Conjectures Now Proved (Theorems)
The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names.
| # | Year | Proved by | Former name | Field | Comments |
|---|---|---|---|---|---|
| 1 | 1962 | Walter Feit, John G. Thompson | Burnside conjecture | Finite simple groups | Feit–Thompson theorem ⇔ “odd order theorem” that finite groups of odd order are solvable |
| 2 | 1968 | Gerhard Ringel, J. W. T. Youngs | Heawood conjecture | Graph theory | Ringel-Youngs theorem |
| 3 | 1971 | Daniel Quillen | Adams conjecture | Algebraic topology | On the J-homomorphism, proposed 1963 by Frank Adams |
| 4 | 1973 | Pierre Deligne | Weil conjectures | Algebraic geometry | ⇒ Ramanujan–Petersson conjecture. Proposed by André Weil. Deligne’s theorems completed ~15 years of work on the general case. |
| 5 | 1975 | Henryk Hecht, Wilfried Schmid | Blattner’s conjecture | Representation theory | For semisimple groups |
| 6 | 1975 | William Haboush | Mumford conjecture | Geometric invariant theory | Haboush’s theorem |
| 7 | 1976 | Kenneth Appel, Wolfgang Haken | Four color theorem | Graph colouring | Traditionally called a “theorem”, long before the proof |
| 8 | 1976 | Daniel Quillen; independently Andrei Suslin | Serre’s conjecture on projective modules | Polynomial rings | Quillen–Suslin theorem |
| 9 | 1977 | Alberto Calderón | Denjoy’s conjecture | Rectifiable curves | A result claimed in 1909 by Arnaud Denjoy, proved by Calderón as a by-product of work on Cauchy singular operators |
| 10 | 1978 | Roger Heath-Brown, Samuel James Patterson | Kummer’s conjecture on cubic Gauss sums | Equidistribution | |
| 11 | 1983 | Gerd Faltings | Mordell conjecture | Number theory | ⇐ Faltings’s theorem, the Shafarevich conjecture on finiteness of isomorphism classes of abelian varieties. Reduction step by Alexey Parshin. |
| 12 | 1983+ | Neil Robertson, Paul D. Seymour | Wagner’s conjecture | Graph theory | Now generally known as the graph minor theorem |
| 13 | 1983 | Michel Raynaud | Manin–Mumford conjecture | Diophantine geometry | The Tate–Voloch conjecture is a quantitative derived conjecture for p-adic varieties |
| 14 | c.1984 | Collective work | Smith conjecture | Knot theory | Based on work of William Thurston on hyperbolic structures on 3-manifolds, with results by William Meeks and Shing-Tung Yau on minimal surfaces, also with Hyman Bass, Cameron Gordon, Peter Shalen, and Rick Litherland |
| 15 | 1984 | Louis de Branges de Bourcia | Bieberbach conjecture (1916) | Complex analysis | ⇐ Robertson conjecture ⇐ Milin conjecture ⇐ de Branges’s theorem |
| 16 | 1984 | Gunnar Carlsson | Segal’s conjecture | Homotopy theory | |
| 17 | 1984 | Haynes Miller | Sullivan conjecture | Classifying spaces | Miller proved the version on mapping BG to a finite complex |
| 18 | 1987 | Grigory Margulis | Oppenheim conjecture | Diophantine approximation | Proved with ergodic theory methods |
| 19 | 1989 | Vladimir I. Chernousov | Weil’s conjecture on Tamagawa numbers | Algebraic groups | Based on Siegel’s theory for quadratic forms; submitted to a long series of case analysis steps |
| 20 | 1990 | Ken Ribet | Epsilon conjecture | Modular forms | |
| 21 | 1992 | Richard Borcherds | Conway–Norton conjecture | Sporadic groups | Usually called monstrous moonshine |
| 22 | 1994 | David Harbater, Michel Raynaud | Abhyankar’s conjecture | Algebraic geometry | |
| 23 | 1994 | Andrew Wiles | Fermat’s Last Theorem | Number theory | ⇔ The modularity theorem for semistable elliptic curves. Proof completed with Richard Taylor. |
| 24 | 1994 | Fred Galvin | Dinitz conjecture | Combinatorics | |
| 25 | 1995 | Doron Zeilberger | Alternating sign matrix conjecture | Enumerative combinatorics | |
| 26 | 1996 | Vladimir Voevodsky | Milnor conjecture | Algebraic K-theory | Voevodsky’s theorem ⇐ norm residue isomorphism theorem ⇔ Beilinson–Lichtenbaum conjecture, Quillen–Lichtenbaum conjecture |
