This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative, and the problems listed here vary widely in both difficulty and importance.
Lists of unsolved problems in mathematics
edit
Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.
The Kourovka Notebook (Russian: Коуровская тетрадь) is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.[16]
The Sverdlovsk Notebook (Russian: Свердловская тетрадь) is a collection of unsolved problems in semigroup theory, first published in 1965 and updated every 2 to 4 years since.[17][18][19]
Pierce–Birkhoff conjecture: every piecewise-polynomial is the maximum of a finite set of minimums of finite collections of polynomials.
Rota's basis conjecture: for matroids of rank with disjoint bases , it is possible to create an matrix whose rows are and whose columns are also bases.
Burnside problem: for which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems[24]
Herzog–Schönheim conjecture: if a finite system of left cosets of subgroups of a group form a partition of , then the finite indices of said subgroups cannot be distinct.
The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
The Brennan conjecture: estimating the integral of powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of
The Dittert conjecture concerning the maximum achieved by a particular function of matrices with real, nonnegative entries satisfying a summation condition
The lonely runner conjecture – if runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance from each other runner) at some time?[29]
Map folding – various problems in map folding and stamp folding.
No-three-in-line problem – how many points can be placed in the grid so that no three of them lie on a line?
The sunflower conjecture – can the number of size sets required for the existence of a sunflower of sets be bounded by an exponential function in for every fixed ?
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[31]
Does every positive integer generate a juggler sequence terminating at 1?
Lyapunov function: Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the (asymptotical) stability of motion?
Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also Hales–Jewett theorem and nd game)[41]
Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a boundedn-dimensional set.
The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?[45]
Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[48]
Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
Closed curve problem: find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.[52]
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[53]
The Hopf conjectures relating the curvature and Euler characteristic of higher-dimensional Riemannian manifolds[54]
Bellman's lost-in-a-forest problem – find the shortest route that is guaranteed to reach the boundary of a given shape, starting at an unknown point of the shape with unknown orientation[66]
Borromean rings — are there three unknotted space curves, not all three circles, which cannot be arranged to form this link?[67]
Danzer's problem and Conway's dead fly problem – do Danzer sets of bounded density or bounded separation exist?[68]
The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem[73]
The Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.[114]
Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs[125])
Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter[126]
The second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?[129]
The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for -saturated models of a countable theory.[133]
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.[133]
Shelah's eventual categoricity 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 .[133][134]
The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
The stable forking conjecture for simple theories[135]
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?[136]
The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[137]
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?[138]
Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?[139][140]
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.)
Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?[141]
Is the theory of the field of Laurent series over decidable? of the field of polynomials over ?
Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[142]
Determine the structure of Keisler's order.[143][144]
Casas-Alvero conjecture: if a polynomial of degree defined over a field of characteristic has a factor in common with its first through -th derivative, then must be the -th power of a linear polynomial?
n conjecture: a generalization of the abc conjecture to more than three integers.
abc conjecture: for any , is true for only finitely many positive such that .
Szpiro's conjecture: for any , there is some constant such that, for any elliptic curve defined over with minimal discriminant and conductor , we have .
Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set
How well can non-quadratic irrational numbers be approximated? What is the irrationality measure of specific (suspected) transcendental numbers such as and ?[154]
Bunyakovsky conjecture: if an integer-coefficient polynomial has a positive leading coefficient, is irreducible over the integers, and has no common factors over all where is a positive integer, then is prime infinitely often.
Dickson's conjecture: for a finite set of linear forms with each , there are infinitely many for which all forms are prime, unless there is some congruence condition preventing it.
Dubner's conjecture: every even number greater than is the sum of two primes which both have a twin.
The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
New Mersenne conjecture: for any odd natural number, if any two of the three conditions or , is prime, and is prime are true, then the third condition is also true.
Schinzel's hypothesis H that for every finite collection of nonconstant irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers for which are all primes, or there is some fixed divisor which, for all , divides some .
For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[160]
For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?
For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b?
For any given integers , with gcd(k, c) = 1 and gcd(b, c) = 1, are there infinitely many primes of the form with integer n ≥ 1?
McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)[173][174]
Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (Jeff Cheeger, Aaron Naber, 2014)[267]
^Guy, Richard (1994), Unsolved Problems in Number Theory (2nd ed.), Springer, p. vii, ISBN 978-1-4899-3585-4, archived from the original on 2019-03-23, retrieved 2016-09-22.
^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.
^Friedl, Stefan (2014). "Thurston's vision and the virtual fibering theorem for 3-manifolds". Jahresbericht der Deutschen Mathematiker-Vereinigung. 116 (4): 223–241. doi:10.1365/s13291-014-0102-x. MR 3280572. S2CID 56322745.
^Thurston, William P. (1982). "Three-dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. New Series. 6 (3): 357–381. doi:10.1090/S0273-0979-1982-15003-0. MR 0648524.
^ ab"Millennium Problems". claymath.org. Archived from the original on 2017-06-06. Retrieved 2015-01-20.
^Bellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Archived from the original on 2016-10-21. Retrieved 2018-07-07.
^Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 978-90-5199-490-2.
^"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.
^"Math Problems Guide: From Simple to Hardest Math Problems Tips & Examples". blendedlearningmath. Retrieved 2024-11-28.
^"Poincaré Conjecture". Clay Mathematics Institute. Archived from the original on 2013-12-15.
^rybu (November 7, 2009). "Smooth 4-dimensional Poincare conjecture". Open Problem Garden. Archived from the original on 2018-01-25. Retrieved 2019-08-06.
^Khukhro, Evgeny I.; Mazurov, Victor D. (2019), Unsolved Problems in Group Theory. The Kourovka Notebook, arXiv:1401.0300v16
^RSFSR, MV i SSO; Russie), Uralʹskij gosudarstvennyj universitet im A. M. Gorʹkogo (Ekaterinbourg (1969). Свердловская тетрадь: нерешенные задачи теории подгрупп (in Russian). S. l.
^Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп. Свердловск: Уральский государственный университет. 1979.
^Свердловская тетрадь: Сб. нерешённых задач по теории полугрупп. Свердловск: Уральский государственный университет. 1989.
