Zeeman collapsibility conjecture
Does every contractible two-dimensional polyhedron become collapsible after taking its product with one interval?
Explore all problems.
Bochner–Riesz conjecture
How much smoothing of a spherical frequency cutoff guarantees bounded Fourier reconstruction in higher dimensions?
Tuza’s triangle packing conjecture
Can all triangles in any finite simple graph be destroyed by removing at most twice its maximum number of edge-disjoint triangles?
Ryser’s hypergraph conjecture
Does every finite multipartite uniform hypergraph have a vertex cover at most r−1 times its largest matching?
List edge-coloring conjecture
Do separate allowed-color lists on edges always need no more colors than ordinary edge coloring?
Elliott–Halberstam conjecture
How far can averaged control of prime-counting errors in arithmetic progressions extend as the modulus grows?
Fermi bubbles
Giant gamma-ray lobes rise above and below the Milky Way’s center. Their formation history and radiation mechanisms remain debated.
Chiral magnetic effect in nuclear collisions
Hot quark matter may separate electric charges along a magnetic field. Similar particle patterns can also arise from ordinary collision dynamics.
Polymer contact electrification
Plastic surfaces can acquire charge when they touch and separate. Which carriers dominate depends on material chemistry and contact conditions.
Marine ice-cliff instability
A tall Antarctic ice cliff could break up faster than ice flows forward. Whether this can sustain accelerating retreat depends on competing deformation and failure processes.
Deterministic polynomial identity testing
Can an algorithm efficiently determine, without random choices, whether an arithmetic circuit computes the zero polynomial?
Explicit linear circuit lower bounds
Can we explicitly construct binary matrices whose linear maps require more than a constant number of XOR gates per input?
P versus NC
Can every decision problem with a polynomial-time sequential algorithm also be solved with polynomial-size parallel circuits of polylogarithmic depth?
Exact edit-distance complexity
Can the exact number of edits between two strings be computed in O(n^(2−ε)) worst-case time for some fixed ε > 0?
Binary deletion channel capacity
What is the exact reliable information rate when bits disappear independently and the receiver cannot see where they were removed?
Directed feedback vertex set approximation
Can a polynomial-time algorithm remove all directed cycles while deleting at most a fixed multiple of the minimum possible vertex cost?
#BIS approximability
Can we efficiently estimate how many independent vertex sets an arbitrary bipartite graph has, with a guaranteed small relative error?
Constant-factor treewidth approximation
Can a polynomial-time algorithm find a tree decomposition whose largest bag is within a fixed factor of the smallest possible?
Deterministic distributed maximal independent set
Can a network choose a maximal group of nonadjacent nodes in logarithmically many communication rounds without randomness?
Worst-case trace reconstruction
Do polynomially many independently damaged copies always suffice to recover every unknown binary string exactly?
Motion-induced blindness
A dot can disappear from view while it remains on the screen. A moving pattern around it changes whether it reaches awareness.
Déjà vu
A new situation can feel familiar while you know that it is new. How does that clash arise?
Hypnotic suggestion
A suggestion can change how an arm feels, how an action seems to happen, or what someone perceives. The mechanisms differ across people and suggestions.
Placebo analgesia
Expectations and learned treatment associations can reduce pain. Which mechanisms determine the effect and how long it lasts?