Conjectures are usually told as ideas without a place: abstract statements floating in books. But each one was born somewhere, in a specific year, at the hands of someone who was looking at a pattern. This atlas puts 303 conjectures on a world map to reveal that geography: where mathematical activity was concentrated, in which centuries, and how many of those bets remain unresolved.
The map
Drag to rotate the globe, use the scroll wheel to zoom in, and click a point to see the full conjecture. The Play button runs through the timeline from Antiquity to the present day.
Antiquity
Doubling the cube (Delian problem)
Greek geometers · Athens, Greece
Angle trisection
Greek geometers · Athens, Greece
Squaring the circle
Anaxagoras, Hippocrates of Chios and others · Athens, Greece
The fifth postulate (parallel postulate)
Euclid · Alexandria, Egypt
Do odd perfect numbers exist?
Euclid / Nicomachus · Alexandria, Egypt
Honeycomb conjecture
Pappus of Alexandria · Alexandria, Egypt
Middle Ages
Congruent number problem
Al-Karaji (and Arab mathematicians) · Baghdad, present-day Iraq
17th century
Kepler conjecture (sphere packing)
Johannes Kepler · Prague (imperial court of Rudolf II)
Fermat's Last Theorem
Pierre de Fermat · Toulouse, France
Fermat polygonal number theorem
Pierre de Fermat · Toulouse, France
Primality of Fermat numbers
Pierre de Fermat · Toulouse, France
Infinitude of Mersenne primes
Marin Mersenne · Paris, France
Kissing number problem
Isaac Newton and David Gregory · Cambridge, England
18th century
Goldbach's conjecture
Christian Goldbach · Moscow, Russia
Weak (ternary) Goldbach conjecture
Christian Goldbach · Moscow, Russia
Euler's sum of powers conjecture
Leonhard Euler · Saint Petersburg, Russia
Waring's problem
Edward Waring · Cambridge, England
Perfect cuboid conjecture (Euler brick)
Leonhard Euler · Saint Petersburg, Russia
19th century
Gauss class number problem
Carl Friedrich Gauss · Brunswick, Germany
Prime number theorem
A.-M. Legendre and C. F. Gauss · Paris, France
Legendre's conjecture
Adrien-Marie Legendre · Paris, France
Infinitude of Sophie Germain primes
Sophie Germain · Paris, France
Gauss circle problem
Carl Friedrich Gauss · Göttingen, Germany
Catalan's conjecture
Eugène Catalan · Paris, France
Bertrand's postulate
Joseph Bertrand · Paris, France
Twin prime conjecture (de Polignac's conjecture)
Alphonse de Polignac · Paris, France
Dirichlet divisor problem
Peter Gustav Lejeune Dirichlet · Berlin, Germany
Four color conjecture
Francis Guthrie · London (University College London)
Bunyakovsky conjecture
Viktor Bunyakovsky · Saint Petersburg, Russia
Riemann hypothesis
Bernhard Riemann · Göttingen, Germany
Generalized Riemann hypothesis (GRH)
Bernhard Riemann · Göttingen, Germany
Brocard's problem
Henri Brocard · Algiers, Algeria
Tait conjectures (alternating knots)
Peter Guthrie Tait · Edinburgh, Scotland
Continuum hypothesis
Georg Cantor · Halle, Germany
Oppermann's conjecture
Ludvig Oppermann · Copenhagen, Denmark
Kelvin problem (optimal foam)
William Thomson (Lord Kelvin) · Glasgow, United Kingdom
Catalan–Dickson conjecture (aliquot sequences)
Eugène Catalan · Liège, Belgium
Inverse Galois problem
David Hilbert · Göttingen, Germany
Lemoine's conjecture
Émile Lemoine · Quimper, France
Mertens conjecture
Franz Mertens · Vienna, Austria
Painlevé conjecture
Paul Painlevé · Stockholm, Sweden
Hilbert's tenth problem
David Hilbert · Paris, France
Hilbert's third problem
David Hilbert · Paris, France
Hilbert's sixteenth problem (limit cycles)
David Hilbert · Paris, France
No-three-in-line problem
Henry Dudeney · London, England