^ДНЕСТРОВСКАЯ ТЕТРАДЬ [DNIESTER NOTEBOOK] (PDF) (in Russian), The Russian Academy of Sciences, 1993
^"DNIESTER NOTEBOOK: Unsolved Problems in the Theory of Rings and Modules" (PDF), University of Saskatchewan, retrieved 2019-08-15
^Эрлагольская тетрадь [Erlagol notebook] (PDF) (in Russian), The Novosibirsk State University, 2018
^Dowling, T. A. (February 1973). "A class of geometric lattices based on finite groups". Journal of Combinatorial Theory. Series B. 14 (1): 61–86. doi:10.1016/S0095-8956(73)80007-3.
^Smyth, Chris (2008), "The Mahler measure of algebraic numbers: a survey", in McKee, James; Smyth, Chris (eds.), Number Theory and Polynomials, London Mathematical Society Lecture Note Series, vol. 352, Cambridge University Press, pp. 322–349, ISBN 978-0-521-71467-9
^Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, CiteSeerX10.1.1.38.7841, doi:10.1007/BF01110378, MR 1368815, S2CID 14793475.
^Tao, Terence (2018). "Some remarks on the lonely runner conjecture". Contributions to Discrete Mathematics. 13 (2): 1–31. arXiv:1701.02048. doi:10.11575/cdm.v13i2.62728.
^González-Jiménez, Enrique; Xarles, Xavier (2014). "On a conjecture of Rudin on squares in arithmetic progressions". LMS Journal of Computation and Mathematics. 17 (1): 58–76. arXiv:1301.5122. doi:10.1112/S1461157013000259. S2CID 11615385.
^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, S2CID 17531822, archived (PDF) from the original on 2017-08-08, retrieved 2017-07-18
^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, JSTOR 2371542, MR 1507301, PMC1076971, PMID 16577800
^"Dedekind Numbers and Related Sequences" (PDF). Archived from the original (PDF) on 2015-03-15. Retrieved 2020-04-30.
^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. S2CID 33806100.
^S. M. Ulam, Problems in Modern Mathematics. Science Editions John Wiley & Sons, Inc., New York, 1964, page 76.
^Kaloshin, Vadim; Sorrentino, Alfonso (2018). "On the local Birkhoff conjecture for convex billiards". Annals of Mathematics. 188 (1): 315–380. arXiv:1612.09194. doi:10.4007/annals.2018.188.1.6. S2CID 119171182.
^Paul Halmos, Ergodic theory. Chelsea, New York, 1956.
^Kari, Jarkko (2009). "Structure of reversible cellular automata". Structure of Reversible Cellular Automata. International Conference on Unconventional Computation. Lecture Notes in Computer Science. Vol. 5715. Springer. p. 6. Bibcode:2009LNCS.5715....6K. doi:10.1007/978-3-642-03745-0_5. ISBN 978-3-642-03744-3.
^ abc"Open Q - Solving and rating of hard Sudoku". english.log-it-ex.com. Archived from the original on 10 November 2017.
^"Higher-Dimensional Tic-Tac-Toe". PBS Infinite Series. YouTube. 2017-09-21. Archived from the original on 2017-10-11. Retrieved 2018-07-29.
^Barlet, Daniel; Peternell, Thomas; Schneider, Michael (1990). "On two conjectures of Hartshorne's". Mathematische Annalen. 286 (1–3): 13–25. doi:10.1007/BF01453563. S2CID 122151259.
^Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, JSTOR 2324212, MR 1252928
^Conway, John H.; Neil J.A. Sloane (1999), Sphere Packings, Lattices and Groups (3rd ed.), New York: Springer-Verlag, pp. 21–22, ISBN 978-0-387-98585-5
^Hales, Thomas (2017), The Reinhardt conjecture as an optimal control problem, arXiv:1703.01352
^Brass, Peter; Moser, William; Pach, János (2005), Research Problems in Discrete Geometry, New York: Springer, p. 45, ISBN 978-0387-23815-9, MR 2163782
^Gardner, Martin (1995), New Mathematical Diversions (Revised Edition), Washington: Mathematical Association of America, p. 251
^Musin, Oleg R.; Tarasov, Alexey S. (2015). "The Tammes Problem for N = 14". Experimental Mathematics. 24 (4): 460–468. doi:10.1080/10586458.2015.1022842. S2CID 39429109.
^Katz, Mikhail G. (2007), Systolic geometry and topology, Mathematical Surveys and Monographs, vol. 137, American Mathematical Society, Providence, RI, p. 57, doi:10.1090/surv/137, ISBN 978-0-8218-4177-8, MR 2292367
^Rosenberg, Steven (1997), The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds, London Mathematical Society Student Texts, vol. 31, Cambridge: Cambridge University Press, pp. 62–63, doi:10.1017/CBO9780511623783, ISBN 978-0-521-46300-3, MR 1462892
^Ghosh, Subir Kumar; Goswami, Partha P. (2013), "Unsolved problems in visibility graphs of points, segments, and polygons", ACM Computing Surveys, 46 (2): 22:1–22:29, arXiv:1012.5187, doi:10.1145/2543581.2543589, S2CID 8747335
^Boltjansky, V.; Gohberg, I. (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
^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 (4): 1047–1053, arXiv:1604.08657, doi:10.1090/jams/869, S2CID 15732134
^Moreno, José Pedro; Prieto-Martínez, Luis Felipe (2021). "El problema de los triángulos de Kobon" [The Kobon triangles problem]. La Gaceta de la Real Sociedad Matemática Española (in Spanish). 24 (1): 111–130. hdl:10486/705416. MR 4225268.
