2014), "Petersen-colorings and some families of snarks", Ars Mathematica Contemporanea, 7 (1): 161–173, MR 3047618.
  64. ^ Jensen, Tommy R.; Toft, Bjarne (1995), "12.20 List-Edge-Chromatic Numbers", Graph Coloring Problems, New York: Wiley-Interscience, pp. 201–202, ISBN 0-471-02865-7.
  65. ^ Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 0668845.
  66. ^ Molloy, Michael; Reed, Bruce (1998), "A bound on the total chromatic number", Combinatorica, 18 (2): 241–280, doi:10.1007/PL00009820, MR 1656544.
  67. ^ Barát, János; Tóth, Géza (2010), "Towards the Albertson Conjecture", Electronic Journal of Combinatorics, 17 (1): R73, arXiv:0909.0413, Bibcode:2009arXiv0909.0413B.
  68. ^ Wood, David (January 19, 2009), "Book Thickness of Subdivisions", Open Problem Garden, retrieved 2013-02-05.
  69. ^ Fulek, R.; Pach, J. (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry, 44 (6–7): 345–355, doi:10.1007/978-3-642-18469-7_21, MR 2785903.
  70. ^ Hartsfield, Nora; Ringel, Gerhard (2013), Pearls in Graph Theory: A Comprehensive Introduction, Dover Books on Mathematics, Courier Dover Publications, p. 247, ISBN 978-0-486-31552-2, MR 2047103.
  71. ^ Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, doi:10.1007/s00373-010-0934-9, MR 2669457.
  72. ^ Nöllenburg, Martin; Prutkin, Roman; Rutter, Ignaz (2016), "On self-approaching and increasing-chord drawings of 3-connected planar graphs", Journal of Computational Geometry, 7 (1): 47–69, arXiv:1409.0315, doi:10.20382/jocg.v7i1a3, MR 3463906
  73. ^ Pach, János; Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, 152, American Mathematical Society, pp. 126–127.
  74. ^ Demaine, E.; O'Rourke, J. (2002–2012), "Problem 45: Smallest Universal Set of Points for Planar Graphs", The Open Problems Project, retrieved 2013-03-19.
  75. ^ Chudnovsky, Maria (2014), "The Erdös–Hajnal conjecture—a survey" (PDF), Journal of Graph Theory, 75 (2): 178–190, arXiv:1606.08827, doi:10.1002/jgt.21730, MR 3150572, Zbl 1280.05086.
  76. ^ Spinrad, Jeremy P. (2003), "2. Implicit graph representation", Efficient Graph Representations, pp. 17–30, ISBN 0-8218-2815-0.
  77. ^ "Jorgensen's Conjecture", Open Problem Garden, retrieved 2016-11-13.
  78. ^ Baird, William; Bonato, Anthony (2012), "Meyniel's conjecture on the cop number: a survey", Journal of Combinatorics, 3 (2): 225–238, arXiv:1308.3385, doi:10.4310/JOC.2012.v3.n2.a6, MR 2980752
  79. ^ Ducey, Joshua E. (2017), "On the critical group of the missing Moore graph", Discrete Mathematics, 340 (5): 1104–1109, doi:10.1016/j.disc.2016.10.001, MR 3612450
  80. ^ Kühn, Daniela; Mycroft, Richard; Osthus, Deryk (2011), "A proof of Sumner's universal tournament conjecture for large tournaments", Proceedings of the London Mathematical Society, Third Series, 102 (4): 731–766, arXiv:1010.4430, doi:10.1112/plms/pdq035, MR 2793448, Zbl 1218.05034.
  81. ^ Brešar, Boštjan; Dorbec, Paul; Goddard, Wayne; Hartnell, Bert L.; Henning, Michael A.; Klavžar, Sandi; Rall, Douglas F. (2012), "Vizing's conjecture: a survey and recent results", Journal of Graph Theory, 69 (1): 46–76, doi:10.1002/jgt.20565, MR 2864622.
  82. ^ a b c Shelah S, Classification Theory, North-Holland, 1990
  83. ^ Keisler, HJ (1967). "Ultraproducts which are not saturated". J. Symb. Log. 32: 23–46. doi:10.2307/2271240.