20th century
General Burnside problem (periodic groups)
William Burnside · Greenwich, London, United Kingdom
Poincaré conjecture
Henri Poincaré · Paris, France
Thomson problem (charges on a sphere)
J. J. Thomson · Cambridge, England
Denjoy conjecture
Arnaud Denjoy · Paris, France
Carmichael's totient function conjecture
Robert Carmichael · Hartselle, Alabama, United States
Lindelöf hypothesis
Ernst Lindelöf · Helsinki, Finland
Hauptvermutung of topology
Ernst Steinitz and Heinrich Tietze · Vienna, Austria
Torsion conjecture (elliptic curves)
Beppo Levi · Turin, Italy
Infinitude of Wieferich primes
Arthur Wieferich · Münster, Germany
Unknotting problem
Max Dehn · Göttingen, Germany
Burnside's conjecture (odd order)
William Burnside · London (Royal Naval College, Greenwich)
Inscribed square problem (Toeplitz)
Otto Toeplitz · Göttingen, Germany
Hilbert–Pólya conjecture
David Hilbert and George Pólya · Göttingen, Germany
Lebesgue's universal covering problem
Henri Lebesgue · Paris, France
Bieberbach conjecture
Ludwig Bieberbach · Frankfurt am Main, Germany
Ramanujan conjecture
Srinivasa Ramanujan · Cambridge, United Kingdom
Kakeya conjecture
Sōichi Kakeya · Sendai, Japan
Goormaghtigh conjecture
René Goormaghtigh · Ostend, Belgium
Classification of convex pentagonal tilings
Karl Reinhardt · Frankfurt am Main, Germany
Pólya conjecture
George Pólya · Zurich, Switzerland
Vandiver's conjecture
Harry Vandiver · Ithaca, New York, United States
Suslin's hypothesis
Mikhail Suslin · Moscow, Russia
Carathéodory conjecture
Constantin Carathéodory · Munich, Germany
Mordell conjecture
Louis Mordell · Manchester, England
Hardy–Littlewood conjectures (prime k-tuples)
G. H. Hardy and J. E. Littlewood · Oxford, United Kingdom
Second Hardy–Littlewood conjecture
G. H. Hardy and J. E. Littlewood · Cambridge, England
Artin's conjecture on primitive roots
Emil Artin · Hamburg, Germany
Birkhoff conjecture (billiards)
George David Birkhoff · Overisel, Michigan, United States
Entscheidungsproblem (decision problem)
David Hilbert · Göttingen, Germany
Von Neumann conjecture
John von Neumann · Berlin, Germany
Oppenheim conjecture
Alexander Oppenheim · Salford, England
Keller's conjecture
Ott-Heinrich Keller · Berlin, Germany
Littlewood conjecture
John Edensor Littlewood · Cambridge, England
Hilbert–Smith conjecture
David Hilbert and Paul Althaus Smith · Göttingen, Germany
Köthe conjecture
Gottfried Köthe · Münster, Germany
Hopf invariant one problem
Heinz Hopf · Zurich, Switzerland
Pillai's conjecture
Subbayya Sivasankaranarayana Pillai · Chidambaram, Tamil Nadu, India
Hopf conjecture on positive curvature (S²×S²)
Heinz Hopf · Zurich, Switzerland
Erdős discrepancy problem
Paul Erdős · Budapest, Hungary
Borsuk's conjecture
Karol Borsuk · Warsaw, Poland
Lehmer's totient problem
Derrick Henry Lehmer · Stanford, California, United States
Hopf conjecture (Euler characteristic and positive curvature)
Heinz Hopf · Zurich, Switzerland
Lehmer's Mahler measure problem
Derrick Henry Lehmer · Princeton, United States
Invariant subspace problem
John von Neumann · Princeton, United States
Happy ending problem (Erdős–Szekeres)
Esther Klein, Paul Erdős and George Szekeres · Budapest, Hungary
Cramér's conjecture
Harald Cramér · Stockholm, Sweden
Bochner–Riesz conjecture
Salomon Bochner · Princeton, United States
Erdős conjecture on arithmetic progressions
Paul Erdős · Manchester, England
Collatz conjecture
Lothar Collatz · Berlin, Germany
Scholz conjecture (addition chains)
Arnold Scholz · Berlin, Germany
Waring–Goldbach problem
Ivan Vinogradov and Hua Luogeng · Velikiye Luki, Russia