| 27 | 1998 | Thomas Callister Hales | Kepler conjecture | Sphere packing | |
| 28 | 1998 | Thomas Callister Hales, Sean McLaughlin | Dodecahedral conjecture | Voronoi decompositions | |
| 29 | 2000 | Krzysztof Kurdyka, Tadeusz Mostowski, Adam Parusiński | Gradient conjecture | Gradient vector fields | Attributed to René Thom, c.1970 |
| 30 | 2001 | Christophe Breuil, Brian Conrad, Fred Diamond, Richard Taylor | Taniyama–Shimura conjecture | Elliptic curves | Now the modularity theorem for elliptic curves. Once known as the “Weil conjecture”. |
| 31 | 2001 | Mark Haiman | n! conjecture | Representation theory | |
| 32 | 2001 | Daniel Frohardt, Kay Magaard | Guralnick–Thompson conjecture | Monodromy groups | |
| 33 | 2002 | Preda Mihăilescu | Catalan’s conjecture (1844) | Exponential Diophantine equations | ⇐ Pillai’s conjecture ⇐ abc conjecture. Mihăilescu’s theorem. |
| 34 | 2002 | Maria Chudnovsky, Neil Robertson, Paul D. Seymour, Robin Thomas | Strong perfect graph conjecture | Perfect graphs | Chudnovsky–Robertson–Seymour–Thomas theorem |
| 35 | 2002 | Grigori Perelman | Poincaré conjecture (1904) | 3-manifolds | |
| 36 | 2003 | Grigori Perelman | Geometrization conjecture of Thurston | 3-manifolds | ⇒ spherical space form conjecture |
| 37 | 2003 | Ben Green; independently Alexander Sapozhenko | Cameron–Erdős conjecture | Sum-free sets | |
| 38 | 2003 | Nils Dencker | Nirenberg–Treves conjecture | Pseudo-differential operators | |
| 39 | 2004 | Nobuo Iiyori, Hiroshi Yamaki | Frobenius conjecture | Group theory | A consequence of the classification of finite simple groups, completed in 2004 |
| 40 | 2004 | Adam Marcus, Gábor Tardos | Stanley–Wilf conjecture | Permutation classes | Marcus–Tardos theorem |
| 41 | 2004 | Ualbai U. Umirbaev, Ivan P. Shestakov | Nagata’s conjecture on automorphisms | Polynomial rings | |
| 42 | 2004 | Ian Agol; independently Danny Calegari, David Gabai | Tameness conjecture | Geometric topology | ⇒ Ahlfors measure conjecture |
| 43 | 2008 | Avraham Trahtman | Road coloring conjecture | Graph theory | |
| 44 | 2008 | Chandrashekhar Khare, Jean-Pierre Wintenberger | Serre’s modularity conjecture | Modular forms | |
| 45 | 2009 | Jeremy Kahn, Vladimir Markovic | Surface subgroup conjecture | 3-manifolds | ⇒ Ehrenpreis conjecture on quasiconformality |
| 46 | 2009 | Jeremie Chalopin, Daniel Gonçalves | Scheinerman’s conjecture | Intersection graphs | |
| 47 | 2010 | Terence Tao, Van H. Vu | Circular law | Random matrix theory | |
| 48 | 2011 | Joel Friedman; independently Igor Mineyev | Hanna Neumann conjecture | Group theory | |
| 49 | 2012 | Simon Brendle | Hsiang–Lawson’s conjecture | Differential geometry | |
| 50 | 2012 | Fernando Codá Marques, André Neves | Willmore conjecture | Differential geometry | |
| 51 | 2013 | Yitang Zhang | Bounded gap conjecture | Number theory | The sequence of gaps between consecutive prime numbers has a finite lim inf. See Polymath8 for quantitative results. |
| 52 | 2013 | Adam Marcus, Daniel Spielman, Nikhil Srivastava | Kadison–Singer problem | Functional analysis | The original problem was not a conjecture: its authors believed it false. Reformulated as the “paving conjecture”, then a question on random polynomials. |
| 53 | 2015 | Jean Bourgain, Ciprian Demeter, Larry Guth | Main conjecture in Vinogradov’s mean-value theorem | Analytic number theory | Bourgain–Demeter–Guth theorem ⇐ decoupling theorem |
| 54 | 2018 | Karim Adiprasito | g-conjecture | Combinatorics | |
| 55 | 2019 | Dimitris Koukoulopoulos, James Maynard | Duffin–Schaeffer conjecture | Number theory | Rational approximation of irrational numbers |
Additional proved conjectures
- Deligne’s conjecture on 1-motives (proved by Deligne)
- Goldbach’s weak conjecture (proved in 2013)
- Sensitivity conjecture (proved in 2019)