Collatz conjecture
Choose a positive whole number. Halve it if it is even; otherwise triple it and add one. Repeat. Does every starting number reach one?
Analysis and dynamics
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Yang–Mills existence and mass gap
A successful framework for particle physics still needs a rigorous four-dimensional construction with a positive gap above its vacuum energy.
Littlewood conjecture
Two real numbers can each be approximated by fractions. The conjecture predicts especially good simultaneous approximations using a common denominator.
Hilbert’s sixteenth problem: limit cycles
A polynomial rule for motion in the plane can create isolated repeating trajectories. How many can occur when the polynomial degree is fixed?
Falconer distance conjecture
A sufficiently large fractal set is predicted to determine a set of pairwise distances with positive ordinary length.
Sarnak’s conjecture
Does the arithmetic Möbius sequence average to zero against every sequence generated by a system with zero topological entropy?
Kakeya conjecture
A set can contain a unit line segment pointing in every direction and still have zero volume. Must it nevertheless have full fractal dimension?
Euler–Mascheroni constant
The difference between the harmonic series and a logarithm approaches a famous constant. Is that constant a ratio of integers?
Fourier Entropy–Influence conjecture
Does a yes-or-no rule’s sensitivity to flipped inputs limit how widely its Fourier weight can spread?
Aaronson–Ambainis influence conjecture
Must a bounded low-degree polynomial that varies appreciably have an input bit with noticeable influence?
Invariant subspace problem
Must every bounded linear operator on complex Hilbert space preserve a nontrivial closed subspace?
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?
Dynamical uniform boundedness conjecture
Can finite arithmetic orbits be bounded by map degree and number-field degree alone?
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?
Amenability of Thompson’s group F
Can Thompson’s group F admit an average that is unchanged by every group translation?
Baum–Connes conjecture
Can the proper-action geometry of a countable discrete group recover the K-theory of its reduced operator algebra?
Planar first-passage percolation fluctuations
How widely does the fastest travel time vary across a square grid with independent random edge times?
Relativistic Vlasov–Maxwell global regularity
Can smooth three-dimensional collisionless plasma equations develop a singularity from regular initial data?
Cartan–Hadamard conjecture
Must a region in a nonpositively curved space have at least the boundary area of a Euclidean ball with the same volume?
Bloch’s constant
What is the largest disk radius guaranteed on a one-to-one image patch of every normalized holomorphic map?
Green–Griffiths–Lang conjecture
Must all entire complex curves in a general-type projective variety lie in one proper algebraic subset?
Positivity problem for linear recurrences
Can an algorithm decide whether an integer sequence defined by a linear recurrence never becomes negative?