Every theorem begins as A Question unanswered.

A living atlas of mathematics' open problems — from Riemann to Kakeya, with provenance, status, and prize.

270+Open conjectures tracked 9Mathematical fields 7Millennium prize problems

The Atlas

Open problems by field

Each entry includes statement, significance, why it is hard, latest progress, prize money and an attention rating. Click any row to expand the full record.

research/problems_summary.html

Open Problems · Overview

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Number theory

Number Theory

Riemann, Goldbach, abc, BSD, Schanuel — analytic, additive, Diophantine.

52 problems3 millennium
↗ Geometry / Topology

Geometry & Topology

Hodge, smooth 4D Poincaré, Kakeya, sphere packing, Hopf positive curvature, Singer.

44 problems1 millennium
↗ Algebra / Alg-Geom

Algebra & Algebraic Geometry

Jacobian, Burnside, Standard conjectures, Connes embedding, Zilber–Pink.

30 problems
↗ PDE / Physics

PDE & Math Physics

Navier–Stokes, Yang–Mills, Onsager, Hilbert 16(b), turbulence.

24 problems2 millennium
↗ Combinatorics

Combinatorics

Ramsey R(5,5), Hadwiger, sunflower, Frankl, polynomial Freiman–Ruzsa.

32 problems
↗ CS / Logic

CS & Logic

P vs NP, UGC, ETH, Continuum Hypothesis, Ultimate-L, Černý.

26 problems1 millennium
↗ Probability / Stochastic

Probability & Stochastic

KPZ, SLE, LQG, Tracy–Widom, SK spin glass, Φ⁴_3 SPDE, stochastic NS.

30 problems3 cross-listed
↗ Analysis

Analysis

Stein restriction, Bochner–Riesz, invariant subspace, Furstenberg ×2 ×3, Sarnak Möbius.

26 problems2 cross-listed
↗ Others / Long-tail

Others (Placeholder)

Applied, numerical, special functions, constructive, historical niche. Low-priority long-tail placeholder.

12 problemsplaceholder
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Explorations

Multi-agent research projects

Deeper dives where multiple agents collaborate on a single open problem — from literature triage to proposition decomposition.

explore / mersenne

Mersenne primes infinitude

Five propositions decomposed and analyzed by parallel research agents — primitive factor density, Bateman–Horn singular series, Chebotarev contradictions, and arithmetic incompressibility.

Read the explore → explore / weil_analogy

Weil analogy / number-field geometrization

Meta-program to geometrize Spec ℤ so RH becomes a special case of the Lefschetz fixed-point formula. 10 candidate routes + 11 precise propositions (~70 agents + 5 paper writers/reviewers). 1 strong-GO (P7), 2 NO-GO (P5, P8), 1 warn-defer (P10).

Read the explore → explore / multi-agent-research

Multi-agent recipe

Templates and recipes for running brainstorm + deep-dive agent pipelines on any open problem — verbatim prompts, sample outputs, and orchestration scripts.

Read the recipe →