Autonomous soft-robot repair
A soft gripper can heal a clean cut. Can it detect an unexpected dirty, gaping injury, restore contact, repair itself and verify that it can carry a load again?
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Microfluidic artificial lungs
Tiny blood channels can exchange enough oxygen to support a premature infant in a bench test. Can the complete device keep those channels open and compatible with blood during prolonged use?
Sponge contraction coordination
A freshwater sponge expands and squeezes its water canals to flush out irritants. How does a body without conventional neurons organize the wave and bring it to an end?
Diatom silica pore morphogenesis
A diatom builds a glass-like shell pierced by patterned pores. How do proteins outside its silica-making compartment control the holes formed inside?
Agaricus somatic incompatibility
Two button-mushroom colonies can meet and form a zone of dead cells. Which recognition products make their shared cytoplasm incompatible?
Pyrocystis bioluminescence mechanosensing
A single Pyrocystis lunula cell can flash when pushed or deformed. Which molecular sensor turns that physical change into the first signal for light?
Cuttlefish camouflage feedback
A cuttlefish can adjust its skin pattern through a meandering series of changes. What tells it whether the current pattern is getting closer to the background?
RNA silencing inheritance duration
A worm can stop making a fluorescent protein, and its descendants can remain dark after the original trigger is gone. What determines when the inherited silencing ends?
Lovász Hamiltonian path conjecture
Does complete symmetry among a connected graph’s vertices guarantee a route visiting every vertex exactly once?
Chvátal’s toughness conjecture
Is there one fixed level of resistance to vertex removal that guarantees a round trip visiting every vertex exactly once?
Blaschke conjecture
If shortest paths stay uniquely determined out to the full diameter, must the entire metric be one of the standard symmetric geometries?
Mazur’s separable quotient problem
Can every infinite-dimensional Banach space be collapsed to an infinite-dimensional space described by a countable dense set?
Cherlin–Zilber algebraicity conjecture
Must every infinite simple group with finite logical dimension come from algebraic geometry?
Cabling conjecture
Can surgery on a knot create a genuinely decomposable 3-manifold only through the known cable-knot construction?
Alperin weight conjecture
Can each block’s irreducible representations in prime characteristic be counted entirely from local subgroup data?
Broué’s abelian defect group conjecture
Does an abelian-defect representation block retain its derived structure when replaced by its local normalizer block?
Pillai conjecture
For each fixed positive gap, are there only finitely many distinct pairs of integer perfect powers separated by that gap?
Landau’s constant
How large a round disk must every normalized holomorphic map of the unit disk cover somewhere in its image?
Erdős distinct distances problem
Must every set of n distinct points in the plane determine at least a constant times n/√log n different distances?
Solving chess
If both sides played perfectly from the ordinary starting position, would chess end in a draw or a forced win? An unbeatable-looking engine does not yet supply that proof.
Integer multiplication complexity
Long multiplication can be improved to O(n log n) bit operations. Is that logarithmic overhead unavoidable, or could multiplying arbitrary n-bit integers get closer to linear time?
Online bin packing competitive ratio
Items arrive one at a time and must be placed immediately into unit-capacity bins. What is the smallest guaranteed long-run overhead compared with packing the whole list after seeing it?
Mid-Pleistocene transition
Why did the dominant rhythm of ice ages change about a million years ago?
Dansgaard–Oeschger events
What drove Greenland’s repeated rapid warmings during the last ice age?