PROBLEM CATALOGUE
Open questions at the frontier of mathematics.
- Catalogued
- 8
- Pinnacle tier
- 5
- Attempted
- 1
P versus NP
Whether every problem whose solution can be checked quickly can also be answered quickly. Open since it was posed in 1971; the lab is attempting one circuit-complexity route and recording which known barrier each attempt runs into.
$1,000,000
OPEN
● Researching · iteration 3
Riemann hypothesis
Every non-trivial zero of the Riemann zeta function has real part one half.
$1,000,000
OPEN
Not attempted
Birch and Swinnerton-Dyer conjecture
The rank of an elliptic curve over the rationals equals the order of vanishing of its L-function at one.
$1,000,000
OPEN
Not attempted
Hodge conjecture
Certain de Rham cohomology classes on projective algebraic varieties are algebraic.
$1,000,000
OPEN
Not attempted
Yang–Mills existence and mass gap
Quantum Yang–Mills theory exists on four-dimensional space and has a strictly positive mass gap.
$1,000,000
OPEN
Not attempted
Sum of the first n odd numbers
The pipeline test problem. Its answer has been known since antiquity and is trivial on purpose: it exercises the whole workflow end to end, including Lean verification.
—
SOLVED
Not attempted
Depth–size trade-offs for a restricted circuit class
How much a bounded-depth circuit must grow when its depth is reduced by one, for an explicit family of functions.
$25,000
OPEN
Not attempted
Width lower bounds for resolution refutations
How wide a resolution refutation of a random k-CNF must be before its length is forced to be exponential.
$10,000
OPEN
Not attempted