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.
36 shown
Explain why the test loss of neural networks follows power laws in model size, data and compute, and predict the exponents from properties of the data and architecture. Reproducible small-scale experiments serve as evidence.
Assign biological function to the genes of the minimal synthetic cell JCVI-syn3A that are still uncharacterised, several of which are essential for growth.
Explain and overcome the degradation of metal-halide perovskite solar cells so that their high efficiencies persist for decades under real operating conditions.
Determine which source populations and emission mechanisms produce fast radio bursts, and whether repeating and apparently non-repeating bursts share a common origin, using public CHIME/FRB and other catalogues.
Determine whether marine ice-sheet and ice-cliff instabilities can drive rapid Antarctic retreat this century, and how much they widen sea-level projections.
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).
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.
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.
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.
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.
Identify the pairing mechanism and the minimal theory that explains superconductivity, the pseudogap and the strange-metal normal state of the copper-oxide superconductors.
Establish a coherent, geochemically plausible route from simple feedstocks to activated ribonucleotides and self-replicating RNA under one consistent set of early-Earth conditions.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
Decide whether every problem whose solutions can be verified in polynomial time can also be solved in polynomial time (Clay Millennium Prize Problem). A full solution is not expected here; the goal is mapped barriers and verifiable partial results.
Prove that the rank of an elliptic curve over Q equals the order of vanishing of its L-function at s = 1, together with the refined leading-term formula (Clay Millennium Prize Problem). A full solution is not expected here.
Prove that on a non-singular complex projective variety every rational Hodge class is a rational linear combination of classes of algebraic cycles (Clay Millennium Prize Problem). A full solution is not expected here.
Prove that every non-trivial zero of the Riemann zeta function has real part 1/2 (Clay Millennium Prize Problem). A full solution is not expected here; the goal is verifiable partial progress.
Prove that for every compact simple gauge group a non-trivial quantum Yang–Mills theory exists on R^4 and has a mass gap Δ > 0 (Clay Millennium Prize Problem).
