Antarctic marine ice-sheet and ice-cliff instability
Determine whether marine ice-sheet and ice-cliff instabilities can drive rapid Antarctic retreat this century, and how much they widen sea-level projections.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
26 shown
Approximation ratio and integrality gap for metric TSP
Find better polynomial-time approximation algorithms for the metric Traveling Salesman Problem and prove the conjectured 4/3 integrality gap of the subtour LP. The best known ratio is 3/2 − ε with ε > 10^−36 (Karlin–Klein–Oveis Gharan).
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Electrochemical ammonia synthesis under ambient conditions
Find a catalyst and cell design that reduce N2 to ammonia electrochemically at ambient conditions with verified, contamination-free rates — and separate real signals from false positives.
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Explicit rigid matrices (Valiant's rigidity problem)
Construct explicit n×n matrices that stay high-rank even after many entry changes, with parameters strong enough for Valiant's circuit lower bounds. Random matrices are highly rigid, but no explicit matrix is known to meet the required parameters.
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Frankl's union-closed sets conjecture
Every finite union-closed family of sets other than {∅} has an element lying in at least half of its sets. Since Gilmer's 2022 entropy breakthrough the best proven fraction is about 0.38; closing the gap to 1/2 is open.
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Legendre's conjecture
Prove that there is always a prime between n^2 and (n+1)^2. For consecutive cubes the analogue is known beyond an explicit (astronomically large) threshold.
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Mechanism of high-temperature superconductivity in the cuprates
Identify the pairing mechanism and the minimal theory that explains superconductivity, the pseudogap and the strange-metal normal state of the copper-oxide superconductors.
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Prebiotic routes to nucleotides and the RNA world
Establish a coherent, geochemically plausible route from simple feedstocks to activated ribonucleotides and self-replicating RNA under one consistent set of early-Earth conditions.
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Small-scale problems of cold dark matter
Decide whether the cusp–core, too-big-to-fail and rotation-curve diversity problems of ΛCDM on galaxy scales are explained by baryonic physics, by modified dark-matter properties (e.g. self-interactions), or by observational systematics.
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The 1/3–2/3 conjecture for balanced pairs in posets
Every finite poset that is not a chain has elements x, y such that x precedes y in between 1/3 and 2/3 of its linear extensions. The best general constant is (5−√5)/10 ≈ 0.276; all posets with up to 14 elements have been verified.
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The cosmological lithium problem
Explain why the lithium-7 abundance observed in old metal-poor halo stars is a factor of about 3–4 below the prediction of standard Big Bang nucleosynthesis with the CMB baryon density.
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The equity premium puzzle
Why have stock returns historically exceeded safe-asset returns by far more than standard consumption-based models with plausible risk aversion predict? Reproducible calibrations on public data count as B-style evidence.
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The Erdős unit distance problem in the plane
Determine the growth of u(n), the maximum number of unit distances among n points in the plane. Erdős's conjecture u(n) = n^{1+o(1)} was disproved in May 2026; the true exponent now lies between about 1.014 (Sawin) and 4/3 (Spencer–Szemerédi–Trotter).
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The Erdős–Rado sunflower conjecture
Show that every family of more than C_k^n sets of size n contains a k-sunflower, for a constant C_k depending only on k. The best bound, about (Ck log n)^n, follows the 2019 breakthrough of Alweiss, Lovett, Wu and Zhang.
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The exponent ω of matrix multiplication
Determine ω, the smallest exponent such that n×n matrices can be multiplied with n^(ω+o(1)) arithmetic operations. The best published bound is ω < 2.371339, a 2026 preprint claims ω < 2.371177, and it is conjectured that ω = 2.
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The graph reconstruction conjecture
Every finite simple graph on at least three vertices is determined up to isomorphism by its deck, the multiset of its vertex-deleted subgraphs. Verified by computer for all graphs up to 13 vertices; open in general.
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The invariant subspace problem for Hilbert spaces
Does every bounded linear operator on a separable infinite-dimensional complex Hilbert space have a non-trivial closed invariant subspace? The answer is negative for some Banach spaces and positive for many operator classes. The Hilbert space case is open.
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The Jacobian conjecture in two variables
Prove or disprove that a polynomial map C^2 → C^2 with non-zero constant Jacobian determinant has a polynomial inverse. The conjecture was disproved in dimension 3 (and higher) in July 2026; the plane case remains open.
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The Kakeya conjecture in dimensions n ≥ 4
Show that every Kakeya (Besicovitch) set in R^n has Hausdorff and Minkowski dimension n. The plane is classical and R^3 was settled by Wang and Zahl in 2025; all dimensions n ≥ 4 remain open.
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The log-rank conjecture in communication complexity
Is the deterministic communication complexity of every Boolean matrix M bounded by a polynomial in log rank(M)? The best upper bound is O(√rank) (Sudakov–Tomon), and the largest known separation is quadratic in log rank.
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The paradox of the plankton
Explain why many plankton species competing for the same few limiting resources coexist, and show quantitatively which mechanisms account for observed diversity in real communities.
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The solar coronal heating problem
Determine which physical mechanisms heat the solar corona to millions of kelvin above a photosphere of about 5,800 K, and in what proportion, using public spacecraft data and simulations.
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The strong CP problem
Explain why the CP-violating θ parameter of QCD is experimentally smaller than about 10^-10 when nothing in the Standard Model requires it to be small.
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The twin prime conjecture and bounded prime gaps
Prove that there are infinitely many primes p with p + 2 prime. Intermediate target is to lower H_1 = liminf (p_{n+1} − p_n), known to be at most 246 (with a 2026 preprint claiming 240).
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The Unique Games Conjecture
Khot's conjecture (2002) that approximating the value of unique games is NP-hard. Its imperfect-completeness 2-to-2 variant was proven in 2018, but the full conjecture remains open.
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Theory of the glass transition
Determine whether the dramatic slowdown of supercooled liquids reflects an underlying thermodynamic phase transition (such as an ideal glass at a Kauzmann temperature) or is purely dynamical.
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