^Matoušek, Jiří (2002), Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, p. 206, doi:10.1007/978-1-4613-0039-7, ISBN 978-0-387-95373-1, MR 1899299
^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 978-0-387-23815-9, MR 2163782
^Aronov, Boris; Dujmović, Vida; Morin, Pat; Ooms, Aurélien; Schultz Xavier da Silveira, Luís Fernando (2019), "More Turán-type theorems for triangles in convex point sets", Electronic Journal of Combinatorics, 26 (1): P1.8, arXiv:1706.10193, Bibcode:2017arXiv170610193A, doi:10.37236/7224, archived from the original on 2019-02-18, retrieved 2019-02-18
^Atiyah, Michael (2001), "Configurations of points", Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 359 (1784): 1375–1387, Bibcode:2001RSPTA.359.1375A, doi:10.1098/rsta.2001.0840, ISSN 1364-503X, MR 1853626, S2CID 55833332
^Howards, Hugh Nelson (2013), "Forming the Borromean rings out of arbitrary polygonal unknots", Journal of Knot Theory and Its Ramifications, 22 (14): 1350083, 15, arXiv:1406.3370, doi:10.1142/S0218216513500831, MR 3190121, S2CID 119674622
^Solomon, Yaar; Weiss, Barak (2016), "Dense forests and Danzer sets", Annales Scientifiques de l'École Normale Supérieure, 49 (5): 1053–1074, arXiv:1406.3807, doi:10.24033/asens.2303, MR 3581810, S2CID 672315; Conway, John H., Five $1,000 Problems (Update 2017)(PDF), On-Line Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12
^Brandts, Jan; Korotov, Sergey; Křížek, Michal; Šolc, Jakub (2009), "On nonobtuse simplicial partitions" (PDF), SIAM Review, 51 (2): 317–335, Bibcode:2009SIAMR..51..317B, doi:10.1137/060669073, MR 2505583, S2CID 216078793, archived (PDF) from the original on 2018-11-04, retrieved 2018-11-22. See in particular Conjecture 23, p. 327.
^Arutyunyants, G.; Iosevich, A. (2004), "Falconer conjecture, spherical averages and discrete analogs", in Pach, János (ed.), Towards a Theory of Geometric Graphs, Contemp. Math., vol. 342, Amer. Math. Soc., Providence, RI, pp. 15–24, doi:10.1090/conm/342/06127, ISBN 978-0-8218-3484-8, MR 2065249
^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, pp. 161–179, CiteSeerX10.1.1.241.5335, doi:10.5565/PUBLMAT_Esco02_07, MR 1964819, S2CID 77088
^Weaire, Denis, ed. (1997), The Kelvin Problem, CRC Press, p. 1, ISBN 978-0-7484-0632-6
^Brass, Peter; Moser, William; Pach, János (2005), Research problems in discrete geometry, New York: Springer, p. 457, ISBN 978-0-387-29929-7, MR 2163782
^Mahler, Kurt (1939). "Ein Minimalproblem für konvexe Polygone". Mathematica (Zutphen) B: 118–127.
^Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete & Computational Geometry, 7 (2): 153–162, doi:10.1007/BF02187832, MR 1139077
^Wagner, Neal R. (1976), "The Sofa Problem" (PDF), The American Mathematical Monthly, 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022, archived (PDF) from the original on 2015-04-20, retrieved 2014-05-14
^Chai, Ying; Yuan, Liping; Zamfirescu, Tudor (June–July 2018), "Rupert Property of Archimedean Solids", The American Mathematical Monthly, 125 (6): 497–504, doi:10.1080/00029890.2018.1449505, S2CID 125508192
^Steininger, Jakob; Yurkevich, Sergey (December 27, 2021), An algorithmic approach to Rupert's problem, arXiv:2112.13754
^Ghomi, Mohammad (2018-01-01). "Dürer's Unfolding Problem for Convex Polyhedra". Notices of the American Mathematical Society. 65 (1): 25–27. doi:10.1090/noti1609. ISSN 0002-9920.
^Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly, 59 (9): 606–611, doi:10.2307/2306764, JSTOR 2306764, MR 0050303
^ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, archived from the original on October 5, 2016, retrieved 2016-10-04.
^Pleanmani, Nopparat (2019), "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph", Discrete Mathematics, Algorithms and Applications, 11 (6): 1950068, 7, doi:10.1142/s179383091950068x, MR 4044549, S2CID 204207428
^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, S2CID 18942362
^Bousquet, Nicolas; Bartier, Valentin (2019), "Linear Transformations Between Colorings in Chordal Graphs", in Bender, Michael A.; Svensson, Ola; Herman, Grzegorz (eds.), 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, LIPIcs, vol. 144, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 24:1–24:15, doi:10.4230/LIPIcs.ESA.2019.24, ISBN 978-3-95977-124-5, S2CID 195791634
^Gethner, Ellen (2018), "To the Moon and beyond", in Gera, Ralucca; Haynes, Teresa W.; Hedetniemi, Stephen T. (eds.), Graph Theory: Favorite Conjectures and Open Problems, II, Problem Books in Mathematics, Springer International Publishing, pp. 115–133, doi:10.1007/978-3-319-97686-0_11, ISBN 978-3-319-97684-6, MR 3930641
^Chung, Fan; Graham, Ron (1998), Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, pp. 97–99.
^Toft, Bjarne (1996), "A survey of Hadwiger's conjecture", Congressus Numerantium, 115: 249–283, MR 1411244.
^Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991), Unsolved Problems in Geometry, Springer-Verlag, Problem G10.
^Hägglund, Jonas; Steffen, Eckhard (2014), "Petersen-colorings and some families of snarks", Ars Mathematica Contemporanea, 7 (1): 161–173, doi:10.26493/1855-3974.288.11a, MR 3047618, archived from the original on 2016-10-03, retrieved 2016-09-30.
^Jensen, Tommy R.; Toft, Bjarne (1995), "12.20 List-Edge-Chromatic Numbers", Graph Coloring Problems, New York: Wiley-Interscience, pp. 201–202, ISBN 978-0-471-02865-9.
^Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, CiteSeerX10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645, archived (PDF) from the original on 2016-03-04, retrieved 2016-10-04.
^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, S2CID 1500695
^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, vol. 152, American Mathematical Society, pp. 126–127.
^Demaine, E.; O'Rourke, J. (2002–2012), "Problem 45: Smallest Universal Set of Points for Planar Graphs", The Open Problems Project, archived from the original on 2012-08-14, retrieved 2013-03-19.
^Conway, John H., Five $1,000 Problems (Update 2017)(PDF), Online Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12
^mdevos; Wood, David (December 7, 2019), "Jorgensen's Conjecture", Open Problem Garden, archived from the original on 2016-11-14, retrieved 2016-11-13.
^Ducey, Joshua E. (2017), "On the critical group of the missing Moore graph", Discrete Mathematics, 340 (5): 1104–1109, arXiv:1509.00327, doi:10.1016/j.disc.2016.10.001, MR 3612450, S2CID 28297244
^Broersma, Hajo; Patel, Viresh; Pyatkin, Artem (2014), "On toughness and Hamiltonicity of $2K_2$-free graphs" (PDF), Journal of Graph Theory, 75 (3): 244–255, doi:10.1002/jgt.21734, MR 3153119, S2CID 1377980
^Jaeger, F. (1985), "A survey of the cycle double cover conjecture", Annals of Discrete Mathematics 27 – Cycles in Graphs, North-Holland Mathematics Studies, vol. 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1, ISBN 978-0-444-87803-8.
^Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7, doi:10.37236/3252.
^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, S2CID 985458, Zbl 1280.05086, archived (PDF) from the original on 2016-03-04, retrieved 2016-09-22.
^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.
^Babai, László (June 9, 1994). "Automorphism groups, isomorphism, reconstruction". Handbook of Combinatorics. Archived from the original (PostScript) on 13 June 2007.
^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
^Fomin, Fedor V.; Høie, Kjartan (2006), "Pathwidth of cubic graphs and exact algorithms", Information Processing Letters, 97 (5): 191–196, doi:10.1016/j.ipl.2005.10.012, MR 2195217
^Schwenk, Allen (2012). Some History on the Reconstruction Conjecture(PDF). Joint Mathematics Meetings. Archived from the original (PDF) on 2015-04-09. Retrieved 2018-11-26.
^Ramachandran, S. (1981), "On a new digraph reconstruction conjecture", Journal of Combinatorial Theory, Series B, 31 (2): 143–149, doi:10.1016/S0095-8956(81)80019-6, MR 0630977
^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, S2CID 119169562, Zbl 1218.05034.
^Tuza, Zsolt (1990). "A conjecture on triangles of graphs". Graphs and Combinatorics. 6 (4): 373–380. doi:10.1007/BF01787705. MR 1092587. S2CID 38821128.
^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, CiteSeerX10.1.1.159.7029, doi:10.1002/jgt.20565, MR 2864622, S2CID 9120720.
^ abcdeKitaev, Sergey; Lozin, Vadim (2015). Words and Graphs. Monographs in Theoretical Computer Science. An EATCS Series. doi:10.1007/978-3-319-25859-1. ISBN 978-3-319-25857-7. S2CID 7727433 – via link.springer.com.
^ abcdeKitaev, S. V.; Pyatkin, A. V. (April 1, 2018). "Word-Representable Graphs: a Survey". Journal of Applied and Industrial Mathematics. 12 (2): 278–296. doi:10.1134/S1990478918020084. S2CID 125814097 – via Springer Link.
^ abcdeKitaev, Sergey V.; Pyatkin, Artem V. (2018). "Графы, представимые в виде слов. Обзор результатов" [Word-representable graphs: A survey]. Дискретн. анализ и исслед. опер. (in Russian). 25 (2): 19–53. doi:10.17377/daio.2018.25.588.
^Marc Elliot Glen (2016). "Colourability and word-representability of near-triangulations". arXiv:1605.01688 [math.CO].
^Kitaev, Sergey (2014-03-06). "On graphs with representation number 3". arXiv:1403.1616v1 [math.CO].
^Glen, Marc; Kitaev, Sergey; Pyatkin, Artem (2018). "On the representation number of a crown graph". Discrete Applied Mathematics. 244: 89–93. arXiv:1609.00674. doi:10.1016/j.dam.2018.03.013. S2CID 46925617.
^Spinrad, Jeremy P. (2003), "2. Implicit graph representation", Efficient Graph Representations, American Mathematical Soc., pp. 17–30, ISBN 978-0-8218-2815-1.
^"Seymour's 2nd Neighborhood Conjecture". faculty.math.illinois.edu. Archived from the original on 11 January 2019. Retrieved 17 August 2022.
^mdevos (May 4, 2007). "5-flow conjecture". Open Problem Garden. Archived from the original on November 26, 2018.
^mdevos (March 31, 2010). "4-flow conjecture". Open Problem Garden. Archived from the original on November 26, 2018.
^Hrushovski, Ehud (1989). "Kueker's conjecture for stable theories". Journal of Symbolic Logic. 54 (1): 207–220. doi:10.2307/2275025. JSTOR 2275025. S2CID 41940041.
^ abcShelah S (1990). Classification Theory. North-Holland.
^Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN 978-1-904987-71-0.
^Peretz, Assaf (2006). "Geometry of forking in simple theories". Journal of Symbolic Logic. 71 (1): 347–359. arXiv:math/0412356. doi:10.2178/jsl/1140641179. S2CID 9380215.
^Baldwin, John T. (July 24, 2009). Categoricity(PDF). American Mathematical Society. ISBN 978-0-8218-4893-7. Archived (PDF) from the original on July 29, 2010. Retrieved February 20, 2014.
^Shelah, Saharon (2009). "Introduction to classification theory for abstract elementary classes". arXiv:0903.3428 [math.LO].
^Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
^Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
^Keisler, HJ (1967). "Ultraproducts which are not saturated". J. Symb. Log. 32 (1): 23–46. doi:10.2307/2271240. JSTOR 2271240. S2CID 250345806.
^Malliaris, Maryanthe; Shelah, Saharon (10 August 2012). "A Dividing Line Within Simple Unstable Theories". arXiv:1208.2140 [math.LO]. Malliaris, M.; Shelah, S. (2012). "A Dividing Line within Simple Unstable Theories". arXiv:1208.2140 [math.LO].
^Guo, Song; Sun, Zhi-Wei (2005), "On odd covering systems with distinct moduli", Advances in Applied Mathematics, 35 (2): 182–187, arXiv:math/0412217, doi:10.1016/j.aam.2005.01.004, MR 2152886, S2CID 835158
^"Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key". Archived from the original on 2016-03-27. Retrieved 2016-03-18.
^Robertson, John P. (1996-10-01). "Magic Squares of Squares". Mathematics Magazine. 69 (4): 289–293. doi:10.1080/0025570X.1996.11996457. ISSN 0025-570X.
^ abWaldschmidt, Michel (2013), Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Springer, pp. 14, 16, ISBN 978-3-662-11569-5
^Waldschmidt, Michel (2008). An introduction to irrationality and transcendence methods(PDF). 2008 Arizona Winter School. Archived from the original (PDF) on 16 December 2014. Retrieved 15 December 2014.