  84. ^ Malliaris M, Shelah S, "A dividing line in simple unstable theories." https://arxiv.org/abs/1208.2140
  85. ^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
  86. ^ Peretz, Assaf (2006). "Geometry of forking in simple theories". Journal of Symbolic Logic Volume. 71 (1): 347–359. arXiv:math/0412356. doi:10.2178/jsl/1140641179.
  87. ^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae. 159 (1): 1–50. arXiv:math/9802134. Bibcode:1998math......2134S.
  88. ^ Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN 978-1-904987-71-0.
  89. ^ Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
  90. ^ Baldwin, John T. (July 24, 2009). Categoricity (PDF). American Mathematical Society. ISBN 978-0-8218-4893-7. Retrieved February 20, 2014.
  91. ^ Shelah, Saharon. "Introduction to classification theory for abstract elementary classes".
  92. ^ Hrushovski, Ehud (1989). "Kueker's conjecture for stable theories". Journal of Symbolic Logic. 54 (1): 207–220. doi:10.2307/2275025.
  93. ^ 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.
  94. ^ Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
  95. ^ "Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key".
  96. ^ https://arxiv.org/pdf/1604.07746v1.pdf
  97. ^ Guo, Song; Sun, Zhi-Wei (2005), "On odd covering systems with distinct moduli", Advances in Applied Mathematics, 35 (2): 182–187, doi:10.1016/j.aam.2005.01.004, MR 2152886
  98. ^ Aigner, Martin (2013), Markov's theorem and 100 years of the uniqueness conjecture, Cham: Springer, doi:10.1007/978-3-319-00888-2, ISBN 978-3-319-00887-5, MR 3098784
  99. ^ Ribenboim, P. (2006). Die Welt der Primzahlen (in German) (2nd ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
  100. ^ Dobson, J. B. (1 April 2017), "On Lerch's formula for the Fermat quotient", p. 23, arXiv:1103.3907v6
  101. ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society, 125: 1503–1509, JSTOR 2162098
  102. ^ Wolchover, Natalie (July 11, 2017), "Pentagon Tiling Proof Solves Century-Old Math Problem", Quanta Magazine
  103. ^ Lee, Choongbum (2017). "Ramsey numbers of degenerate graphs". Annals of Mathematics. 185 (3): 791–829. arXiv:1505.04773. doi:10.4007/annals.2017.185.3.2.
  104. ^ Lamb, Evelyn (26 May 2016), "Two-hundred-terabyte maths proof is largest ever", Nature, 534: 17–18, Bibcode:2016Natur.534...17L, doi:10.1038/nature.2016.19990, PMID 27251254; Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016), "Solving and verifying the Boolean Pythagorean triples problem via cube-and-conquer", Theory and applications of satisfiability testing—SAT 2016, Lecture Notes in Comput. Sci., 9710, Springer, [Cham], pp. 228–245, arXiv:1605.00723, doi:10.1007/978-3-319-40970-2_15, MR 3534782
  105. ^ Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398.
  106. ^ Bourgain, Jean; Ciprian, Demeter; Larry, Guth (2015). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". Annals of Mathematics. 184: 633–682. doi:10.4007/annals.2016.184.2.7.
  107. ^ https://arxiv.org/pdf/1509.05363v5.pdf
  108. ^ Duncan, John F. R.; Griffin, Michael J.; Ono, Ken (1 December 2015). "Proof of the umbral moonshine conjecture". Research in the Mathematical Sciences. 2 (1): 26. doi:10.1186/s40687-015-0044-7 – via link.springer.com.
  109. ^ https://arxiv.org/pdf/1406.6534v10.pdf
  110. ^ "A Long-Sought Proof, Found and Almost Lost". Quanta Magazine. Natalie Wolchover. March 28, 2017. Retrieved May 2, 2017.
  111. ^ Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
  112. ^ Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
  113. ^ Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
  114. ^ Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric (2006). "The Kadison-Singer problem in mathematics and engineering: A detailed account". In Han, Deguang; Jorgensen, Palle E. T.; Larson, David Royal. Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida. Contemporary Mathematics. 414. American Mathematical Society. pp. 299–355. doi:10.1090/conm/414/07820. ISBN 978-0-8218-3923-2. Retrieved 24 April 2015.
  115. ^ Mackenzie, Dana. "Kadison–Singer Problem Solved" (PDF). SIAM News (January/February 2014). Society for Industrial and Applied Mathematics. Retrieved 24 April 2015.
