Mathematics
Open problems, research attempts and community review.
Provisional editorial order for Top Open Problems: Millennium Prize Problems first, followed by assessed difficulty tiers; the unassessed remainder keeps its previous catalogue order. About rankings
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- Math-1 P versus NP Problem
- Math-2 The Riemann Hypothesis
- Math-3 Yang–Mills Existence and Mass Gap
- Math-4 Hodge Conjecture
- Math-5 Navier–Stokes Existence and Smoothness
- Math-6 Birch and Swinnerton-Dyer Conjecture
- Math-7 Global Langlands Functoriality Conjecture
- Math-8 NP versus P/poly
- Math-9 NP versus coNP Problem
- Math-10 Tate Conjecture for Algebraic Cycles
- Math-11 Generalized Hodge Conjecture
- Math-12 Extended Riemann hypothesis for Dedekind zeta functions
- Math-13 Generalized Riemann Hypothesis for Dirichlet L-functions
- Math-14 Schanuel's Conjecture
- Math-15 Vojta's Diophantine Approximation Conjecture
- Math-16 Grothendieck Period Conjecture
- Math-17 Smooth 4-Dimensional Poincaré Conjecture
- Math-18 Hilbert's Tenth Problem over the Rationals
- Math-19 Grothendieck's Standard Conjecture D
- Math-20 Fontaine–Mazur geometric Galois-representation conjecture
- Math-21 Hilbert's 12th Problem: Extension of Kronecker-Weber Theorem
- Math-22 Navier-Stokes Existence and Smoothness: Fefferman A
- Math-23 Birch & Swinnerton-Dyer conjecture
- Math-24 Inverse Galois Problem
- Math-25 Abundance Conjecture for the minimal model program
- Math-26 Termination of Flips Conjecture in the Minimal Model Program
- Math-27 The Standard Conjectures on Algebraic Cycles
- Math-28 Grothendieck Standard Conjecture of Hodge Type
- Math-29 Deligne's Absolute Hodge Conjecture
- Math-30 Grothendieck's Variational Hodge Conjecture
- Math-31 Equivariant Tamagawa number conjecture: Burns-Flach Conjecture4 package
- Math-32 Artins Conjecture on the Holomorphy of Non-Abelian Artin L-Functions
- Math-33 Woodins Ultimate L Conjecture for Supercompact Cardinals
- Math-34 Grothendiecks Section Conjecture in Anabelian Geometry
- Math-35 ABC Conjecture
- Math-36 Farrell-Jones Conjecture
- Math-37 Baum-Connes Conjecture
- Math-38 Novikov Homotopy Invariance Conjecture
- Math-39 The Quantum PCP Conjecture
- Math-40 Generalized Ramanujan conjecture for automorphic representations
- Math-41 The Borel Conjecture on Topological Rigidity of Aspherical Manifolds
- Math-42 Kakeya Conjecture in Dimensions Four and Higher
- Math-43 Sharp Exponents in the Fourier Restriction Conjecture
- Math-44 Invariant subspace problem
- Math-45 Hilbert's Sixteenth Problem
- Math-46 P vs. PSPACE
- Math-47 One-way functions exist
- Math-48 Unconditional Separation of Quantum and Classical Bounded-Error Polynomial Time (BQP != BPP)
- Math-49 Unconditional Separation of BQP from the Polynomial Hierarchy
- Math-50 BQP Versus NP: Complete Language-Class Relation