List of unsolved problems in mathematics

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] 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. 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.

Lists of unsolved problems in mathematics

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
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

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved, as of 2016:[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

Algebra

Algebraic geometry

Analysis

Combinatorics

Discrete geometry

Euclidean geometry

Dynamical systems

Graph theory

Paths and cycles in graphs

Graph coloring and labeling

Graph drawing

Miscellaneous graph theory

Group theory

Model theory

Number theory

General

Additive number theory

Algebraic number theory

Combinatorial number theory

Prime numbers

Partial differential equations

Ramsey theory

Set theory

Other

Problems solved since 1995

References

  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 9781489935854.
  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
  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 (), e (), Khinchin's Constant (), irrational numbers (), transcendental numbers (), and irrationality measures () 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 , 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 , accessed 15 December 2014.
  16. Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419Freely accessible, doi:10.1007/s00283-011-9255-y, MR 2902144
  17. Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–253, doi:10.1090/noti1100
  18. 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
  19. Wagner, Neal R. (1976), "The Sofa Problem" (PDF), The American Mathematical Monthly, 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022
  20. 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
  21. Bellos, Alex (11 August 2015), "Attack on the pentagon results in discovery of new mathematical tile", The Guardian
  22. ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, retrieved 2016-10-04.
  23. 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, doi:10.1007/978-3-642-03745-0_5
  24. Florek, Jan (2010), "On Barnette's conjecture", Discrete Mathematics, 310 (10-11): 1531–1535, doi:10.1016/j.disc.2010.01.018, MR 2601261.
  25. 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
  26. 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.
  27. Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7.
  28. 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 608921.
  29. L. Babai, Automorphism groups, isomorphism, reconstruction, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447–1540.
  30. Chung, Fan; Graham, Ron (1998), Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, pp. 97–99.
  31. Toft, Bjarne (1996), "A survey of Hadwiger's conjecture", Congressus Numerantium, 115: 249–283, MR 1411244.
  32. Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991), Unsolved Problems in Geometry, Springer-Verlag, Problem G10.
  33. 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.
  34. Hägglund, Jonas; Steffen, Eckhard (2014), "Petersen-colorings and some families of snarks", Ars Mathematica Contemporanea, 7 (1): 161–173, MR 3047618.
  35. 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.
  36. Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 668845.
  37. Molloy, Michael; Reed, Bruce (1998), "A bound on the total chromatic number", Combinatorica, 18 (2): 241–280, doi:10.1007/PL00009820, MR 1656544.
  38. Barát, János; Tóth, Géza (2010), "Towards the Albertson Conjecture", Electronic Journal of Combinatorics, 17 (1): R73, arXiv:0909.0413Freely accessible.
  39. Wood, David (January 19, 2009), "Book Thickness of Subdivisions", Open Problem Garden, retrieved 2013-02-05.
  40. 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.
  41. Hartsfield, Nora; Ringel, Gerhard (2013), Pearls in Graph Theory: A Comprehensive Introduction, Dover Books on Mathematics, Courier Dover Publications, p. 247, ISBN 9780486315522, MR 2047103.
  42. 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.
  43. 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, doi:10.20382/jocg.v7i1a3, MR 3463906
  44. 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.
