Ranking axis: structural centrality × how much a proof would transform mathematics — not raw fame, not difficulty alone. Fame is a tiebreaker. This is why some household names land surprisingly low.
Inspired by the July 2026 counterexample to the Jacobian conjecture (Levent Alpöge + Claude, a 216-character polynomial map ℂ³→ℂ³ with constant Jacobian determinant that is not injective — n ≥ 3 refuted, n = 2 still open; verified across the community within a day of posting). One more reminder that "open" is a temporary condition — see the graveyard at the bottom for what "solvable" has looked like lately.
How to read an entry. Links are marked by favicon, where they exist:
Wikipedia — or
arXiv where no decent article exists
a formal statement in Lean 4, almost always in Google DeepMind's formal-conjectures repo (statements, not proofs!)
Manifold prediction markets, usually the "is it true?" market — probabilities are snapshots from July 2026 and will drift
Metaculus (search links — their API is login-walled) and other public forecasting or notable expert discussion
Corrections and additions welcome — PRs open. Especially wanted: markets and formalizations that exist but aren't linked here.
- Riemann Hypothesis (+ GRH for L-functions) — thousands of theorems are conditional on it; secretly a statement about the "spectrum" of the primes. The one problem on both Hilbert's 1900 list and the Millennium list.
- P vs NP — the only problem here whose answer changes what civilization can do, not just what mathematicians know. Also the deepest: we can't even prove weak circuit lower bounds.
- Langlands functoriality / general reciprocity — the grand unification of number theory, representation theory, and harmonic analysis. Technically a program, not a problem, but FLT was a corollary of one corner of it. That's the tier it lives in.
- Birch–Swinnerton-Dyer — the bridge between analysis and arithmetic of elliptic curves; rank part and finiteness of Ш both open in general.
- Hodge Conjecture — which topology is algebraic; the health check for all of algebraic geometry.
- Tate Conjecture + Grothendieck's Standard Conjectures — the motives package; Hodge's arithmetic sibling, arguably more consequential than Hodge itself.
- Navier–Stokes global regularity — do fluids blow up? (Related: interior finite-time singularity for 3D Euler — Hou–Chen's computer-assisted boundary blowup was a landmark.)
- Yang–Mills existence & mass gap — construct a 4D QFT rigorously at all; the gap between physics and math in one problem.
- abc Conjecture — would trivialize half of Diophantine number theory. Status: contested — Mochizuki's IUT claim is not accepted by the broader community (the Scholze–Stix objection stands).
- Hardy–Littlewood prime k-tuples (twin primes as flagship) — post-Zhang/Maynard we have bounded gaps; the full conjecture is the real prize.
- Smooth 4-dimensional Poincaré Conjecture — the last Poincaré. Dimension 4 is the lawless frontier of topology.
- Schanuel's Conjecture — one statement that subsumes essentially all of transcendence theory (e+π irrational? Corollary.). The most underrated entry on this list.
- Bombieri–Lang — geometry governs rational points; the ultimate generalization of Faltings.
- Existence of one-way functions — the foundation of all cryptography; Impagliazzo's five worlds made precise.
- Fontaine–Mazur + Bloch–Kato/Beilinson conjectures — which Galois representations come from geometry, and what L-values mean. The arithmetic engine room.
- Borel & Novikov Conjectures — rigidity of aspherical manifolds; where topology, geometry, and operator algebras (Baum–Connes) meet.
- Hilbert's 16th problem (second part) — a uniform bound H(n) on limit cycles of degree-n polynomial vector fields; not even known finite for n = 2. On Hilbert's list and Smale's list; the central open problem of dynamical systems. (new)
- Cosmic censorship (weak & strong) — are naked singularities generic? Is determinism salvageable in GR? The organizing conjectures of mathematical relativity; strong censorship (C⁰ version) already dented by Dafermos–Luk. Kerr stability only recently settled for slow rotation. (new)
- Goldbach (binary) — famous and genuinely important, but a proof likely refines circle-method tech rather than creating a new world. Ternary case: done (Helfgott).
- Lindelöf Hypothesis — RH's understudy; even this is out of reach.
- Elliott–Halberstam — primes in progressions beyond GRH; would give prime gaps ≤ 6 (Polymath8b).
