Theory of mind
Children learn to reason about what others know and want. How does this grow into understanding minds that influence one another?
Explore all problems.
Furstenberg’s ×2, ×3 conjecture
Must an atomless circle distribution preserved by doubling and tripling be uniform?
Banach–Mazur rotations problem
Does exact unit-sphere symmetry force a separable real Banach space to be Hilbert?
Birkhoff billiard conjecture
Are ellipses the only smooth strictly convex planar billiards that are integrable near the boundary?
Auslander–Reiten conjecture
Do vanishing higher extension groups force a module to be projective?
Finitistic dimension conjecture
For each finite-dimensional algebra, are finite projective dimensions of finitely generated modules bounded?
Tate conjecture over finite fields
Do all Galois-invariant cohomology classes of the relevant degree come from algebraic cycles over a finite field?
Dynamical uniform boundedness conjecture
Can finite arithmetic orbits be bounded by map degree and number-field degree alone?
Meyniel’s conjecture
Can O(√n) pursuers catch a visible evader on every connected n-vertex graph?
Černý conjecture
Can every synchronizing n-state automaton be reset with at most (n−1)² input symbols?
Hypergraph dualization
Can a classical polynomial-time algorithm recognize all minimal hitting sets?
Uniform-distribution PAC learning of DNF
Can ordinary random labeled examples teach a DNF formula in polynomial time under the uniform distribution?
Carmichael’s totient conjecture
Can an Euler totient value come from exactly one positive integer?
Generalized Ramanujan conjecture
Are the local components of every cuspidal automorphic representation of GLn tempered?
Erdős–Turán additive basis conjecture
Must an additive basis of nonnegative integers have arbitrarily many representations of some sums?
Ryser’s Latin square conjecture
Does every odd-order Latin square have a transversal?
Erdős–Lovász Tihany conjecture
Can a graph be split into two parts retaining prescribed coloring needs?
Planar self-avoiding walk scaling limit
Do critical self-avoiding walks on a fixed square lattice converge to SLE₈/₃ as the grid becomes finer?
Developmental dyslexia
Reading difficulty can involve language, sound–symbol learning, and reading circuits. How these factors interact over development remains unsettled.
Developmental dyscalculia
Persistent arithmetic difficulty can involve numbers, memory, strategy execution and strategy selection. Their developmental causal relationship remains incomplete.
Statistical learning
People pick up patterns through exposure. Which learning processes are shared across tasks and senses, and which are specialized, remains unsettled.
Infantile amnesia
Infants can form memories, yet adults recall few early-life episodes. How development changes memory persistence and later access remains unresolved.
Predictive language processing
Readers and listeners anticipate language. How several predictive mechanisms combine across linguistic structure and different people remains unsettled.
Ticklishness
Tickling can produce involuntary laughter and mixed feelings. How human touch processing, context and laughter circuits combine remains incomplete.