^Albert, John, Some unsolved problems in number theory(PDF), archived from the original (PDF) on 17 January 2014, retrieved 15 December 2014
^For some background on the numbers in this problem, see articles by Eric W. Weisstein at WolframMathWorld (all articles accessed 22 August 2024):
Euler's Constant
Catalan's Constant
Apéry's Constant
irrational numbers (Archived 2015-03-27 at the Wayback Machine)
transcendental numbers (Archived 2014-11-13 at the Wayback Machine)
irrationality measures (Archived 2015-04-21 at the Wayback Machine)
^ abWaldschmidt, Michel (2003-12-24). "Open Diophantine Problems". arXiv:math/0312440.
^Kontsevich, Maxim; Zagier, Don (2001), Engquist, Björn; Schmid, Wilfried (eds.), "Periods", Mathematics Unlimited — 2001 and Beyond, Berlin, Heidelberg: Springer, pp. 771–808, doi:10.1007/978-3-642-56478-9_39, ISBN 978-3-642-56478-9, retrieved 2024-08-22
^Weisstein, Eric W. "Khinchin's Constant". mathworld.wolfram.com. Retrieved 2024-09-22.
^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
^Huisman, Sander G. (2016). "Newer sums of three cubes". arXiv:1604.07746 [math.NT].
^Dobson, J. B. (1 April 2017), "On Lerch's formula for the Fermat quotient", p. 23, arXiv:1103.3907v6 [math.NT]
^Ribenboim, P. (2006). Die Welt der Primzahlen. Springer-Lehrbuch (in German) (2nd ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
^Mazur, Barry (1992), "The topology of rational points", Experimental Mathematics, 1 (1): 35–45, doi:10.1080/10586458.1992.10504244, S2CID 17372107, archived from the original on 2019-04-07, retrieved 2019-04-07
^Dimitrov, Vessilin; Gao, Ziyang; Habegger, Philipp (2021). "Uniformity in Mordell–Lang for curves" (PDF). Annals of Mathematics. 194: 237–298. arXiv:2001.10276. doi:10.4007/annals.2021.194.1.4. S2CID 210932420.
^Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an extension problem with optimal estimate and applications". Annals of Mathematics. 181 (3): 1139–1208. arXiv:1310.7169. doi:10.4007/annals.2015.181.3.6. JSTOR 24523356. S2CID 56205818.
^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. S2CID 3590991.
^Cohen, Stephen D.; Fried, Michael D. (1995), "Lenstra's proof of the Carlitz–Wan conjecture on exceptional polynomials: an elementary version", Finite Fields and Their Applications, 1 (3): 372–375, doi:10.1006/ffta.1995.1027, MR 1341953
^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 (eds.). 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. Vol. 414. American Mathematical Society. pp. 299–355. doi:10.1090/conm/414/07820. ISBN 978-0-8218-3923-2. Retrieved 24 April 2015.
^Mackenzie, Dana. "Kadison–Singer Problem Solved" (PDF). SIAM News. No. January/February 2014. Society for Industrial and Applied Mathematics. Archived (PDF) from the original on 23 October 2014. Retrieved 24 April 2015.
^ abAgol, Ian (2004). "Tameness of hyperbolic 3-manifolds". arXiv:math/0405568.
^Kurdyka, Krzysztof; Mostowski, Tadeusz; Parusiński, Adam (2000). "Proof of the gradient conjecture of R. Thom". Annals of Mathematics. 152 (3): 763–792. arXiv:math/9906212. doi:10.2307/2661354. JSTOR 2661354. S2CID 119137528.
^Moreira, Joel; Richter, Florian K.; Robertson, Donald (2019). "A proof of a sumset conjecture of Erdős". Annals of Mathematics. 189 (2): 605–652. arXiv:1803.00498. doi:10.4007/annals.2019.189.2.4. S2CID 119158401.
^Stanley, Richard P. (1994), "A survey of Eulerian posets", in Bisztriczky, T.; McMullen, P.; Schneider, R.; Weiss, A. Ivić (eds.), Polytopes: abstract, convex and computational (Scarborough, ON, 1993), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 440, Dordrecht: Kluwer Academic Publishers, pp. 301–333, MR 1322068. See in particular p. 316.
^Kalai, Gil (2018-12-25). "Amazing: Karim Adiprasito proved the g-conjecture for spheres!". Archived from the original on 2019-02-16. Retrieved 2019-02-15.
^Santos, Franciscos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics. 176 (1): 383–412. arXiv:1006.2814. doi:10.4007/annals.2012.176.1.7. S2CID 15325169.
^Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica. Documenta Mathematica Series. 6 (Extra Volume "Optimization Stories"): 75–85. doi:10.4171/dms/6/13. ISBN 978-3-936609-58-5.
^Chung, Fan; Greene, Curtis; Hutchinson, Joan (April 2015). "Herbert S. Wilf (1931–2012)". Notices of the AMS. 62 (4): 358. doi:10.1090/noti1247. ISSN 1088-9477. OCLC 34550461. The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004.
^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. S2CID 119615076.
^"News from 2007". American Mathematical Society. AMS. 31 December 2007. Archived from the original on 17 November 2015. Retrieved 2015-11-13. The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..."
^Brown, Aaron; Fisher, David; Hurtado, Sebastian (2017-10-07). "Zimmer's conjecture for actions of SL(𝑚,ℤ)". arXiv:1710.02735 [math.DS].
^Xue, Jinxin (2014). "Noncollision Singularities in a Planar Four-body Problem". arXiv:1409.0048 [math.DS].
^Xue, Jinxin (2020). "Non-collision singularities in a planar 4-body problem". Acta Mathematica. 224 (2): 253–388. doi:10.4310/ACTA.2020.v224.n2.a2. S2CID 226420221.
^
Richard P Mann. "Known Historical Beggar-My-Neighbour Records". Retrieved 2024-02-10.
^Bowditch, Brian H. (2006). "The angel game in the plane" (PDF). School of Mathematics, University of Southampton: warwick.ac.uk Warwick University. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.
^Kloster, Oddvar. "A Solution to the Angel Problem" (PDF). Oslo, Norway: SINTEF ICT. Archived from the original (PDF) on 2016-01-07. Retrieved 2016-03-18.
^Mathe, Andras (2007). "The Angel of power 2 wins" (PDF). Combinatorics, Probability and Computing. 16 (3): 363–374. doi:10.1017/S0963548306008303 (inactive 1 November 2024). S2CID 16892955. Archived (PDF) from the original on 2016-10-13. Retrieved 2016-03-18.{{cite journal}}: CS1 maint: DOI inactive as of November 2024 (link)
^Gacs, Peter (June 19, 2007). "THE ANGEL WINS" (PDF). Archived from the original (PDF) on 2016-03-04. Retrieved 2016-03-18.