Smith conjecture
Paul A. Smith · New York, United States
Jacobian conjecture
Ott-Heinrich Keller · Berlin, Germany
Mahler conjecture
Kurt Mahler · Manchester, United Kingdom
Duffin–Schaeffer conjecture
Richard J. Duffin and Albert C. Schaeffer · West Lafayette, Indiana, United States
Reconstruction conjecture
Stanislaw Ulam and Paul Kelly · Madison, United States
Whitehead asphericity conjecture
J. H. C. Whitehead · Oxford, United Kingdom
Erdős–Turán conjecture on additive bases
Paul Erdős and Pál Turán · Budapest, Hungary
Hadwiger conjecture
Hugo Hadwiger · Bern, Switzerland
Erdős–Ulam problem (points at rational distances)
Paul Erdős and Stanisław Ulam · Los Alamos, New Mexico, United States
Erdős unit distance problem
Paul Erdős · Budapest, Hungary
Erdős distinct distances problem
Paul Erdős · Budapest, Hungary
Lehmer's tau conjecture
Derrick Henry Lehmer · Berkeley, California, United States
Growth of diagonal Ramsey numbers
Paul Erdős · Budapest, Hungary
Erdős–Straus conjecture
Paul Erdős · Budapest, Hungary
Weil conjectures
André Weil · Chicago, United States
Hodge conjecture
William Hodge · Cambridge, United Kingdom
Seifert conjecture
Herbert Seifert · Heidelberg, Germany
Hadwiger–Nelson problem
Edward Nelson and Hugo Hadwiger · Chicago, United States
Giuga's conjecture
Giuseppe Giuga · Rome, Italy
Kaplansky's conjectures on group rings
Irving Kaplansky · Chicago, United States
Whitehead problem
J. H. C. Whitehead · Oxford, United Kingdom
Ore conjecture
Øystein Ore · New Haven, Connecticut, United States
Borel conjecture
Armand Borel · Princeton, United States
Erdős–Moser conjecture
Paul Erdős · Budapest, Hungary
Calabi conjecture
Eugenio Calabi · Amsterdam, Netherlands
Tutte's 5-flow conjecture
William Thomas Tutte · Toronto, Canada
Zarankiewicz crossing number conjecture
Kazimierz Zarankiewicz · Warsaw, Poland
Taniyama–Shimura conjecture (modularity)
Yutaka Taniyama · Tokyo, Japan
Brauer's height zero conjecture
Richard Brauer · Cambridge, Massachusetts, United States
Chern conjecture (flat affine manifolds)
Shiing-Shen Chern · Chicago, United States
Kneser–Poulsen conjecture
Martin Kneser and Ernst Poulsen · Berlin, Germany
Jacobson's conjecture
Nathan Jacobson · New Haven, Connecticut, United States
Hirsch conjecture
Warren M. Hirsch · New York, United States
Strengthened Hanna Neumann conjecture
Hanna Neumann · Hull, United Kingdom
Eilenberg–Ganea conjecture
Samuel Eilenberg and Tudor Ganea · New York, United States
Gilbreath's conjecture
Norman Gilbreath · Los Angeles, United States
Schinzel's hypothesis H
Andrzej Schinzel · Warsaw, Poland
Nakayama conjecture (representation theory)
Tadasi Nakayama · Nagoya, Japan
Sendov's conjecture
Blagovest Sendov · Sofia, Bulgaria
Kadison–Singer problem
Richard Kadison and Isadore Singer · New York, United States
Nagata's conjecture
Masayoshi Nagata · Kyoto, Japan
Sato–Tate conjecture
Mikio Sato and John Tate · Tokyo, Japan
Kervaire invariant problem
Michel Kervaire · New York, United States
Schanuel's conjecture
Stephen Schanuel · New York, United States
Sierpiński problem
Wacław Sierpiński · Warsaw, Poland
Lang's conjectures
Serge Lang · New York, United States
Sunflower conjecture
Paul Erdős and Richard Rado · Budapest, Hungary
Wall–Sun–Sun primes (Fibonacci–Wieferich)
Donald Dines Wall · United States
Generalized Poincaré conjecture
Stephen Smale · Berkeley, United States
Strong perfect graph conjecture
Claude Berge · Paris, France
Aperiodic monotile problem (the “hat” / einstein)
David Smith, J. S. Myers, C. S. Kaplan and C. Goodman-Strauss · Bridlington, United Kingdom