- Chowla & Sarnak conjectures — Möbius randomness; Tao's logarithmic results are the beachhead.
- Are elliptic curve ranks unbounded? — heuristics now say bounded (!), reversing decades of folklore.
- Hilbert's 12th (explicit class field theory) — real recent progress (Dasgupta–Kakde), still open.
- Hilbert's 10th over ℚ — is there an algorithm for rational points? (Over ℤ: no — MRDP. Over ℚ: open; over some big rings recently resolved.)
- Inverse Galois Problem — is every finite group a Galois group over ℚ?
- Artin's primitive root conjecture — follows from GRH; unconditional = open.
- Lehmer's Mahler measure problem — a spectral gap for algebraic numbers.
- Littlewood Conjecture (simultaneous approximation) — exceptions have measure zero (Einsiedler–Katok–Lindenstrauss); full statement open.
- Zilber–Pink — the unlikely-intersections master conjecture (André–Oort was its solved special case).
- Serre's uniformity question — Galois images of elliptic curves over ℚ: is 37 the last exceptional prime?
- Vandiver / Leopoldt — old, stubborn, structurally meaningful.
- Grothendieck's section conjecture — anabelian geometry's central open question: rational points = sections of the fundamental exact sequence. (new)
- Selberg's 1/4 eigenvalue conjecture / Ramanujan–Petersson for Maass forms — the archimedean edge of Langlands; everything in analytic number theory wants it. (new)
- Slice-ribbon conjecture — knot concordance's central mystery (the Conway knot episode was a warning shot).
- L-space conjecture — the conjectural trinity (Floer homology / taut foliations / left-orderability) of 3-manifolds.
- 4D Schoenflies — even more embarrassing than smooth Poincaré, arguably.
- Volume Conjecture — quantum invariants see hyperbolic geometry.
- Cannon Conjecture — group theory determines when a boundary is a sphere.
- Hopf conjectures — positive curvature on S²×S²; sign of the Euler characteristic.
- Andrews–Curtis — widely suspected false; a disproof would be spectacular.
- Whitehead asphericity — 80+ years, elementary statement, nothing.
- SYZ conjecture & homological mirror symmetry — why mirror symmetry works; proved in families of examples, open as a general principle. (new)
- Restriction Conjecture — the summit of harmonic analysis; Kakeya was its foothill (3D Kakeya fell in 2025 — Wang–Zahl; higher dimensions and restriction itself open).
- Falconer distance conjecture — fractal geometry's flagship.
- Invariant subspace problem (Hilbert space) — Enflo's 2023 claim remains unaccepted; cursed energy.
- Crouzeix's conjecture — the constant is 2, everyone believes it, nobody can do it.
- Fuglede-type spectral questions in low dimensions — false in general (Tao), alive in ℝ², ℝ³ (the conjecture was fully proved for convex domains).
- Furstenberg's ×2, ×3 measure conjecture — rigidity of arithmetic dynamics; Rudolph's theorem is the partial result everyone wants to remove the entropy hypothesis from.
- Soliton resolution conjecture — every reasonable dispersive evolution decomposes into solitons + radiation; proved for energy-critical waves in special cases (Duyckaerts–Kenig–Merle), wide open in general. (new)
- Quantum unique ergodicity — arithmetic case is a Fields-medal theorem (Lindenstrauss); general negatively-curved manifolds open. (new)
- Kaplansky zero-divisor & idempotent conjectures — especially spicy since the unit conjecture was refuted (Gardam, 2021).
- Köthe Conjecture — ring theory's oldest open wound (1930).
- Alperin weight & Broué's abelian defect conjectures — modular representation theory's core (McKay itself fell in '24, Brauer's height zero in '24 too — the local-global program is on a run).
- Brauer's remaining problems on blocks and characters — the 1963 list that keeps giving.
- Is Thompson's group F amenable? — the most contested question in geometric group theory; multiple contradictory claimed proofs in both directions have died. (new)
- Hadwiger's Conjecture (graph minors vs. coloring) — the deepest thing in graph theory.
- Erdős's conjecture on APs in sets of divergent reciprocal sum — Bloom–Sisask did density; full conjecture (primes ⇒ Green–Tao as corollary) open.