^Larson, Eric (2017). "The Maximal Rank Conjecture". arXiv:1711.04906 [math.AG].
^Kerz, Moritz; Strunk, Florian; Tamme, Georg (2018), "Algebraic K-theory and descent for blow-ups", Inventiones Mathematicae, 211 (2): 523–577, arXiv:1611.08466, Bibcode:2018InMat.211..523K, doi:10.1007/s00222-017-0752-2, MR 3748313, S2CID 253741858
^Song, Antoine. "Existence of infinitely many minimal hypersurfaces in closed manifolds" (PDF). www.ams.org. Retrieved 19 June 2021. ..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves..
^"Antoine Song | Clay Mathematics Institute". ...Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality
^Wolchover, Natalie (July 11, 2017), "Pentagon Tiling Proof Solves Century-Old Math Problem", Quanta Magazine, archived from the original on August 6, 2017, retrieved July 18, 2017
^Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. 179 (2): 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6. S2CID 50742102.
^Guth, Larry; Katz, Nets Hawk (2015). "On the Erdos distinct distance problem in the plane". Annals of Mathematics. 181 (1): 155–190. arXiv:1011.4105. doi:10.4007/annals.2015.181.1.2.
^Henle, Frederick V.; Henle, James M. "Squaring the Plane" (PDF). www.maa.org Mathematics Association of America. Archived (PDF) from the original on 2016-03-24. Retrieved 2016-03-18.
^Brock, Jeffrey F.; Canary, Richard D.; Minsky, Yair N. (2012). "The classification of Kleinian surface groups, II: The Ending Lamination Conjecture". Annals of Mathematics. 176 (1): 1–149. arXiv:math/0412006. doi:10.4007/annals.2012.176.1.1.
^Faber, C.; Pandharipande, R. (2003), "Hodge integrals, partition matrices, and the conjecture", Ann. of Math., 2, 157 (1): 97–124, arXiv:math.AG/9908052, doi:10.4007/annals.2003.157.97
^Shestakov, Ivan P.; Umirbaev, Ualbai U. (2004). "The tame and the wild automorphisms of polynomial rings in three variables". Journal of the American Mathematical Society. 17 (1): 197–227. doi:10.1090/S0894-0347-03-00440-5. MR 2015334.
^Hutchings, Michael; Morgan, Frank; Ritoré, Manuel; Ros, Antonio (2002). "Proof of the double bubble conjecture". Annals of Mathematics. Second Series. 155 (2): 459–489. arXiv:math/0406017. doi:10.2307/3062123. hdl:10481/32449. JSTOR 3062123. MR 1906593.
^Teixidor i Bigas, Montserrat; Russo, Barbara (1999). "On a conjecture of Lange". Journal of Algebraic Geometry. 8 (3): 483–496. arXiv:alg-geom/9710019. Bibcode:1997alg.geom.10019R. ISSN 1056-3911. MR 1689352.
^Ullmo, E (1998). "Positivité et Discrétion des Points Algébriques des Courbes". Annals of Mathematics. 147 (1): 167–179. arXiv:alg-geom/9606017. doi:10.2307/120987. JSTOR 120987. S2CID 119717506. Zbl 0934.14013.
^Zhang, S.-W. (1998). "Equidistribution of small points on abelian varieties". Annals of Mathematics. 147 (1): 159–165. doi:10.2307/120986. JSTOR 120986.
^Hales, Thomas; Adams, Mark; Bauer, Gertrud; Dang, Dat Tat; Harrison, John; Hoang, Le Truong; Kaliszyk, Cezary; Magron, Victor; McLaughlin, Sean; Nguyen, Tat Thang; Nguyen, Quang Truong; Nipkow, Tobias; Obua, Steven; Pleso, Joseph; Rute, Jason; Solovyev, Alexey; Ta, Thi Hoai An; Tran, Nam Trung; Trieu, Thi Diep; Urban, Josef; Ky, Vu; Zumkeller, Roland (2017). "A formal proof of the Kepler conjecture". Forum of Mathematics, Pi. 5: e2. arXiv:1501.02155. doi:10.1017/fmp.2017.1.
^Hales, Thomas C.; McLaughlin, Sean (2010). "The dodecahedral conjecture". Journal of the American Mathematical Society. 23 (2): 299–344. arXiv:math/9811079. Bibcode:2010JAMS...23..299H. doi:10.1090/S0894-0347-09-00647-X.
^Park, Jinyoung; Pham, Huy Tuan (2022-03-31). "A Proof of the Kahn-Kalai Conjecture". arXiv:2203.17207 [math.CO].
^Huang, C.; Kotzig, A.; Rosa, A. (1982). "Further results on tree labellings". Utilitas Mathematica. 21: 31–48. MR 0668845..
^Hartnett, Kevin (19 February 2020). "Rainbow Proof Shows Graphs Have Uniform Parts". Quanta Magazine. Retrieved 2020-02-29.
^Shitov, Yaroslav (1 September 2019). "Counterexamples to Hedetniemi's conjecture". Annals of Mathematics. 190 (2): 663–667. arXiv:1905.02167. doi:10.4007/annals.2019.190.2.6. JSTOR 10.4007/annals.2019.190.2.6. MR 3997132. S2CID 146120733. Zbl 1451.05087. Retrieved 19 July 2021.
^He, Dawei; Wang, Yan; Yu, Xingxing (2019-12-11). "The Kelmans-Seymour conjecture I: Special separations". Journal of Combinatorial Theory, Series B. 144: 197–224. arXiv:1511.05020. doi:10.1016/j.jctb.2019.11.008. ISSN 0095-8956. S2CID 29791394.
^He, Dawei; Wang, Yan; Yu, Xingxing (2019-12-11). "The Kelmans-Seymour conjecture II: 2-Vertices in K4−". Journal of Combinatorial Theory, Series B. 144: 225–264. arXiv:1602.07557. doi:10.1016/j.jctb.2019.11.007. ISSN 0095-8956. S2CID 220369443.