Vaught conjecture
Robert Lawson Vaught · Berkeley, United States
Kato square root problem
Tosio Kato · Berkeley, California, United States
Vector fields on spheres problem
Frank Adams · Cambridge, United Kingdom
Slice-ribbon conjecture
Ralph Fox · Princeton, United States
Leopoldt's conjecture
Heinrich-Wolfgang Leopoldt · Erlangen, Germany
Bateman–Horn conjecture
Paul Bateman and Roger Horn · Urbana, Illinois, United States
Gaussian moat problem
Basil Gordon · Los Angeles, United States
Erdős–Sós conjecture
Paul Erdős and Vera Sós · Budapest, Hungary
Feit–Thompson conjecture
Walter Feit and John Thompson · Vienna, Austria
Ringel's conjecture
Gerhard Ringel · Berlin, Germany
Tate conjecture
John Tate · Cambridge, Massachusetts, United States
Zeeman conjecture (collapsibility)
Christopher Zeeman · Coventry, United Kingdom
Edge reconstruction conjecture
Frank Harary · Ann Arbor, Michigan, United States
Novikov conjecture
Sergei Novikov · Moscow, Russia
Birch and Swinnerton-Dyer conjecture
Bryan Birch and Peter Swinnerton-Dyer · Cambridge, England
Chowla conjecture
Sarvadaman Chowla · University Park, Pennsylvania, United States
Willmore conjecture
Thomas Willmore · Durham, United Kingdom
Andrews–Curtis conjecture
James Andrews and Morton Curtis · Houston, United States
Arnold conjecture (symplectic topology)
Vladimir Arnold · Moscow, Russia
Total coloring conjecture
Mehdi Behzad and Vadim G. Vizing · East Lansing, Michigan, United States
Wall's D2 problem
C. T. C. Wall · Cambridge, United Kingdom
Zariski–Lipman conjecture
Oscar Zariski and Joseph Lipman · Cambridge, Massachusetts, United States
Hedetniemi's conjecture
Stephen Hedetniemi · Ann Arbor, United States
Moving sofa problem
Leo Moser · Edmonton, Canada
Birch–Tate conjecture
Bryan Birch and John Tate · Oxford, England
Moser's worm problem
Leo Moser · Edmonton, Canada
Lonely runner conjecture
Jörg Wills · Berlin, Germany
Langlands program
Robert Langlands · Princeton, United States
Ryser's conjecture
Herbert Ryser · Pasadena, United States
Graceful tree conjecture (Kotzig–Ringel–Rosa)
Gerhard Ringel and Anton Kotzig · Bratislava, Czechoslovakia
Stein's restriction conjecture
Elias M. Stein · Princeton, United States
Ryser–Brualdi–Stein conjecture
Herbert Ryser, Richard Brualdi and Sherman Stein · Pasadena, United States
Oberwolfach problem
Gerhard Ringel · Oberwolfach, Germany
Elliott–Halberstam conjecture
Peter D. T. A. Elliott and Heini Halberstam · Nottingham, England
Standard conjectures on algebraic cycles
Alexander Grothendieck · Bures-sur-Yvette, France
Dixmier conjecture
Jacques Dixmier · Paris, France
Virtually Haken conjecture
Friedhelm Waldhausen · Bielefeld, Germany
Surface subgroup conjecture
Friedhelm Waldhausen · Bielefeld, Germany
Vizing's conjecture (domination)
Vadim G. Vizing · Novosibirsk, Russia
Grimm's conjecture
Carl Albert Grimm · Rapid City, South Dakota, United States
Main conjecture of Iwasawa theory
Kenkichi Iwasawa · Princeton, United States
Lovász conjecture
László Lovász · Budapest, Hungary
Barnette's conjecture
David Barnette · Davis, United States
Cosmic censorship hypothesis
Roger Penrose · London, United Kingdom
Milnor conjecture
John Milnor · Princeton, United States
Optimal sphere packing in dimensions 8 and 24
Classical problem (optimality of E8 and Leech) · Berlin, Germany
Singular cardinals hypothesis (SCH)
Set theory · Berkeley, United States
Rota's conjecture on matroids
Gian-Carlo Rota · Cambridge, Massachusetts, United States
Ehrenpreis conjecture
Leon Ehrenpreis · New York, United States
Is the Conway knot slice?
