OP-0001Number theoryOpen
The Riemann hypothesis
Do all non-trivial zeros of the Riemann zeta function lie on the critical line?
Millennium Prize Problems· 1859$1,000,000 prize
An atlas of open scientific questions
The next frontier
starts with a question.
Explore the problems that move science forward. Precise statements, original sources, and the progress that brings us closer to an answer.
12151
research problems
6516
open or partially solved
32
research fields
62
source collections
The paper behind the index
OpenFrontiers: A Collection of Open Research Problems and a Benchmark for Scientific RediscoveryJiarui Yao, Ruida Wang, Tong Zhang · arXiv:2610.37739 · 2026
Find your field
Explore all fields ↗∑11349 ↗
Mathematics
From prime numbers to the shape of space.
⌘285 ↗
Computer science
The possibilities and limits of computation.
∇70 ↗
Machine learning
How we learn, generalize, and optimize.
ψ339 ↗
Physics
Quantum information and the laws of matter.
∫34 ↗
Statistics
Finding structure in uncertainty and data.
β36 ↗
Biology
Models of life and natural systems
∑26 ↗
Economics
Equilibria, games and social systems
☉5 ↗
Astronomy
Stars, planets and their physical origins.
⚗3 ↗
Chemistry
Molecules, reactions and chemical structure.
⊕4 ↗
Earth sciences
Earth's interior, landscapes and climate.
Editor's selection
Questions worth sitting with
Landmark conjectures alongside focused, recent research questions.
OP-0026Computational complexityOpen
P versus NP
Can every problem whose solution is quickly checkable also be quickly solved?
Millennium Prize Problems· 1971$1,000,000 prize
OP-0085Learning theoryPartially solved
Data Selection for Regression Tasks
How much can carefully choosing a small training set improve regression? Determine sharp loss ratios under a fixed data budget.
COLT Open Problems· Status reported by the supplied archive· 2025
OP-0091Geometry and topologyOpen
The Hopf product conjecture
Can the product of two spheres carry a metric with strictly positive sectional curvature?
ICM Conjectures· Source-reported status· 1970
OP-0092Quantum informationPartially solved
NPT bound entanglement
Do there exist entangled states with a negative partial transpose from which no pure entanglement can be distilled?
Open Quantum Problems· Source-reported status
OP-0096AlgebraOpen
Andrews–Curtis conjecture
Can every balanced presentation of the trivial group be reduced to the standard one by elementary relator moves?
TheoremDB· Source-reported status· 1965
Connected knowledge
Follow the original sources
ICM Conjectures
2898 ↗The saved International Congress of Mathematicians catalogue, with original statements, remaining questions, and proceedings links.
COLT Open Problems
30 ↗30 papers from the supplied COLT archive, with original text, figures, and dated status notes.
TheoremDB
3909 ↗Mathematical problems with explicit statements, research context, and a record of progress.
Open Quantum Problems
114 ↗Ten categories spanning quantum information, many-body theory, quantum fields, spectral theory, and foundations.
AIM Problem Lists
3470 ↗Research questions developed in American Institute of Mathematics workshops.
Millennium Prize Problems
6 ↗The foundational mathematical challenges selected by the Clay Mathematics Institute.
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