Non-Abelian anyons in the 5/2 fractional quantum Hall state
Do the excitations of the 5/2 fractional quantum Hall state exhibit non-Abelian exchange statistics?
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Many-body localization in two dimensions
Can an isolated two-dimensional disordered interacting system avoid thermalization indefinitely?
Kitaev spin liquid in α-RuCl3
Is the field-induced state of α-RuCl3 a Kitaev spin liquid with fractional excitations?
Granular layering at large particle-size ratios
Why do flowing mixtures with very different grain sizes form repeated layers?
Earthquake nucleation
How does slow fault slip develop into the unstable rupture of a natural earthquake?
Deep-focus earthquake mechanism
How do earthquakes occur hundreds of kilometres down, where ordinary brittle failure is suppressed?
Early geodynamo energy budget
What supplied the energy to sustain Earth’s magnetic dynamo before the solid inner core grew?
Deep mantle water cycle
How much water do subducting rocks carry deep into the mantle, and how is it exchanged?
Lightning initiation
How does fast electrical breakdown become a lightning leader in natural thunderstorm fields?
Low-cloud climate feedback
How will low clouds change under sustained warming, and how will that change affect Earth’s energy balance?
Fourier Entropy–Influence conjecture
Does a yes-or-no rule’s sensitivity to flipped inputs limit how widely its Fourier weight can spread?
Hadwiger’s illumination conjecture
Can every convex body in n dimensions have its entire boundary illuminated from at most 2ⁿ directions?
Symmetric Mahler conjecture
Do cubes and their dual shapes minimize a symmetric body’s volume multiplied by its polar’s volume?
Kissing number in five dimensions
How many equal five-dimensional balls can touch one central ball without overlapping each other?
Zariski cancellation for affine three-space
If adding one affine coordinate makes a complex variety look like four-dimensional affine space, was it already affine three-space?
Borel conjecture
For certain closed manifolds, does matching their homotopy structure force them to be the same topological shape?
Novikov conjecture
Do a manifold’s higher signatures stay unchanged when its homotopy structure is preserved?
Brocard–Ramanujan problem
Can a factorial plus one be a perfect square beyond the three known examples?
Singmaster’s conjecture
Is there a fixed limit on how often any number greater than one can appear in Pascal’s triangle?
Erdős–Ulam problem
Can points be dense throughout the plane while every distance between two of them is rational?
Moser’s convex worm problem
What is the smallest-area convex region that can contain every planar curve of length one after repositioning it?
Minimum opaque barrier for a square
How little total barrier length can intercept every straight line that passes through a unit square?
Seymour’s second-neighborhood conjecture
Must a directed graph without two-way edges have a vertex reaching at least as many new vertices in two steps as in one?
Borsuk’s conjecture in dimension four
Can every bounded four-dimensional set be split into five pieces of strictly smaller diameter?