Kaplansky’s zero-divisor conjecture
Can two nonzero finite algebraic combinations of elements of a torsion-free group multiply to zero?
Explore all problems.
Aaronson–Ambainis influence conjecture
Must a bounded low-degree polynomial that varies appreciably have an input bit with noticeable influence?
Deterministic prime construction
Can an algorithm reliably construct a prime of any requested bit length in polynomial time without randomness?
Polynomial hierarchy infinitude
Does each additional level of alternating computational choices add power, forever?
Metric TSP subtour integrality gap
How much can the best traveling-salesperson tour cost exceed its standard fractional relaxation in the worst metric instance?
NP versus coNP
Whenever yes answers have short efficiently checkable certificates, must no answers have them too?
Small-Set Expansion hypothesis
Is it computationally hard to distinguish a graph with a small nearly isolated set from one where every equally small set strongly expands?
Optimal propositional proof system
Can one efficiently checkable proof system match every other system’s proof lengths within a polynomial factor?
Densest-k-subgraph approximation
Can a fast algorithm always choose k vertices spanning within a fixed factor of the largest possible number of edges?
Polynomial-time parity-game solving
Can the winner of every finite parity game be determined in polynomial time?
P versus PSPACE
Can every problem solvable with polynomial memory also be solved in polynomial time?
Mast seeding
Some plants produce huge seed crops together, then little for years. How do weather cues and stored resources set that rhythm?
Maintenance of sexual reproduction
Sexual reproduction can be costly compared with reproduction without a partner. Why does it persist so widely?
Genome size evolution
Closely related organisms can carry very different amounts of DNA. What determines how genomes expand or shrink?
Bacterial speciation
Bacteria exchange DNA across lineages. How do distinct evolutionary populations form and remain separate?
Cambrian explosion
Animal body plans diversified rapidly around the Ediacaran–Cambrian transition. What combination of changes drove that radiation?
Rangeomorph nutrition
Rangeomorphs were branching Ediacaran organisms. How did they obtain nutrients in the deep-sea settings where some lived?
Evolution of menopause
Some mammals live long after reproduction ends. Which evolutionary conditions make that unusual life history favorable?
Intra-Golgi transport
The Golgi modifies cargo as it moves through the organelle. How is that traffic coordinated with the recycling of its resident proteins?
Bulky cargo export from the ER
Some cargo is too large for a conventional small transport vesicle. How does it leave the endoplasmic reticulum?
Nuclear pore selectivity
Nuclear pores admit some molecules and exclude others. How does their dynamic central barrier adjust to the cargo passing through?
Mammalian cell size homeostasis
Different mammalian cells maintain different sizes. What sets their target size and coordinates growth with division?
Organ growth termination
Organs slow their growth and reach characteristic sizes. How do local signals and the rest of the body coordinate that stopping point?
Mitochondrial DNA bottleneck
A mother’s mix of mitochondrial DNA variants can change sharply between generations. What physical sampling process drives that change?