- Diagonal Ramsey asymptotics — the exponential barrier finally cracked (Campos–Griffiths–Morris–Sahasrabudhe 2023); true growth rate still open.
- Sunflower Conjecture — big progress (Alweiss–Lovett–Wu–Zhang 2019), gap remains.
- Reconstruction Conjecture — can you rebuild a graph from its vertex-deleted deck?
- Cycle double cover / Berge–Fulkerson / Tutte's flow conjectures — the snark-infested waters.
- Hadwiger–Nelson (chromatic number of the plane) — now 5 ≤ χ ≤ 7 thanks to a hobbyist (de Grey, 2018).
- Inscribed square (Toeplitz) — every Jordan curve? Smooth case done; continuous case open since 1911.
- Sphere packing / kissing numbers in general dimension — 8 and 24 fell to Viazovska; everything else is wilderness (and dim 10+ lower bounds just moved for the first time in decades).
- Caccetta–Häggkvist — digraph girth; simple statement, no traction.
- Erdős–Hajnal conjecture — forbidding any one induced subgraph forces polynomial-size cliques or independent sets; structural graph theory's north star. (new)
- Sidorenko's conjecture — bipartite graphs are "quasirandomness-minimal"; deceptively innocent, tied to Gowers norms and graph limits. (new)
- Hadamard matrix conjecture — a Hadamard matrix in every order 4k; smallest open order 668. (new)
- Erdős–Turán conjecture on additive bases — must representation counts of an additive basis be unbounded? (new)
- The Continuum Problem, post-Cohen — independence didn't kill it: Woodin's Ultimate-L vs. the forcing-axiom camp (which says 2^ℵ⁰ = ℵ₂!) make it a live research program about which axioms are true.
- Vaught's Conjecture — model theory's white whale.
- Unique Games Conjecture — half-proved (2-to-2 games); would settle optimal inapproximability across the board.
- Matrix multiplication exponent ω = 2? — inching down for 50 years; current record ω < 2.3714.
- P = BPP (derandomization) — everyone believes it; a proof requires circuit lower bounds we can't touch.
- Graph isomorphism in P? — Babai got quasipolynomial; the last step is stuck.
- Log-rank conjecture — communication complexity's oldest embarrassment.
- VP vs VNP (Valiant's hypothesis) — permanent vs. determinant; the algebraic P vs NP, plausibly easier, still untouched. (new)
- Quantum PCP conjecture — hardness of approximating ground-state energy; the NLTS breakthrough (2022) was the first real step. (new)
- 3D Ising / percolation critical exponents & conformal invariance — 2D is a triumph (SLE); 3D is a desert with physics predictions (bootstrap!) and no proofs.
- KPZ universality in full generality — proved for exactly solvable models only.
- Ergodic/quantitative theory of turbulence — beyond even Navier–Stokes regularity. (Meanwhile Hilbert's 6th — deriving fluid equations from Newtonian particles — took a giant leap with Deng–Hani–Ma 2025.)
- Kannan–Lovász–Simonovits (KLS) conjecture — one spectral constant governing isoperimetry of all log-concave measures; Yuansi Chen's 2020 near-resolution electrified the field, the constant-factor question remains. (new)
- Mahler volume conjecture — cubes minimize the volume product; proved in ℝ³ (2020), open in general. Symplectic connection: it followed from Viterbo's conjecture — which was just refuted, so the route died but the conjecture didn't. (new)
- Odd perfect numbers — open for ~2300 years, the oldest problem in mathematics. A proof would be a fireworks show, not an earthquake.
- Infinitely many Mersenne primes (and: finitely many Fermat primes?) — no tools exist. None.
- Normality of π, e, √2 — we cannot prove a single natural constant is normal. Humbling, isolated.
- Irrationality of γ (Euler–Mascheroni) and ζ(5) — ζ(3) took Apéry magic; the magic didn't generalize.
- Lehmer's totient problem — does φ(n) | n−1 force primality?
- Ramsey number R(5,5) — known to be in [43, 46]; finite computation, cosmically infeasible. ("Aliens demand R(6,6): attack." — Erdős)
- Union-closed sets (Frankl) — Gilmer's 2022 breakthrough got a constant (now ≈ 0.38); the ½ remains.
