Where the map runs out

Open as of 2026. A Clay Millennium Prize Problem.

Open as of 2026. Known when the analytic rank is 0 or 1. A Clay Millennium Prize Problem.

Proved over fields of characteristic zero (Hironaka, 1964) and in dimension up to three in positive characteristic. Open beyond.

Open. Stable groups are computed through about dimension 90 as of 2026, with no general formula.

Hill, Hopkins and Ravenel ruled out every dimension above 126 (announced 2009, published 2016). In 2024 Lin, Wang and Xu showed that dimension 126 does have such manifolds (preprint). So they exist in exactly the dimensions 2, 6, 14, 30, 62 and 126.

Proved by Fernando Codá Marques and André Neves in 2012, published 2014.

Open in general when the curvature changes sign in a degenerate way.

Proof announced 1998 and published 2005. A machine-checked formal proof was completed in 2014 and published in 2017.

Open for general continuous closed curves as of 2026.

Open in general. Proved for some classes of minimal hypersurfaces with extra symmetry.

Proved in the time-symmetric (Riemannian) case in 1997–2001. The general case is open.

Proved by Perelman in 2002–2003, verified by 2006, Clay prize awarded (and declined) in 2010.

Open as of 2026. Dimension four is the one case the higher-dimensional methods cannot reach.

Proved for the figure-eight knot and a few other families. Open in general.

Open as of 2026. Order 12 is the smallest unsettled case.

Open as of 2026.

Proved in many special cases; the general conjectures remain open as of 2026.

Conjectured by Gauss; open as of 2026.

Open since 1859; a Clay Millennium Prize Problem and one of Hilbert's 1900 problems.

Open as of 2026. Bounded gaps proved in 2013; the best unconditional gap is 246.

A claimed proof (Mochizuki, 2012; published 2021) is not accepted by most number theorists as of 2026.

Open as of 2026. Verified by computer up to 4 × 10¹⁸; the "weak" version for odd numbers was proved by Harald Helfgott (2013 preprint).

Open as of 2026; any odd perfect number would have to exceed 10¹⁵⁰⁰.

Unproven as of 2026; would imply P ≠ NP.

Open as of 2026; the new standards assume so.

Proved by Louis de Branges in 1984 (published 1985), after 68 years.

Proved in three dimensions (Hong Wang and Joshua Zahl, 2025 preprint); open in four or more dimensions as of 2026.

Proved in two dimensions and in high dimensions; open in three dimensions as of 2026.

Open as of 2026; known to be astronomically large.

Open as of 2026; a Clay Millennium Prize Problem. Most researchers believe P ≠ NP.

Decidable if Schanuel's conjecture holds (Macintyre–Wilkie, 1996); open unconditionally as of 2026.

Independent of ZFC. Whether new axioms should, or can, settle it is disputed as of 2026.

Open as of 2026; many published proofs have been wrong.

Open as of 2026; known for all solvable groups and many simple ones, including the Monster.

Open as of 2026.

Proved by Marc Cabanes and Britta Späth, announced in 2023 and published in the Annals of Mathematics in 2026.

The continuous version was solved by Kolmogorov and Arnold (1957); the algebraic version is open as of 2026.

Open as of 2026; one of Stephen Smale's eighteen problems for the twenty-first century.

Open as of 2026. In 2026 a team including Levent Alpöge, working with an AI model, announced matrices for 668 and the other missing orders below 2000, so the smallest unknown order is now above 2000.

Open as of 2026; known to lie between 43 and 46 (upper bound 2024).

Open as of 2026; proved for hexagons in 2006 and nearly proved asymptotically in 2017.

Open as of 2026. Proved for up to six colours, the six-colour case in 1993.

Open as of 2026; explicit constructions have improved greatly since 2016 but remain far from random ones.

Open as of 2026, for every value of the parameter.

Open as of 2026; proved at many parameter values, beginning with Yoccoz's work around 1990 and most recently at the Feigenbaum points (Dudko and Lyubich).

Open as of 2026, even for equations of degree two.

Open in general as of 2026; proved in some special settings.

Open as of 2026; proved in 1990 under an extra assumption of positive entropy.

Open in general as of 2026; sharp answers exist only for special classes of chains.

Open as of 2026; the Gaussian case is known to within one bit (2008).

Open as of 2026; proposed remedies are being tried, and none is agreed as sufficient.

Open as of 2026; partial explanations exist for simplified models.

Open as of 2026 in two, three and four dimensions; solved above four dimensions (1992).

Open as of 2026; understood for some simplified models, not for networks used in practice.

Open as of 2026. Proved when the number of colours exceeds about 1.81 times the maximum degree.

Open as of 2026; proved with the constant 4 in place of 1, and improved only slightly since.

Open as of 2026. A preprint of August 2026 lowered the best upper bound to 2.371177, from 2.371339.

Open as of 2026. Largely settled in one space dimension for data with small total variation, open in two and three.

Open as of 2026; a Clay Millennium Prize Problem.

No confirmed detection as of 2026.

No accepted theory as of 2026; candidates include string theory and loop quantum gravity.

Recent calculations suggest information does escape, but how is still debated as of 2026.

