Millennium Prize Problems
Five open problems from Clay’s famous prize program, linking mathematics with computation and mathematical physics.
5 problemsCollections
Selected problems that share a subject or a research tradition. Each leads to its own overview and sources.
The observed universe
Unseen matter, cosmic expansion, energetic particles, and brief radio flashes.
40 problemsOrigins and complexity
From the first living systems to complex cells and multicellular organisms, explore several transitions in life’s history.
10 problemsConsciousness and sleep
Experience, dreams, and the roles of sleep: connected subjects with distinct questions and evidence.
4 problemsPrimes and integers
Patterns in prime numbers, and equations whose solutions must be whole numbers.
14 problemsGraphs and coloring
Colors, subgraphs, labels and paths that visit every vertex.
20 problemsQuantum foundations
Measurement, gravity, and black holes challenge our account of quantum physics.
10 problemsLearning and intelligence
Human cognition and machine learning face different questions about learning, generalization, and goals.
32 problemsGenes and development
How do genomes and regulatory systems shape traits, reliable development, and repair?
32 problemsEnergy and materials
Generating, storing, and converting energy demands both physical understanding and workable devices.
33 problemsInside planets and stars
Composition, structure and internal dynamics of planets and stars.
37 problemsLimits of computation
Algorithms, computational limits and the guarantees behind decisions.
49 problemsPerception and speech
How does the brain combine senses and build fluent speech?
23 problemsGeometric extremes
Packing, covering, distances and other limits set by geometry.
11 problemsInside cells
Molecular traffic, cell machinery, patterned shells and signals between cells.
14 problemsWater and waste
Harvesting freshwater, preserving salt barriers, destroying persistent chemicals, and sorting layered plastics.
4 problemsFields and rational points
Field symmetries and rational points on curves and higher-dimensional varieties.
7 problemsAlgebras and modules
Representations, extensions and resolutions that reveal algebraic structure.
4 problemsDynamics
Repeated maps, billiard trajectories and periodic motion.
5 problemsBases and transversals
Rearranging bases, choosing distinct symbols and listing minimal hitting sets.
4 problemsEcosystems and their environment
Species interactions, geographic patterns, symbiotic partnerships and carbon moving through the ocean.
22 problemsAlgebraic geometry
Polynomial equations define spaces whose shapes, singularities and algebraic cycles pose different open problems.
8 problemsGroups and operators
Structure in groups, representations and infinite-dimensional spaces.
7 problemsElectronics and photonics
Lasers, clocks, analog circuits and devices for moving quantum information through optical networks.
7 problems