John Horton Conway · Cambridge, United Kingdom
Erdős sumset conjecture
Paul Erdős · Budapest, Hungary
P versus NP
Stephen Cook · Toronto, Canada
Hall's conjecture
Marshall Hall Jr. · Pasadena, California, United States
Singmaster's conjecture
David Singmaster · London, United Kingdom
Berge–Fulkerson conjecture
Delbert Ray Fulkerson · Ithaca, United States
McKay conjecture
John McKay · Montreal, Canada
Stark conjectures
Harold Stark · Cambridge, Massachusetts, United States
Erdős–Faber–Lovász conjecture
Paul Erdős · Budapest, Hungary
Property P for knots
Classical problem (popularized by R. H. Bing) · Madison, United States
Gaussian correlation inequality
Olive Jean Dunn · Los Angeles, United States
Chvátal's conjecture on intersecting families
Václav Chvátal · Columbus, Ohio, United States
Tutte's 3-flow conjecture
William Tutte · Waterloo, Canada
Classification of finite simple groups
Daniel Gorenstein, Richard Brauer, Michael Aschbacher and others · New Brunswick, New Jersey, United States
Cycle double cover conjecture
Paul Seymour and George Szekeres · Sydney, Australia
Montgomery's pair correlation conjecture
Hugh Montgomery · Ann Arbor, Michigan, United States
Chvátal's toughness conjecture
Václav Chvátal · Montreal, Canada
Burr–Erdős conjecture
Stefan Burr and Paul Erdős · Budapest, Hungary
Bombieri–Lang conjecture
Enrico Bombieri and Serge Lang · Pisa, Italy
Fuglede's conjecture (spectral sets)
Bent Fuglede · Copenhagen, Denmark
Tameness conjecture
Albert Marden · Minneapolis, United States
Herzog–Schönheim conjecture
Marcel Herzog and Jochanan Schönheim · Tel Aviv, Israel
List edge-coloring conjecture
Vadim G. Vizing and others · Odesa, Ukraine
Auslander–Reiten conjecture
Maurice Auslander and Idun Reiten · Waltham, Massachusetts, United States
Dürer's conjecture (unfolding polyhedra)
Albrecht Dürer and Geoffrey C. Shephard · Norwich, England
Atiyah conjecture (L² Betti numbers)
Michael Atiyah · Oxford, United Kingdom
Connes embedding problem
Alain Connes · Paris, France
Bass–Quillen conjecture
Hyman Bass and Daniel Quillen · Cambridge, Massachusetts, United States
Greenberg's conjecture
Ralph Greenberg · Philadelphia, Pennsylvania, United States
Kelmans–Seymour conjecture
Paul Seymour (and, independently, Alexander Kelmans) · Oxford, United Kingdom
Erdős–Hajnal conjecture
Paul Erdős and András Hajnal · Budapest, Hungary
Inscribed rectangle problem (rectangular peg)
Herbert Vaughan · Urbana, United States
Berry–Tabor conjecture
Michael Berry and Michael Tabor · Bristol, England
Caccetta–Häggkvist conjecture
Louis Caccetta and Roland Häggkvist · Perth, Australia
Weinstein conjecture
Alan Weinstein · Berkeley, United States
Dinitz problem
Jeff Dinitz · Columbus, United States
Kobon triangle problem
Kobon Fujimura · Tokyo, Japan
Shelah's PCF conjecture
Saharon Shelah · Jerusalem, Israel
Union-closed sets conjecture (Frankl)
Péter Frankl · Paris, France
Goldfeld's conjecture (rank of elliptic curves)
Dorian Goldfeld · Cambridge, Massachusetts, United States
Deligne's conjecture on special values of L-functions
Pierre Deligne · Bures-sur-Yvette, France
Kazhdan–Lusztig conjectures
David Kazhdan and George Lusztig · Cambridge, Massachusetts, United States
Cherlin–Zilber conjecture
Gregory Cherlin and Boris Zilber · New Brunswick, New Jersey, United States
Monstrous moonshine
John Conway and Simon Norton · Cambridge, England
Fundamental lemma (Langlands program)
Robert Langlands and Diana Shelstad · Princeton, United States
Fortune's conjecture (fortunate numbers)
Reo Fortune · Cambridge, England
