Millennium Prize Problems

(A) Global regularity on $\mathbb{R}^3$. For every smooth, rapidly decreasing, divergence-free $u_0$, with $f\equiv 0$, there exist global smooth $u$ and $p$ satisfying the equatio…

Let $M$ be a topological manifold of dimension $3$ (a second-countable Hausdorff space each of whose points has a neighborhood homeomorphic to an open subset of $\mathbb{R}^3$). Su…

For a complex number $s$ with $\operatorname{Re}(s)>1$, let … the series being absolutely convergent, and let $\zeta$ also denote its meromorphic continuation to $\mathbb{C}$, whic…

Let $G$ be a compact simple Lie group with Lie algebra $\mathfrak{g}$ equipped with an invariant inner product $\langle\cdot,\cdot\rangle$, let $g>0$ be a coupling constant, and co…

Fix the alphabet $\{0,1\}$, and call a set $L\subseteq\{0,1\}^{*}$ a language. For $x\in\{0,1\}^{*}$ let $|x|$ denote its length, and let…

Let $X$ be a non-singular complex projective variety of complex dimension $n$ (equivalently, a compact complex manifold admitting a holomorphic embedding into some complex projecti…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with conductor $N$, given by a minimal Weierstrass model … By the Mordell–Weil theorem the group of rational points is finitely gener…