Complex analysis
Holomorphic maps, their images and the geometry they reveal.
3 problemsAnalysis and dynamics
Continuous change, differential equations, and limiting behavior.
Explore Analysis and dynamics
Differential equations
Existence, smoothness and long-time behavior of solutions to evolution equations.
2 problemsCollections
Browse collectionsGeometric extremes
Packing, covering, distances and other limits set by geometry.
11 problemsDynamics
Repeated maps, billiard trajectories and periodic motion.
5 problemsGroups and operators
Structure in groups, representations and infinite-dimensional spaces.
7 problemsA few problems to explore
34 totalCollatz 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?
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?