Boolean Pythagorean triples problem
Ronald Graham · Murray Hill, New Jersey, United States
Szpiro's conjecture
Lucien Szpiro · Paris, France
Gyárfás–Sumner conjecture
András Gyárfás and David Sumner · Budapest, Hungary
Smale's mean value conjecture
Stephen Smale · Berkeley, United States
Alspach's conjecture (cycle decompositions)
Brian Alspach · Burnaby, Canada
Tuza's conjecture
Zsolt Tuza · Budapest, Hungary
Geometrization conjecture
William Thurston · Princeton, United States
Firoozbakht's conjecture
Farideh Firoozbakht · Isfahan, Iran
Baum–Connes conjecture
Paul Baum and Alain Connes · Bures-sur-Yvette, France
Yau's conjecture on minimal surfaces
Shing-Tung Yau · Princeton, United States
Ending lamination theorem
William Thurston · Princeton, New Jersey, United States
Local connectivity of the Mandelbrot set (MLC)
Adrien Douady and John Hubbard · Orsay, France
Section conjecture (anabelian geometry)
Alexander Grothendieck · Montpellier, France
Zimmer's conjecture
Robert Zimmer · Chicago, United States
Telescope conjecture
Douglas Ravenel · Seattle, Washington, United States
abc conjecture
David Masser and Joseph Oesterlé · Paris, France
Falconer's conjecture
Kenneth Falconer · Bristol, United Kingdom
Fujita conjecture
Takao Fujita · Tokyo, Japan
Beilinson conjectures (special values of L-functions)
Alexander Beilinson · Moscow, Russia
Bunkbed conjecture
Pieter Kasteleyn · Leiden, Netherlands
1-factorization conjecture
Amanda Chetwynd and Anthony Hilton · Reading, United Kingdom
Andrica's conjecture
Dorin Andrica · Cluj-Napoca, Romania
Alperin's weight conjecture
Jonathan Alperin · Chicago, United States
Sidorenko's conjecture
Alexander Sidorenko · Moscow, Russia
Nearby Lagrangian conjecture
Vladimir Arnold · Moscow, Russia
Serre's modularity conjecture
Jean-Pierre Serre · Paris, France
Vojta's conjecture
Paul Vojta · New Haven, United States
Log-rank conjecture
László Lovász and Michael Saks · Budapest, Hungary
New Mersenne conjecture (Bateman–Selfridge–Wagstaff)
Paul T. Bateman, John Selfridge and Samuel Wagstaff · Urbana, Illinois, United States
Rota's basis conjecture
Gian-Carlo Rota · Cambridge, Massachusetts, United States
Bloch–Kato conjecture (Tamagawa numbers)
Spencer Bloch and Kazuya Kato · Chicago, United States
Second neighborhood conjecture
Paul Seymour · Princeton, New Jersey, United States
Berge conjecture (knots and Dehn surgery)
John Berge · United States
Superpermutations (minimal length)
Classical combinatorics problem · United States
Double bubble conjecture
Classical problem (3D statement by J. Foisy) · Williamstown, United States
Sensitivity conjecture
Noam Nisan and Mario Szegedy · Jerusalem, Israel
Beal conjecture
Andrew Beal · Dallas, United States
Farrell–Jones conjecture
F. Thomas Farrell and Lowell E. Jones · Binghamton, United States
Cannon's conjecture
James W. Cannon · Provo, United States
Fermat–Catalan conjecture
Henri Darmon and Andrew Granville · Athens, Georgia, United States
André–Oort conjecture
Yves André and Frans Oort · Paris, France
Erdős–Gyárfás conjecture
Paul Erdős and András Gyárfás · Budapest, Hungary
Fontaine–Mazur conjecture
Jean-Marc Fontaine and Barry Mazur · Paris, France
Cobordism hypothesis
John C. Baez and James Dolan · Riverside, California, United States
HRT conjecture (linear independence of time-frequency translates)
Christopher Heil, Jayakumar Ramanathan and Pankaj Topiwala · Atlanta, United States
Angel problem
John Horton Conway · Liverpool, England
Volume conjecture (knots)
Rinat Kashaev and Hitoshi Murakami · Tokyo, Japan
Zauner's conjecture (SIC-POVMs)
