Pólya’s eigenvalue conjecture
Does the area of a planar region impose a sharp lower bound on every Dirichlet vibration frequency, rather than only on high frequencies in the limit?
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Quantum unique ergodicity conjecture
Must every sequence of increasingly high-frequency waves spread uniformly in phase space on a connected compact manifold of negative curvature and dimension at least two?
Gersho’s conjecture in three dimensions
For many representative points in a uniform three-dimensional region, is a body-centered cubic arrangement asymptotically best at minimizing squared distance to the nearest representative?
Kelvin problem
What periodic partition of three-dimensional space into equal-volume cells uses the least shared surface area?
Root hydrotropism
A root can curve toward a wetter patch of soil. What first tells its growing cells that one side has more available water than the other?
S-RNase specificity
A Petunia flower can stop its own pollen while accepting pollen with a different mating type. How does a small team of pollen proteins distinguish so many versions of the same toxic enzyme?
Saccharibacteria host recognition
A tiny Saccharibacterium survives by attaching to another bacterium. Which molecular handshake decides whether that bacterial partner is usable?
Paulinella protein import
A Paulinella amoeba carries a photosynthetic compartment descended from a bacterium. It makes essential proteins outside that compartment—how do those proteins cross its inner membrane to work inside?
Lichen morphogenesis
A yellow lichen on bark has organized layers built by organisms with separate genomes. What makes its fungus and alga assemble that shared body?
Outer membrane vesicle cargo selection
A bacterium releases membrane bubbles whose protein contents differ from its own envelope. What determines which proteins leave in those bubbles and which stay behind?
Periodical cicada timing
A cicada nymph can spend sixteen winters underground before preparing to emerge. What stores that history and opens the developmental gate in the right year?
Hibernation arousal
A hibernating ground squirrel repeatedly warms up in the middle of winter. What internal signal makes a long cold torpor bout end?
Neocalanus diapause termination
A female Neocalanus copepod waits hundreds of metres below the surface, then restarts egg production. What ends its months of dormancy?
Bogong moth stellar navigation
Rotate a projected Australian night sky and a migrating Bogong moth turns with it. Which celestial pattern does its small brain read?
Salmon olfactory imprinting
A salmon learns stream odors as a juvenile and uses them years later when returning to spawn. How does that memory survive the replacement of sensory cells?
Coral larval settlement
A swimming coral larva touches a microbial film and may attach and become a polyp. Which signals in a natural mixed biofilm make both steps happen?
Daphnia life-history kairomones
Water from roach fed Daphnia makes Daphnia reproduce at a smaller size. Which dissolved chemical carries that warning?
Aphid wing induction
A crowded wingless pea-aphid mother can produce daughters with wings. How does a touch on the mother become a developmental instruction to her embryos?
Indistinguishability obfuscation from LWE
A program can reveal how it was written even when its outputs reveal nothing about that choice. Can a compiler hide the choice using only the familiar Learning with Errors hardness assumption?
Public-key encryption from one-way functions
One-way functions support secret-key cryptography. Are they enough to let anyone encrypt a message with a public key while only its owner can decrypt it?
Shannon capacity of the seven-cycle
Seven symbols sit around a circle, and neighboring symbols can be confused. How many messages can long strings of these symbols carry with absolutely no decoding mistakes?
Binary symmetric channel reliability function
If each transmitted bit flips independently with a fixed probability, adding redundancy reduces mistakes. How quickly can the best possible error probability fall as the message block gets longer?
Li–Li network coding conjecture
Several senders share a network of two-way links. Can mixing their messages inside the network ever carry more independent traffic than forwarding and splitting the messages along paths?
Distributed six-cycle detection
Each computer knows only its own place in a network and can send short messages to neighbors. How many communication rounds are needed to discover whether the network contains a cycle through six different computers?