Erdős–Hajnal conjecture
Forbidding one fixed induced graph is predicted to force a large group of vertices that are all connected to each other or all disconnected.
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Ultra-high-energy cosmic rays
Particles arrive at Earth with energies far beyond those reached by human accelerators. Which cosmic objects accelerate them, and by what process?
Long COVID
Some people experience persistent or new health problems after SARS-CoV-2 infection. Which biological processes sustain these different conditions?
Continual learning
An AI system can learn new tasks and lose skills it already had. How can it keep useful knowledge while adapting with limited memory and computation?
Diagonal Ramsey numbers
Color every edge of a complete graph red or blue. How many vertices force k vertices whose connecting edges all have the same color?
Fast radio bursts
Brief radio flashes arrive from across the universe. Magnetars can produce some of them, but what generates their intense emission and explains the range of observed bursts?
Tumor dormancy
Cancer cells can persist quietly for years and later form a new tumor. What keeps them dormant, and what allows them to start growing again?
Out-of-distribution generalization
A model can predict well on familiar data and fail when conditions change. Which changes can it handle reliably, and how can we know?
Graceful tree conjecture
Every tree is predicted to admit vertex numbers whose differences across edges are all distinct and fill the required range.
Proton decay
Protons appear stable in ordinary observations. Could they eventually decay through interactions too rare to have been detected yet?
Antibiotic persistence
Some bacteria survive an antibiotic exposure without inheriting resistance to the drug. What allows these persister cells to survive and later resume growth?
Adversarial robustness
Carefully chosen changes to an input can fool a learned classifier. How can we provide reliable protection against a stated class of such changes?
Square peg problem
Must every simple closed curve in the plane contain four points that are the corners of a square?
Coronal heating problem
The Sun’s outer atmosphere is much hotter than its visible surface. Which processes deliver and dissipate enough energy to maintain it?
Human microbiome causality
Microbial communities differ between people and are associated with many health conditions. Which differences change the host, and which simply reflect other changes?
Machine unlearning
Can a trained model efficiently remove the influence of selected training data, with credible evidence of what was removed?
Smooth four-dimensional Poincaré conjecture
A four-dimensional space can have the topology of a sphere. Must every smooth version be smoothly equivalent to the standard sphere?
Andrews–Curtis conjecture
A set of generators and relations may describe the trivial group in a complicated way. Can a small set of permitted moves always simplify it?
Lehmer’s Mahler measure problem
A polynomial’s roots define a measure of its size. Can integer polynomials have this measure arbitrarily close to one from above?
Littlewood conjecture
Two real numbers can each be approximated by fractions. The conjecture predicts especially good simultaneous approximations using a common denominator.
Schanuel’s conjecture
The conjecture predicts how many algebraically independent numbers must appear when independent complex numbers are paired with their exponentials.
Bombieri–Lang conjecture
Algebraic spaces of general type are predicted to have rational points confined to a smaller algebraic subset, rather than spread densely everywhere.
Hilbert’s tenth problem over the rationals
Can an algorithm always decide whether an equation with integer coefficients has a solution made of rational numbers?
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?