Gerhard Zauner · Vienna, Austria
Navier–Stokes existence and smoothness
Clay Mathematics Institute · Cambridge, Massachusetts, United States
Yang–Mills existence and mass gap
Clay Mathematics Institute · Cambridge, Massachusetts, United States
21st century
Casas-Alvero conjecture
Eduardo Casas-Alvero · Barcelona, Spain
Exponential time hypothesis (ETH)
Russell Impagliazzo and Ramamohan Paturi · San Diego, United States
Strong exponential time hypothesis (SETH)
Russell Impagliazzo, Ramamohan Paturi and Chris Calabro · San Diego, California, United States
Unique games conjecture
Subhash Khot · Princeton, United States
Crouzeix's conjecture
Michel Crouzeix · Rennes, France
1-2-3 conjecture
Maciej Karoński, Tomasz Łuczak and Andrew Thomason · Poznań, Poland
Zilber–Pink conjecture
Boris Zilber and Richard Pink · Oxford, United Kingdom
Kahn–Kalai conjecture
Jeff Kahn and Gil Kalai · Jerusalem, Israel
Sarnak's conjecture
Peter Sarnak · Princeton, United States
Quadrisecant approximation conjecture
Gyo Taek Jin and Hyoung Seop Kim · Daejeon, South Korea
Bounded gaps between primes
Yitang Zhang · Durham, New Hampshire, United States
Each marker shows its status: ◯ for those still open and ★ for those resolved. The color corresponds to the era in which it was formulated.
What you see at first glance
- The 20th century dominates. 247 of the 303 conjectures were formulated in that century. This is partly a genuine reflection (the professionalization of mathematics and the explosion of new fields) and partly a bias in the sources, which document recent work better.
- Most remain open. 192 conjectures have been neither proved nor disproved; 87 are resolved, 18 were disproved by a counterexample, and 6 turned out to be undecidable: it was proved that they can be neither proved nor disproved within the usual axioms.
- A few cities account for a lot. Paris, Budapest, Princeton, Berlin, Göttingen and Cambridge hold a large share of the map. Budapest stands out thanks to the Hungarian school of combinatorics and, in particular, the enormous number of problems posed by Paul Erdős.
How it was built
The data was gathered in waves. First, from books on the history of mathematics (Burton, Kline, Boyer); then by consulting biographies in the MacTutor archive and the lists of unsolved problems on Wikipedia and Open Problem Garden, cross-referencing them to find the missing ones. For each conjecture I recorded who formulated it, where, in what year, its current status and, where applicable, when and how it was resolved.
Some judgment calls had to be made. For example, an announced proof that the community has not yet fully verified is kept as open, with a note, rather than as resolved. And when only the city is known, the marker goes at its coordinates: many conjectures share a location, so the map spreads them out slightly so that zooming in can separate them.
What the map doesn't tell you
A map like this is only as good as its sources. Mathematics done outside Europe and North America, or published in other languages, is underrepresented: not because it didn't exist, but because the sources consulted record it less often. The place assigned to a conjecture is also a simplification: many emerged from correspondence between people in different countries, or were formulated years before being published.
That's why this is a living project. The Atlas page also has the Mathematicians layer, which shows where more than twenty thousand mathematicians were trained according to Wikidata. If you have data or corrections, write to me.