^He, Dawei; Wang, Yan; Yu, Xingxing (2019-12-09). "The Kelmans-Seymour conjecture III: 3-vertices in K4−". Journal of Combinatorial Theory, Series B. 144: 265–308. arXiv:1609.05747. doi:10.1016/j.jctb.2019.11.006. ISSN 0095-8956. S2CID 119625722.
^He, Dawei; Wang, Yan; Yu, Xingxing (2019-12-19). "The Kelmans-Seymour conjecture IV: A proof". Journal of Combinatorial Theory, Series B. 144: 309–358. arXiv:1612.07189. doi:10.1016/j.jctb.2019.12.002. ISSN 0095-8956. S2CID 119175309.
^Zang, Wenan; Jing, Guangming; Chen, Guantao (2019-01-29). "Proof of the Goldberg–Seymour Conjecture on Edge-Colorings of Multigraphs". arXiv:1901.10316v1 [math.CO].
^Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398. S2CID 117651702.
^Huh, June (2012). "Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs". Journal of the American Mathematical Society. 25 (3): 907–927. arXiv:1008.4749. doi:10.1090/S0894-0347-2012-00731-0.
^Chalopin, Jérémie; Gonçalves, Daniel (2009). "Every planar graph is the intersection graph of segments in the plane: extended abstract". In Mitzenmacher, Michael (ed.). Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, Bethesda, MD, USA, May 31 - June 2, 2009. ACM. pp. 631–638. doi:10.1145/1536414.1536500.
^Aharoni, Ron; Berger, Eli (2009). "Menger's theorem for infinite graphs". Inventiones Mathematicae. 176 (1): 1–62. arXiv:math/0509397. Bibcode:2009InMat.176....1A. doi:10.1007/s00222-008-0157-3.
^Seigel-Itzkovich, Judy (2008-02-08). "Russian immigrant solves math puzzle". The Jerusalem Post. Retrieved 2015-11-12.
^Diestel, Reinhard (2005). "Minors, Trees, and WQO" (PDF). Graph Theory (Electronic Edition 2005 ed.). Springer. pp. 326–367.
^Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2002). "The strong perfect graph theorem". Annals of Mathematics. 164: 51–229. arXiv:math/0212070. Bibcode:2002math.....12070C. doi:10.4007/annals.2006.164.51. S2CID 119151552.
^Klin, M. H., M. Muzychuk and R. Poschel: The isomorphism problem for circulant graphs via Schur ring theory, Codes and Association Schemes, American Math. Society, 2001.
^Chen, Zhibo (1996). "Harary's conjectures on integral sum graphs". Discrete Mathematics. 160 (1–3): 241–244. doi:10.1016/0012-365X(95)00163-Q.
^Friedman, Joel (January 2015). "Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: with an Appendix by Warren Dicks" (PDF). Memoirs of the American Mathematical Society. 233 (1100): 0. doi:10.1090/memo/1100. ISSN 0065-9266. S2CID 117941803.
^Mineyev, Igor (2012). "Submultiplicativity and the Hanna Neumann conjecture". Annals of Mathematics. Second Series. 175 (1): 393–414. doi:10.4007/annals.2012.175.1.11. MR 2874647.
^Namazi, Hossein; Souto, Juan (2012). "Non-realizability and ending laminations: Proof of the density conjecture". Acta Mathematica. 209 (2): 323–395. doi:10.1007/s11511-012-0088-0.
^Pila, Jonathan; Shankar, Ananth; Tsimerman, Jacob; Esnault, Hélène; Groechenig, Michael (2021-09-17). "Canonical Heights on Shimura Varieties and the André-Oort Conjecture". arXiv:2109.08788 [math.NT].
^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 (2): 633–682. arXiv:1512.01565. Bibcode:2015arXiv151201565B. doi:10.4007/annals.2016.184.2.7. hdl:1721.1/115568. S2CID 43929329.
^Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
^Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
^Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
^Zhang, Yitang (2014-05-01). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. ISSN 0003-486X.
^"Bounded gaps between primes - Polymath Wiki". asone.ai. Archived from the original on 2020-12-08. Retrieved 2021-08-27.
^Maynard, James (2015-01-01). "Small gaps between primes". Annals of Mathematics: 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7. ISSN 0003-486X. S2CID 55175056.
^Cilleruelo, Javier (2010). "Generalized Sidon sets". Advances in Mathematics. 225 (5): 2786–2807. doi:10.1016/j.aim.2010.05.010. hdl:10261/31032. S2CID 7385280.
^"2011 Cole Prize in Number Theory" (PDF). Notices of the AMS. 58 (4): 610–611. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2015-11-06. Retrieved 2015-11-12.
^"Bombieri and Tao Receive King Faisal Prize" (PDF). Notices of the AMS. 57 (5): 642–643. May 2010. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18. Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem.
^Metsänkylä, Tauno (5 September 2003). "Catalan's conjecture: another old diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. 41 (1): 43–57. doi:10.1090/s0273-0979-03-00993-5. ISSN 0273-0979. Archived (PDF) from the original on 4 March 2016. Retrieved 13 November 2015. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu.
^Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 1431-0635, MR 1648105, archived from the original on 2018-04-27, retrieved 2016-03-18
^Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. Archived (PDF) from the original on 2011-05-10. Retrieved 2016-03-06.
^Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX10.1.1.128.531. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255. Archived from the original on 16 September 2000.
^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. S2CID 7974973.
^Lamb, Evelyn (26 May 2016). "Two-hundred-terabyte maths proof is largest ever". Nature. 534 (7605): 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". In Creignou, N.; Le Berre, D. (eds.). Theory and Applications of Satisfiability Testing – SAT 2016. Lecture Notes in Computer Science. Vol. 9710. Springer, [Cham]. pp. 228–245. arXiv:1605.00723. doi:10.1007/978-3-319-40970-2_15. ISBN 978-3-319-40969-6. MR 3534782. S2CID 7912943.
^Linkletter, David (27 December 2019). "The 10 Biggest Math Breakthroughs of 2019". Popular Mechanics. Retrieved 20 June 2021.
^Piccirillo, Lisa (2020). "The Conway knot is not slice". Annals of Mathematics. 191 (2): 581–591. doi:10.4007/annals.2020.191.2.5. S2CID 52398890.
^Agol, Ian (2013). "The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves, and Jason Manning)" (PDF). Documenta Mathematica. 18: 1045–1087. arXiv:1204.2810v1. doi:10.4171/dm/421. S2CID 255586740.