  45. 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.
  46. Chudnovsky, Maria (2014), "The Erdös–Hajnal conjecture—a survey" (PDF), Journal of Graph Theory, 75 (2): 178–190, arXiv:1606.08827Freely accessible, doi:10.1002/jgt.21730, MR 3150572, Zbl 1280.05086.
  47. "Jorgensen's Conjecture", Open Problem Garden, retrieved 2016-11-13.
  48. 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:1010.4430Freely accessible, doi:10.1112/plms/pdq035, MR 2793448, Zbl 1218.05034.
  49. 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.
  50. 1 2 3 Shelah S, Classification Theory, North-Holland, 1990
  51. Keisler, HJ, "Ultraproducts which are not saturated." J. Symb Logic 32 (1967) 23—46.
  52. Malliaris M, Shelah S, "A dividing line in simple unstable theories." http://arxiv.org/abs/1208.2140
  53. Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
  54. Peretz, Assaf, "Geometry of forking in simple theories." J. Symbolic Logic Volume 71, Issue 1 (2006), 347–359.
  55. Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae. 159 (1): 1–50. arXiv:math/9802134Freely accessible.
  56. Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN 978-1-904987-71-0.
  57. Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
  58. Baldwin, John T. (July 24, 2009). Categoricity (PDF). American Mathematical Society. ISBN 978-0821848937. Retrieved February 20, 2014.
  59. Shelah, Saharon. "Introduction to classification theory for abstract elementary classes".
  60. Hrushovski, Ehud, "Kueker's conjecture for stable theories." Journal of Symbolic Logic Vol. 54, No. 1 (Mar., 1989), pp. 207–220.
  61. Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B. 97 (3): 293–333. arXiv:math/0512218Freely accessible. doi:10.1016/j.jctb.2006.05.008.
  62. Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
  63. "Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key".
  64. http://arxiv.org/pdf/1604.07746v1.pdf
  65. 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.
  66. Dobson, J. B. (June 2012) [2011], On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907Freely accessible
  67. Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society, 125: 1503–1509, JSTOR 2162098
  68. http://arxiv.org/pdf/1605.00723v1.pdf
  69. Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398.
  70. 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.
  71. http://arxiv.org/pdf/1509.05363v5.pdf
  72. Proof of the umbral moonshine conjecture – Springer
  73. http://arxiv.org/pdf/1406.6534v10.pdf
  74. Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897Freely accessible [math.NT].
  75. Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252Freely accessible [math.NT].
  76. Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748Freely accessible [math.NT].
  77. Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric (2006). Han, Deguang; Jorgensen, Palle E. T.; Larson, David Royal, eds. "The Kadison-Singer problem in mathematics and engineering: A detailed account". Contemporary Mathematics. Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida. American Mathematical Society. 414: 299–355. doi:10.1090/conm/414/07820. ISBN 978-0-8218-3923-2. Retrieved 24 April 2015.
  78. Mackenzie, Dana. "Kadison–Singer Problem Solved" (PDF). SIAM News (January/February 2014). Society for Industrial and Applied Mathematics. Retrieved 24 April 2015.
  79. http://arxiv.org/pdf/1204.2810v1.pdf
  80. http://www.math.jhu.edu/~js/Math646/brendle.lawson.pdf
  81. Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. arXiv:1202.6036Freely accessible. doi:10.4007/annals.2014.179.2.6.
  82. http://arxiv.org/pdf/1101.1330v4.pdf
  83. http://www.math.uiuc.edu/~mineyev/math/art/submult-shnc.pdf
  84. http://annals.math.princeton.edu/wp-content/uploads/annals-v174-n1-p11-p.pdf
  85. https://www.uni-due.de/~bm0032/publ/BlochKato.pdf
  86. page 359
  87. "motivic cohomology – Milnor–Bloch–Kato conjecture implies the Beilinson-Lichtenbaum conjecture – MathOverflow".
  88. http://arxiv.org/pdf/1011.4105v3.pdf
  89. https://www.researchgate.net/profile/Juan_Souto3/publication/228365532_Non-realizability_and_ending_laminations_Proof_of_the_Density_Conjecture/links/541d85a10cf2218008d1d2e5.pdf
  90. Santos, Franciscos (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.
