Unsolved Problems

Does $P = NP$? More formally: if the solution to a problem can be quickly verified (in polynomial time), can the solution also be quickly found (in po...

Do all non-trivial zeros of the Riemann zeta function $\zeta(s)$ have real part equal to $\frac{1}{2}$?...

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere....

Prove that Yang–Mills theory exists and has a mass gap on $\mathbb{R}^4$, meaning the quantum particles have positive masses....

Prove or give a counterexample: Do solutions to the Navier–Stokes equations in three dimensions always exist and remain smooth for all time?...

The conjecture relates the rank of the abelian group of rational points of an elliptic curve to the order of zero of the associated L-function at $s=1...

On a projective non-singular algebraic variety over $\mathbb{C}$, any Hodge class is a rational linear combination of classes of algebraic cycles....

For any $\epsilon > 0$, there exist only finitely many triples $(a, b, c)$ of coprime positive integers with $a + b = c$ such that $c > \text{rad}(abc...

Is every smooth homotopy 4-sphere diffeomorphic to the standard 4-sphere $S^4$?...

There is no set whose cardinality is strictly between that of the integers and the real numbers....

A collection of conjectures about algebraic cycles on smooth projective varieties, including Lefschetz standard conjecture and Künneth standard conjec...

Extend the Kronecker-Weber theorem on abelian extensions of the rationals to any base number field....

Determine the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$, and investigate the topology of real a...

Develop a mathematical framework that axiomatizes physics, particularly mechanics, thermodynamics, and probability theory....

Create a mathematically consistent, predictive model of brain function that goes beyond biological inspiration....

Apply quantum and statistical field theory methods to model and potentially control pathogen evolution....

Develop the mathematics required to control the quantum world for computation....

Identify governing principles for biological systems, analogous to physical laws....

Extend understanding of symmetries and action principles in biology to include robustness, modularity, evolvability, and variability....

Connect the Langlands program to fundamental physics symmetries....

Explore homotopy theory's role in Langlands programs....

Generalize the reciprocity law of number theory to arbitrary number fields....

Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?...

Are there infinitely many prime numbers of the form $2^p - 1$ where $p$ is prime?...

What is the densest packing of spheres in dimensions 4 through 23? More generally, what is the optimal sphere packing density in dimension $n$?...

Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?...

Is there a set whose cardinality is strictly between that of the integers and the real numbers?...

Are there infinitely many primes $p$ such that $2p + 1$ is also prime?...

On a projective algebraic variety, is every Hodge class a rational linear combination of classes of algebraic cycles?...

For an elliptic curve $E$ over the rationals, does the rank of its group of rational points equal the order of vanishing of its $L$-function at $s=1$?...

In the $n$-body problem with $n \geq 4$, can non-collision singularities occur in finite time?...

If $z_1, \ldots, z_n$ are complex numbers that are linearly independent over the rationals, then the transcendence degree of $\mathbb{Q}(z_1, \ldots, ...

For every even number $n$, are there infinitely many pairs of consecutive primes differing by $n$?...

What is the optimal error term in the formula for the number of lattice points inside a circle of radius $r$?...

Does the order of the center of the Steinberg group of the ring of integers of a number field relate to the value of the Dedekind zeta function at $s=...

What is the maximum number and relative positions of limit cycles for polynomial vector fields of degree $n$ in the plane?...

If a compact set in $\mathbb{R}^d$ has Hausdorff dimension greater than $d/2$, must it determine a set of distances with positive Lebesgue measure?...

For a monotone graph property, is the threshold for a random graph to have this property at most a constant factor away from the expectation threshold...

For a univalent function $f(z) = z + \sum_{n=2}^\infty a_n z^n$ on the unit disk, is $|a_n| \leq n$ for all $n$?...

Do there exist any odd perfect numbers? (A perfect number equals the sum of its proper divisors.)...

For varieties over finite fields, are the $\ell$-adic representations arising from étale cohomology related to algebraic cycles in the expected way?...

If a dense linear order without endpoints is complete and has the countable chain condition, must it be isomorphic to the real numbers?...

If polynomials satisfy certain necessary divisibility conditions, do they simultaneously produce infinitely many primes for integer inputs?...

Is there a bound $B(g, d)$ such that every curve of genus $g$ over a number field of degree $d$ has at most $B(g, d)$ rational points?...

If $R$ is a regular local ring and $P, Q$ are prime ideals with intersecting dimensions satisfying a certain condition, is the intersection multiplici...

For how many prime numbers $p$ is a given integer $a$ (not $\pm 1$ or a perfect square) a primitive root modulo $p$?...

For coprime integers $a, b, c$ with $a + b = c$, is $c$ usually not much larger than the product of distinct primes dividing $abc$?...

What is the minimum number of pieces needed to perform a Banach-Tarski decomposition of the ball?...

For which positive integers $m$ and $n$ is the free Burnside group $B(m,n)$ finite? In particular, is $B(2, 5)$ finite?...

Is every finite group the Galois group of some Galois extension of $\mathbb{Q}$?...