  116. ^ https://arxiv.org/pdf/1204.2810v1.pdf
  117. ^ http://www.math.jhu.edu/~js/Math646/brendle.lawson.pdf
  118. ^ Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. 179: 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6.
  119. ^ https://arxiv.org/pdf/1101.1330v4.pdf
  120. ^ http://www.math.uiuc.edu/~mineyev/math/art/submult-shnc.pdf
  121. ^ http://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf
  122. ^ https://www.uni-due.de/~bm0032/publ/BlochKato.pdf
  123. ^ "page 359" (PDF).
  124. ^ "motivic cohomology – Milnor–Bloch–Kato conjecture implies the Beilinson-Lichtenbaum conjecture – MathOverflow".
  125. ^ https://arxiv.org/pdf/1011.4105v3.pdf
  126. ^ https://www.researchgate.net/profile/Juan_Souto3/publication/228365532_Non-realizability_and_ending_laminations_Proof_of_the_Density_Conjecture/links/541d85a10cf2218008d1d2e5.pdf
  127. ^ Santos, Franciscos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics. Princeton University and Institute for Advanced Study. 176 (1): 383–412. arXiv:1006.2814. doi:10.4007/annals.2012.176.1.7.
  128. ^ Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica. Extra Volume "Optimization Stories": 75–85.
  129. ^ Cilleruelo, Javier (2010). "Generalized Sidon sets". Advances in Mathematics. 225: 2786–2807. doi:10.1016/j.aim.2010.05.010.
  130. ^ https://arxiv.org/pdf/0909.2360v3.pdf
  131. ^ https://arxiv.org/pdf/0906.1612v2.pdf
  132. ^ https://arxiv.org/pdf/0910.5501v5.pdf
  133. ^ http://www.csie.ntu.edu.tw/~hil/bib/ChalopinG09.pdf
  134. ^ Lurie, Jacob (2009). "On the classification of topological field theories". Current developments in mathematics. 2008: 129–280. doi:10.4310/cdm.2008.v2008.n1.a3.
  135. ^ https://arxiv.org/pdf/0809.4040.pdf
  136. ^ a b "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF) (Press release). Clay Mathematics Institute. March 18, 2010. Retrieved November 13, 2015. The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.
  137. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)", Inventiones Mathematicae, 178 (3): 485–504, Bibcode:2009InMat.178..485K, doi:10.1007/s00222-009-0205-7
  138. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505–586, Bibcode:2009InMat.178..505K, doi:10.1007/s00222-009-0206-6
  139. ^ "2011 Cole Prize in Number Theory" (PDF). Notices of the AMS. Providence, Rhode Island, United States: American Mathematical Society. 58 (4): 610–611. ISSN 1088-9477. OCLC 34550461.
  140. ^ http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf
  141. ^ https://arxiv.org/pdf/math/0509397.pdf
  142. ^ Seigel-Itzkovich, Judy (2008-02-08). "Russian immigrant solves math puzzle". The Jerusalem Post. Retrieved 2015-11-12.
  143. ^ http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf
  144. ^ http://home.broadpark.no/~oddvark/angel/Angel.pdf
  145. ^ http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf
  146. ^ http://www.cs.bu.edu/~gacs/papers/angel.pdf
  147. ^ http://www.ams.org/journals/proc/2005-133-09/S0002-9939-05-07752-X/S0002-9939-05-07752-X.pdf
  148. ^ "Fields Medal – Ngô Bảo Châu". International Congress of Mathematicians 2010. ICM. 19 August 2010. Retrieved 2015-11-12. Ngô Bảo Châu is being awarded the 2010 Fields Medal for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods.
  149. ^ https://arxiv.org/pdf/math/0405568v1.pdf
  150. ^ "Graph Theory".
  151. ^ Chung, Fan; Greene, Curtis; Hutchinson, Joan (April 2015). "Herbert S. Wilf (1931–2012)" (PDF). Notices of the AMS. Providence, Rhode Island, United States: American Mathematical Society. 62 (4): 358. ISSN 1088-9477. OCLC 34550461. The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004.