- Second neighborhood conjecture, graceful trees, lonely runner — beloved, bounded blast radius.
- Zaremba's conjecture — continued fractions with bounded partial quotients; Bourgain–Kontorovich got density one.
- Erdős–Straus (4/n = 1/x + 1/y + 1/z) — the classic "looks like homework, isn't."
- Legendre's conjecture & Landau's n²+1 problem — the two Landau problems with zero movement since 1912 (his other two are Goldbach and twin primes, above). (new)
- Beal conjecture — Fermat with mixed exponents and a $1M bounty from a Texas banker. (new)
- Sendov's conjecture — zeros and critical points of polynomials in a disk; Tao settled it for sufficiently high degree, the rest is open. (new)
- Singmaster's conjecture — does any number appear more than 8 times in Pascal's triangle? (new)
- Collatz — "Mathematics is not ready for such problems" (Erdős). Tao's almost-everywhere result is the best in 80 years and it's still nowhere near. Maximal fame, unclear payoff, no attack surface. Its own tier.
- 3×3 magic square of squares — recreational statement, absorbs infinite amateur-hours (a.k.a. the Parker Square problem).
- Perfect cuboid — same energy, 300 years running.
- Brocard's problem (n! + 1 = m²) — three known solutions, no theory.
For a sense of what "solvable" looks like lately:
- Jacobian conjecture — refuted for n ≥ 3 by a 216-character map ℂ³→ℂ³ (
- Bourgain's slicing problem / hyperplane conjecture — Klartag–Lehec '25, pending review
- Moving sofa problem — Baek '24, pending review (
- 3D Kakeya conjecture — Wang–Zahl '25 (
- Kervaire invariant one, dimension 126 — Lin–Wang–Xu '24; the last dimension, settled positively (
- McKay conjecture — Cabanes–Späth '24 (
- Viterbo's conjecture — refuted, Haim-Kislev–Ostrover '24 (
- Consistency of Quine's NF — Holmes' proof verified in Lean, '24 (
- Marton's conjecture / Polynomial Freiman–Ruzsa — Gowers–Green–Manners–Tao '23,
- Aperiodic monotile ("the hat") — Smith–Myers–Kaplan–Goodman-Strauss '23 (
- Telescope conjecture — refuted, Burklund–Hahn–Levy–Schlank '23; the last of Ravenel's conjectures (
- Ryser–Brualdi–Stein (large n) — Montgomery '23 (
- Erdős primitive set conjecture — Lichtman '22
- Erdős–Faber–Lovász — Kang–Kelly–Kühn–Methuku–Osthus '21 · André–Oort — Pila–Shankar–Tsimerman '21 · Kaplansky unit conjecture — refuted, Gardam '21
- Duffin–Schaeffer — Koukoulopoulos–Maynard '19 · Sensitivity conjecture — Huang '19, two pages! · Connes embedding — refuted via MIP*=RE '20
- Sphere packing, dims 8 & 24 — Viazovska '16–'17 · Kadison–Singer — Marcus–Spielman–Srivastava '13 · Bounded prime gaps — Zhang / Maynard '13
Refutation rate in that sample is worth staring at: Connes embedding, Kaplansky units, telescope, Viterbo, Jacobian. Betting "true" on everything is not the free money it looks like.
- Goldbach at B, because fame ≠ fertility.
- Collatz quarantined, because a tier list should measure what a solution buys you.
- Tate above Hodge in spirit if not placement.
- The most underrated entry on the whole list is Schanuel — one sentence that would retire an entire subfield.
- The Manifold "is it true?" markets are thin but surprisingly sane — note the market had the Jacobian conjecture at ~1% before the counterexample, and has smooth 4D Poincaré as a literal coin-flip, matching expert vibes.
- Forecasting coverage is patchy. Manifold has real markets on ~⅓ of this list (linked above). Metaculus has a cluster around the Millennium problems and AI-does-math (
- Lean coverage is better than you'd guess.
- Probabilities are snapshots (July 2026). If you're reading this later, click through.
- Generated graphic:
python3 make_tier_svg.py→tier-list.svg.
List curated in conversation with Claude. Content license: CC BY 4.0.