The journal Science has published a list of the top 125 unanswered questions. The link takes you to a list of the editors’ top 25, with links to essays; here are the remaining 100. A selection:
What is a species?
How did flowers evolve?
Why do some countries grow and others stagnate?
Does Poincare’s test identify spheres in four-dimensional space?
To what extent can we stave off Alzheimer’s?
What powers quasars?
Are we alone in the universe?
Some of them are really "what can’t we do yet" rather than "what don’t we know."
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