Proposed by Penrose in 1969; unproven in general.

Not identified as of 2026, despite decades of direct searches.

Open as of 2026. Some recent galaxy surveys hint that it may change over time, which a pure cosmological constant cannot.

Unresolved as of 2026. The microwave-background and Cepheid–supernova values have stayed apart as measurements improved, though some other local methods land in between.

No violation detected as of 2026, with bounds far tighter than any direct test at the Planck scale.

Resolved for an idealised dilute gas of hard spheres by Yu Deng, Zaher Hani and Xiao Ma (2024–25), work cited in Deng's 2026 Fields Medal. Some argue that dense gases and liquids, and realistic forces between molecules, still lie outside it.

Open as of 2026; exact results exist only in special cases.

Open as of 2026; the critical exponents are known numerically to many digits, but there is no exact solution.

Open as of 2026; there is no accepted theory of the glass transition.

Open as of 2026; there is no consensus explanation.

Open as of 2026. Its value is measured to better than one part in a billion but not explained.

Open as of 2026.

Open as of 2026. One of the Clay Mathematics Institute's Millennium Prize Problems.

Open as of 2026. Small error-corrected logical qubits have been demonstrated.

Open as of 2026. Several interpretations are consistent with all experiments.

Open as of 2026. Revised abundance analyses have narrowed but not closed the gap.

Open as of 2026. X-ray and gravitational-wave measurements are narrowing the range.

Open as of 2026. No biosignature has been confirmed.

Open as of 2026. JWST has found even more early massive black holes than expected.

Open as of 2026. Neutron-star mergers are known to contribute, but whether they are the main source is debated.

Open as of 2026; computer predictions now often succeed for small, rigid molecules, but not reliably.

Open as of 2026; several materials are strong candidates, but none is universally accepted.

Open as of 2026; there is no general theory of strongly correlated electrons.

Open as of 2026, forty years after the discovery.

No confirmed example as of 2026. The highest reproduced results need pressures above a million atmospheres, and a 2025 preprint reporting about 298 K in a lanthanum–scandium hydride at about 2.5 million atmospheres has not yet been independently reproduced.

Open as of 2026. Simpler, Abelian anyons were observed in 2020, and non-Abelian anyons have been simulated on quantum processors since 2023, but interference experiments in real materials have not yet shown the non-Abelian kind beyond doubt.

Open as of 2026. No fusion device has yet produced more electricity than it consumes.

Open as of 2026. Large computer calculations now follow a splitting nucleus, but still rely on fitted forces.

Open as of 2026. Element 118, oganesson, is the heaviest made, and no nucleus in the predicted island of stability has been reached.

Open as of 2026. The averages of the two methods still disagree by about ten seconds, some five times their combined uncertainty.

Open as of 2026. Estimates range from about 4.35 to 4.51 billion years. A 2024 proposal, that tidal heating remelted the Moon's surface 4.35 billion years ago and reset many clocks, favours an old Moon.

Open as of 2026, though laboratory chemistry has made several steps plausible.

Several hypotheses are supported in particular cases; no general answer as of 2026.

Well documented in plants and worms; contested in mammals, including humans, as of 2026.

Narrowed by very large studies but not closed, as of 2026.

Actively disputed as of 2026; estimates of the functional fraction range from under 10% to much higher.

Structure prediction largely solved by AlphaFold 2 (2020–21). How proteins physically fold, and how they misfold, remain open.

Open as of 2026.

Open as of 2026; candidate explanations exist, but none accounts for the full pattern.

Open as of 2026; one fully computational design now rivals natural enzymes on a model reaction, but most are still far slower.

Open as of 2026; several mechanisms have been found, none fully general.

Open as of 2026; the smallest synthetic cell still contains genes of unknown function.

Open as of 2026; every large efficacy trial so far has failed or shown only modest protection.

Open as of 2026; most microbial species are known only from their DNA.

Open as of 2026; several biologically plausible learning rules have been proposed, none established.

Open as of 2026; the rules of homeostatic regulation are only partly known.

Open as of 2026; drugs that clear amyloid slow decline only modestly.

Open as of 2026; memory traces can be tagged in mice, but how they last for decades is unknown.

Open as of 2026; the first large adversarial test of two leading theories, published in 2025, challenged both without settling the question.

Open as of 2026; more than two dozen explanations have been proposed, and none is accepted as sufficient.

Open as of 2026; many mechanisms are known to help, and their relative importance is unresolved.

Open as of 2026; about two million species have been named, and estimates of the total vary more than tenfold.

Open as of 2026; early-warning signals have worked in experiments but are not yet reliable forecasts.

Open as of 2026; a century after Elton described them, no single explanation is accepted.

Open as of 2026; several signalling pathways are known to be involved, but not how size itself is measured.

Open as of 2026; the pattern is supported by gene-expression data, but its cause is debated.

Open as of 2026; case studies exist, a general account does not.

Open as of 2026; the cells involved are increasingly well known, the reasons for the differences are not.

Open as of 2026; the first iPS-derived cell therapies won conditional approval in Japan in 2026, but whole organs are not in reach.