Geometry & Topology — Open Conjectures
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Includes conjectures from differential geometry, discrete geometry, geometric topology, Diophantine geometry, metric geometry, and related areas.
Geometric Topology
| # | Conjecture | Eponym(s) | Comments |
|---|---|---|---|
| 1 | Borel conjecture | Armand Borel | |
| 2 | Bost conjecture | Jean-Benoît Bost | |
| 3 | Farrell–Jones conjecture | F. Thomas Farrell, Lowell E. Jones | |
| 4 | Hilbert–Smith conjecture | David Hilbert, Paul Althaus Smith |
Differential Geometry
| # | Conjecture | Eponym(s) | Comments |
|---|---|---|---|
| 1 | Carathéodory conjecture | Constantin Carathéodory | |
| 2 | Filling area conjecture | — |
Diophantine Geometry
| # | Conjecture | Eponym(s) | Comments |
|---|---|---|---|
| 1 | Bombieri–Lang conjecture | Enrico Bombieri, Serge Lang | |
| 2 | Manin conjecture | Yuri Manin | |
| 3 | Mazur’s conjectures | Barry Mazur | |
| 4 | Uniformity conjecture | — |
Discrete Geometry
| # | Conjecture | Eponym(s) | Comments |
|---|---|---|---|
| 1 | Big-line-big-clique conjecture | — |
Metric Geometry
| # | Conjecture | Eponym(s) | Comments |
|---|---|---|---|
| 1 | Gilbert–Pollack conjecture on the Steiner ratio | Edgar Gilbert, Henry O. Pollak | Steiner ratio of the Euclidean plane |
Other Geometry
| # | Conjecture | Eponym(s) | Comments |
|---|---|---|---|
| 1 | Hopf conjectures | Heinz Hopf | |
| 2 | Toeplitz’ conjecture | Otto Toeplitz | Inscribed square problem for Jordan curves |
| 3 | Ulam’s packing conjecture | Stanislaw Ulam |