^Brendle, Simon (2013). "Embedded minimal tori in and the Lawson conjecture". Acta Mathematica. 211 (2): 177–190. arXiv:1203.6597. doi:10.1007/s11511-013-0101-2.
^Kahn, Jeremy; Markovic, Vladimir (2015). "The good pants homology and the Ehrenpreis conjecture". Annals of Mathematics. 182 (1): 1–72. arXiv:1101.1330. doi:10.4007/annals.2015.182.1.1.
^Austin, Tim (December 2013). "Rational group ring elements with kernels having irrational dimension". Proceedings of the London Mathematical Society. 107 (6): 1424–1448. arXiv:0909.2360. Bibcode:2009arXiv0909.2360A. doi:10.1112/plms/pdt029. S2CID 115160094.
^Lurie, Jacob (2009). "On the classification of topological field theories". Current Developments in Mathematics. 2008: 129–280. arXiv:0905.0465. Bibcode:2009arXiv0905.0465L. doi:10.4310/cdm.2008.v2008.n1.a3. S2CID 115162503.
^ ab"Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF) (Press release). Clay Mathematics Institute. March 18, 2010. Archived from the original on March 22, 2010. Retrieved November 13, 2015. The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.
^Morgan, John; Tian, Gang (2008). "Completion of the Proof of the Geometrization Conjecture". arXiv:0809.4040 [math.DG].
^Rudin, M.E. (2001). "Nikiel's Conjecture". Topology and Its Applications. 116 (3): 305–331. doi:10.1016/S0166-8641(01)00218-8.
^Norio Iwase (1 November 1998). "Ganea's Conjecture on Lusternik-Schnirelmann Category". ResearchGate.
^Tao, Terence (2015). "The Erdős discrepancy problem". arXiv:1509.05363v5 [math.CO].
^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. arXiv:1503.01472. Bibcode:2015arXiv150301472D. doi:10.1186/s40687-015-0044-7. S2CID 43589605.
^Cheeger, Jeff; Naber, Aaron (2015). "Regularity of Einstein Manifolds and the Codimension 4 Conjecture". Annals of Mathematics. 182 (3): 1093–1165. arXiv:1406.6534. doi:10.4007/annals.2015.182.3.5.
^Wolchover, Natalie (March 28, 2017). "A Long-Sought Proof, Found and Almost Lost". Quanta Magazine. Archived from the original on April 24, 2017. Retrieved May 2, 2017.
^Newman, Alantha; Nikolov, Aleksandar (2011). "A counterexample to Beck's conjecture on the discrepancy of three permutations". arXiv:1104.2922 [cs.DM].
^Voevodsky, Vladimir (1 July 2011). "On motivic cohomology with Z/l-coefficients" (PDF). annals.math.princeton.edu. Princeton, NJ: Princeton University. pp. 401–438. Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.
^Geisser, Thomas; Levine, Marc (2001). "The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky". Journal für die Reine und Angewandte Mathematik. 2001 (530): 55–103. doi:10.1515/crll.2001.006. MR 1807268.
^Kahn, Bruno. "Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry" (PDF). webusers.imj-prg.fr. Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.
^Mattman, Thomas W.; Solis, Pablo (2009). "A proof of the Kauffman-Harary Conjecture". Algebraic & Geometric Topology. 9 (4): 2027–2039. arXiv:0906.1612. Bibcode:2009arXiv0906.1612M. doi:10.2140/agt.2009.9.2027. S2CID 8447495.
^Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127–1190. arXiv:0910.5501. doi:10.4007/annals.2012.175.3.4.
^Lu, Zhiqin (September 2011) [2007]. "Normal Scalar Curvature Conjecture and its applications". Journal of Functional Analysis. 261 (5): 1284–1308. arXiv:0711.3510. doi:10.1016/j.jfa.2011.05.002.
^Dencker, Nils (2006), "The resolution of the Nirenberg–Treves conjecture" (PDF), Annals of Mathematics, 163 (2): 405–444, doi:10.4007/annals.2006.163.405, S2CID 16630732, archived (PDF) from the original on 2018-07-20, retrieved 2019-04-07
^"Research Awards". Clay Mathematics Institute. Archived from the original on 2019-04-07. Retrieved 2019-04-07.
^Lewis, A. S.; Parrilo, P. A.; Ramana, M. V. (2005). "The Lax conjecture is true". Proceedings of the American Mathematical Society. 133 (9): 2495–2499. doi:10.1090/S0002-9939-05-07752-X. MR 2146191. S2CID 17436983.
^"Fields Medal – Ngô Bảo Châu". International Congress of Mathematicians 2010. ICM. 19 August 2010. Archived from the original on 24 September 2015. 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.
^Voevodsky, Vladimir (2003). "Reduced power operations in motivic cohomology". Publications Mathématiques de l'IHÉS. 98: 1–57. arXiv:math/0107109. CiteSeerX10.1.1.170.4427. doi:10.1007/s10240-003-0009-z. S2CID 8172797. Archived from the original on 2017-07-28. Retrieved 2016-03-18.
^Baruch, Ehud Moshe (2003). "A proof of Kirillov's conjecture". Annals of Mathematics. Second Series. 158 (1): 207–252. doi:10.4007/annals.2003.158.207. MR 1999922.
^Haas, Bertrand (2002). "A Simple Counterexample to Kouchnirenko's Conjecture" (PDF). Beiträge zur Algebra und Geometrie. 43 (1): 1–8. Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.
^Haiman, Mark (2001). "Hilbert schemes, polygraphs and the Macdonald positivity conjecture". Journal of the American Mathematical Society. 14 (4): 941–1006. doi:10.1090/S0894-0347-01-00373-3. MR 1839919. S2CID 9253880.
^Auscher, Pascal; Hofmann, Steve; Lacey, Michael; McIntosh, Alan; Tchamitchian, Ph. (2002). "The solution of the Kato square root problem for second order elliptic operators on ". Annals of Mathematics. Second Series. 156 (2): 633–654. doi:10.2307/3597201. JSTOR 3597201. MR 1933726.
^Barbieri-Viale, Luca; Rosenschon, Andreas; Saito, Morihiko (2003). "Deligne's Conjecture on 1-Motives". Annals of Mathematics. 158 (2): 593–633. arXiv:math/0102150. doi:10.4007/annals.2003.158.593.