List of conjectures


This is a list of mathematical conjectures.

Open problems

Conjectures now proved (theorems)

The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names.
Priority dateProved byFormer nameFieldComments
1962Walter Feit, John ThompsonBurnside conjecture that, apart from cyclic groups, finite simple groups have even orderfinite simple groupsFeit–Thompson theorem⇔trivially the "odd order theorem" that finite groups of odd order are solvable groups
1968Gerhard Ringel and Ted YoungsHeawood conjecturegraph theoryRingel-Youngs theorem
1971Daniel QuillenAdams conjecturealgebraic topologyOn the J-homomorphism, proposed 1963 by Frank Adams
1973Pierre DeligneWeil conjecturesalgebraic geometry⇒Ramanujan–Petersson conjecture
Proposed by André Weil. Deligne's theorems completed around 15 years of work on the general case.
1975Henryk Hecht and Wilfried SchmidBlattner's conjecturerepresentation theory for semisimple groups
1975William HaboushMumford conjecturegeometric invariant theoryHaboush's theorem
1976Kenneth Appel and Wolfgang HakenFour color theoremgraph colouringTraditionally called a "theorem", long before the proof.
1976Daniel Quillen and Andrei Suslin independentlySerre's conjecture on projective modulespolynomial ringsQuillen–Suslin theorem
1977Alberto CalderónDenjoy's conjecturerectifiable curvesA result claimed in 1909 by Arnaud Denjoy, proved by Calderón as a by-product of work on Cauchy singular operators
1978Roger Heath-Brown and S. J. PattersonKummer's conjecture on cubic Gauss sumsequidistribution
1983Gerd FaltingsMordell conjecturenumber theory⇐Faltings's theorem, the Shafarevich conjecture on finiteness of isomorphism classes of abelian varieties. The reduction step was by Alexey Parshin.
1983 onwardsNeil Robertson and Paul D. SeymourWagner's conjecturegraph theoryNow generally known as the graph minor theorem.
1983Michel RaynaudManin–Mumford conjecturediophantine geometryThe Tate–Voloch conjecture is a quantitative derived conjecture for p-adic varieties.
c.1984Collective workSmith conjectureknot theoryBased on work of William Thurston on hyperbolic structures on 3-manifolds, with results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, also with Hyman Bass, Cameron Gordon, Peter Shalen, and Rick Litherland, written up by Bass and John Morgan.
1984Louis de BrangesBieberbach conjecture, 1916complex analysis⇐Robertson conjecture⇐Milin conjecture⇐de Branges's theorem
1984Gunnar CarlssonSegal's conjecturehomotopy theory
1984Haynes MillerSullivan conjectureclassifying spacesMiller proved the version on mapping BG to a finite complex.
1987Grigory MargulisOppenheim conjecturediophantine approximationMargulis proved the conjecture with ergodic theory methods.
1989V. I. ChernousovWeil's conjecture on Tamagawa numbersalgebraic groupsThe problem, based on Siegel's theory for quadratic forms, submitted to a long series of case analysis steps.
1990Ken Ribetepsilon conjecturemodular forms
1992Richard BorcherdsConway–Norton conjecturesporadic groupsUsually called monstrous moonshine
1994David Harbater and Michel RaynaudAbhyankar's conjecturealgebraic geometry
1994Andrew WilesFermat's Last Theoremnumber theory⇔The modularity theorem for semistable elliptic curves.
Proof completed with Richard Taylor.
1994Fred GalvinDinitz conjecturecombinatorics
1995Doron ZeilbergerAlternating sign matrix conjecture,enumerative combinatorics
1996Vladimir VoevodskyMilnor conjecturealgebraic K-theoryVoevodsky's theorem, ⇐norm residue isomorphism theorem⇔Beilinson–Lichtenbaum conjecture, Quillen–Lichtenbaum conjecture.
The ambiguous term "Bloch-Kato conjecture" may refer to what is now the norm residue isomorphism theorem.
1998Thomas Callister HalesKepler conjecturesphere packing
1998Thomas Callister Hales and Sean McLaughlindodecahedral conjectureVoronoi decompositions
2000Krzysztof Kurdyka, Tadeusz Mostowski and Adam ParusińskiGradient conjecturegradient vector fieldsAttributed to René Thom, c.1970.
2001Christophe Breuil, Brian Conrad, Fred Diamond and Richard TaylorTaniyama–Shimura conjectureelliptic curvesNow the modularity theorem for elliptic curves. Once known as the "Weil conjecture".
2001Mark Haimann! conjecturerepresentation theory
2001Daniel Frohardt and Kay MagaardGuralnick–Thompson conjecturemonodromy groups
2002Preda MihăilescuCatalan's conjecture, 1844exponential diophantine equations⇐Pillai's conjecture⇐abc conjecture
Mihăilescu's theorem
2002Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomasstrong perfect graph conjectureperfect graphsChudnovsky–Robertson–Seymour–Thomas theorem
2002Grigori PerelmanPoincaré conjecture, 19043-manifolds
2003Grigori Perelmangeometrization conjecture of Thurston3-manifolds⇒spherical space form conjecture
2003Ben Green; and independently by Alexander SapozhenkoCameron–Erdős conjecturesum-free sets
2003Nils DenckerNirenberg–Treves conjecturepseudo-differential operators
2004 Nobuo Iiyori and Hiroshi YamakiFrobenius conjecturegroup theoryA consequence of the classification of finite simple groups, completed in 2004 by the usual standards of pure mathematics.
2004Adam Marcus and Gábor TardosStanley–Wilf conjecturepermutation classesMarcus–Tardos theorem
2004Ualbai U. Umirbaev and Ivan P. ShestakovNagata's conjecture on automorphismspolynomial rings
2004Ian Agol and independently by Danny Calegari–David Gabaitameness conjecturegeometric topology⇒Ahlfors measure conjecture
2008Avraham TrahtmanRoad coloring conjecturegraph theory
2008Chandrashekhar Khare, Jean-Pierre WintenbergerSerre's modularity conjecturemodular forms
2009Jeremy Kahn, Vladimir Markovicsurface subgroup conjecture3-manifolds⇒Ehrenpreis conjecture on quasiconformality
2009Jeremie Chalopin and Daniel GonçalvesScheinerman's conjectureintersection graphs
2010Terence Tao and Van H. Vucircular lawrandom matrix theory
2011Joel Friedman and Igor Mineyev, independentlyHanna Neumann conjecturegroup theory
2012Simon BrendleHsiang–Lawson's conjecturedifferential geometry
2012Fernando Codá Marques and André NevesWillmore conjecturedifferential geometry
2013Zhang Yitangbounded gap conjecturenumber theoryThe sequence of gaps between consecutive prime numbers has a finite lim inf. See Polymath Project#Polymath8 for quantitative results.
2013Adam Marcus, Daniel Spielman and Nikhil SrivastavaKadison–Singer problemfunctional analysisThe original problem posed by Kadison and Singer was not a conjecture: its authors believed it false. As reformulated, it became the "paving conjecture" for Euclidean spaces, and then a question on random polynomials, in which latter form it was solved affirmatively.
2015Jean Bourgain, Ciprian Demeter, and Larry GuthMain conjecture in Vinogradov's mean-value theoremanalytic number theoryBourgain–Demeter–Guth theorem, ⇐ decoupling theorem
2019Dimitris Koukoulopoulos and James MaynardDuffin–Schaeffer conjecturenumber theoryRational approximation of irrational numbers