Problem Directoryv1.0

Goldbach Conjecture

The Goldbach Conjecture states every even integer > 2 is a sum of two primes. Current state of progress (June, 2026): The weak conjecture is proven, while t…

math.NT · Number Theory

Generalized Auslander-Reiten Conjecture (Wei, 2010)

GARC asserts that if vanishing self-extensions persist, the module is projective. Current state of progress (June, 2026): settled for many classes including no…

math.CT · Category Theory · math.RA · math.RT

Infinitude of Primes (Euclid, c. 300 BC)

There are infinitely many primes. Euclid's proof (Elements, Book IX, Prop. 20) — the canonical first proof by contradiction every mathematician learns. St…

math.NT · Number Theory

Direct Sum Conjecture for Quantum Communication

The tight direct sum conjecture for quantum communication is falsified for general distributions, shifting focus to weak versions. Current state of progress (J…

math.QA · Quantum Algebra · math.IT · math.CO

Density Hales–Jewett (k=3)

DHJ(3) yields by reducing the k=3 density increment directly to the Corners Theorem.

math.CO · Combinatorics · math.NT · math.DS

De Giorgi's Conjecture

Is every monotone bounded solution to the Allen-Cahn equation in R^n one-dimensional? Current state of progress (June, 2026): True for n ≤ 3, resolved for 4 ≤ …

math.AP · Analysis of PDEs · math.FA · math.DG

Classification of Surfaces of General Type

Current state of progress (June, 2026): Surfaces are bounded by the Noether and Bogomolov-Miyaoka-Yau lines. Density of Chern slopes in [1, 3] is proven, but a…

math.AG · Algebraic Geometry · math.CV

Borel Conjecture

Current state of progress (June, 2026). The topological Borel Conjecture remains a guiding rigidity question for aspherical manifolds, while its smooth version…

math.AT · Algebraic Topology · math.GT · math.OA

Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer conjecture relates the algebraic rank of an elliptic curve to the order of vanishing of its L-function at s=1. Current state of p…

math.NT · Number Theory · math.AG

Berry–Keating Conjecture

The conjecture proposes a self-adjoint operator H = xp whose spectrum corresponds to the Riemann zeros. Current state of progress (June, 2026): foundational se…

math.SP · Spectral Theory · math.NT · math.MP

Baum–Connes Conjecture with Coefficients

The Baum–Connes Conjecture with Coefficients, proposed by Baum, Connes, and Higson (1994), posits the analytic assembly map is an isomorphism. Current state of…

math.OA · Operator Algebras · math.KT · math.GT

Auslander-Reiten Conjecture

The Auslander-Reiten Conjecture (1975) asserts that modules with vanishing self-extensions are projective. Current state of progress (June, 2026): settled for …

math.RA · Rings and Algebras · math.RT

ABC Conjecture (Masser–Oesterlé)

The abc conjecture relates the additive and multiplicative structure of coprime triples a+b=c. Current state of progress (June, 2026): the conjecture is unreso…

math.NT · Number Theory · math.AG

Finite-Time Blow-up in 3D Navier-Stokes

Do smooth, unforced 3D Navier-Stokes solutions blow up in finite time? Current state of progress (June, 2026): non-uniqueness is now verified for unforced Lera…

math.AP · Analysis of PDEs · math.FA · math.PR