  91. Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica. Extra Volume "Optimization Stories": 75–85.
  92. "Generalized Sidon sets".
  93. http://arxiv.org/pdf/0909.2360v3.pdf
  94. http://arxiv.org/pdf/0906.1612v2.pdf
  95. http://arxiv.org/pdf/0910.5501v5.pdf
  96. http://www.csie.ntu.edu.tw/~hil/bib/ChalopinG09.pdf
  97. http://arxiv.org/pdf/0809.4040.pdf
  98. 1 2 "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.
  99. Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)", Inventiones Mathematicae, 178 (3): 485–504, doi:10.1007/s00222-009-0205-7
  100. Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505–586, doi:10.1007/s00222-009-0206-6
  101. "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.
  102. http://www.ww.amc12.org/sites/default/files/pdf/pubs/SquaringThePlane.pdf
  103. http://arxiv.org/pdf/math/0509397.pdf
  104. Seigel-Itzkovich, Judy (2008-02-08). "Russian immigrant solves math puzzle". The Jerusalem Post. Retrieved 2015-11-12.
  105. http://homepages.warwick.ac.uk/~masgak/papers/bhb-angel.pdf
  106. http://home.broadpark.no/~oddvark/angel/Angel.pdf
  107. http://homepages.warwick.ac.uk/~masibe/angel-mathe.pdf
  108. http://www.cs.bu.edu/~gacs/papers/angel.pdf
  109. http://www.ams.org/journals/proc/2005-133-09/S0002-9939-05-07752-X/S0002-9939-05-07752-X.pdf
  110. "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.
  111. http://arxiv.org/pdf/math/0405568v1.pdf
  112. "Graph Theory".
  113. 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.
  114. "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.
  115. http://arxiv.org/pdf/math/0412006v2.pdf
  116. 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
  117. Green, Ben (2004), "The Cameron–Erdős conjecture", The Bulletin of the London Mathematical Society, 36 (6): 769–778, arXiv:math.NT/0304058Freely accessible, doi:10.1112/S0024609304003650, MR 2083752
  118. "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..."
  119. http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_2003__98_/PMIHES_2003__98__1_0/PMIHES_2003__98__1_0.pdf
  120. "Kemnitz' conjecture revisited".
  121. http://www.ams.org/journals/jams/2004-17-01/S0894-0347-03-00440-5/S0894-0347-03-00440-5.pdf
  122. http://annals.math.princeton.edu/wp-content/uploads/annals-v158-n1-p04.pdf
  123. "The strong perfect graph theorem".
  124. http://www.emis.de/journals/BAG/vol.43/no.1/b43h1haa.pdf
  125. Knight, R. W. (2002), The Vaught Conjecture: A Counterexample, manuscript
  126. http://www.ugr.es/~ritore/preprints/0406017.pdf
  127. 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.
  128. http://www.ams.org/journals/jams/2001-14-04/S0894-0347-01-00373-3/S0894-0347-01-00373-3.pdf
  129. http://junon.u-3mrs.fr/monniaux/AHLMT02.pdf
  130. http://arxiv.org/pdf/math/0102150v4.pdf
  131. 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
  132. http://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01178-9/S0025-5718-00-01178-9.pdf
  133. http://intlpress.com/site/pub/files/_fulltext/journals/sdg/2002/0007/0001/SDG-2002-0007-0001-a001.pdf
  134. 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/0311421Freely accessible, doi:10.4007/annals.2003.157.545
  135. http://arxiv.org/pdf/math/9906042v2.pdf
  136. http://arxiv.org/pdf/math/9906212v2.pdf
  137. 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
  138. Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties", Annals of Mathematics 147 (1): 159–165, doi:10.2307/120986
  139. Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 1431-0635, MR 1648105
  140. http://arxiv.org/pdf/1501.02155.pdf
  141. http://arxiv.org/pdf/math/9811079v3.pdf
  142. Norio Iwase (1 November 1998). "Ganea's Conjecture on Lusternik-Schnirelmann Category". ResearchGate.
  143. 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 (in French) 124 (1): 437–449. doi:10.1007/s002220050059. MR 1369424
  144. https://www.researchgate.net/profile/Zhibo_Chen/publication/220188021_Harary's_conjectures_on_integral_sum_graphs/links/5422b2490cf290c9e3aac7fe.pdf
  145. 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.
  146. 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

Books discussing recently solved problems


Books discussing unsolved problems

External links

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