  152. ^ "Bombieri and Tao Receive King Faisal Prize" (PDF). Notices of the AMS. Providence, Rhode Island, United States: American Mathematical Society. 57 (5): 642–643. May 2010. ISSN 1088-9477. OCLC 34550461. Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem.
  153. ^ https://arxiv.org/pdf/math/0412006v2.pdf
  154. ^ Connelly, Robert; Demaine, Erik D.; Rote, Günter (2003), "Straightening polygonal arcs and convexifying polygonal cycles" (PDF), Discrete and Computational Geometry, 30 (2): 205–239, doi:10.1007/s00454-003-0006-7, MR 1931840
  155. ^ 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
  156. ^ "News from 2007". American Mathematical Society. AMS. 31 December 2007. Retrieved 2015-11-13. The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..."
  157. ^ "Reduced power operations in motivic cohomology" (PDF). archive.numdam.org.
  158. ^ Savchev, Svetoslav (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297: 196–201. doi:10.1016/j.disc.2005.02.018.
  159. ^ http://www.ams.org/journals/jams/2004-17-01/S0894-0347-03-00440-5/S0894-0347-03-00440-5.pdf
  160. ^ http://annals.math.princeton.edu/wp-content/uploads/annals-v158-n1-p04.pdf
  161. ^ "The strong perfect graph theorem".
  162. ^ http://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf
  163. ^ Knight, R. W. (2002), The Vaught Conjecture: A Counterexample, manuscript
  164. ^ http://www.ugr.es/~ritore/preprints/0406017.pdf
  165. ^ Metsänkylä, Tauno (5 September 2003). "Catalan's conjecture: another old diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. American Mathematical Society. 41 (1): 43–57. doi:10.1090/s0273-0979-03-00993-5. ISSN 0273-0979. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu.
  166. ^ http://www.ams.org/journals/jams/2001-14-04/S0894-0347-01-00373-3/S0894-0347-01-00373-3.pdf
  167. ^ http://junon.u-3mrs.fr/monniaux/AHLMT02.pdf
  168. ^ https://arxiv.org/pdf/math/0102150v4.pdf
  169. ^ Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3-adic exercises", Journal of the American Mathematical Society, 14 (4): 843–939, doi:10.1090/S0894-0347-01-00370-8, ISSN 0894-0347, MR 1839918
  170. ^ http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf
  171. ^ http://intlpress.com/site/pub/files/_fulltext/journals/sdg/2002/0007/0001/SDG-2002-0007-0001-a001.pdf
  172. ^ Croot, Ernest S., III (2000), Unit Fractions, Ph.D. thesis, University of Georgia, Athens. Croot, Ernest S., III (2003), "On a coloring conjecture about unit fractions", Annals of Mathematics, 157 (2): 545–556, arXiv:math.NT/0311421, doi:10.4007/annals.2003.157.545
  173. ^ https://arxiv.org/pdf/math/9906042v2.pdf
  174. ^ https://arxiv.org/pdf/math/9906212v2.pdf
  175. ^ Ullmo, E (1998). "Positivité et Discrétion des Points Algébriques des Courbes". Annals of Mathematics. 147 (1): 167–179. doi:10.2307/120987. Zbl 0934.14013.
  176. ^ Zhang, S.-W. (1998). "Equidistribution of small points on abelian varieties". Annals of Mathematics. 147 (1): 159–165. doi:10.2307/120986.
  177. ^ Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 1431-0635, MR 1648105
  178. ^ https://arxiv.org/pdf/1501.02155.pdf
  179. ^ https://arxiv.org/pdf/math/9811079v3.pdf
  180. ^ Norio Iwase (1 November 1998). "Ganea's Conjecture on Lusternik-Schnirelmann Category". ResearchGate.
  181. ^ Merel, Loïc (1996). ""Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]". Inventiones Mathematicae. 124 (1): 437–449. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424.
  182. ^ https://www.researchgate.net/profile/Zhibo_Chen/publication/220188021_Harary's_conjectures_on_integral_sum_graphs/links/5422b2490cf290c9e3aac7fe.pdf
  183. ^ Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. Annals of Mathematics. 141 (3): 443–551. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.
  184. ^ Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. Annals of Mathematics. 141 (3): 553–572. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255.

Further reading[edit]

Books discussing recently solved problems[needs update][edit]

Books discussing unsolved problems[edit]

External links[edit]