50 problems

Conjecture (strong, or binary, Goldbach conjecture). Every even integer $n>2$ is the sum of two primes: …

Poincaré conjecture. Every such $M$ is homeomorphic to the three-sphere …

Hodge conjecture. For every non-singular complex projective variety $X$ and every $k$, … That is, every rational cohomology class of type $(k,k)$ on $X$ is a $\mathbb Q$-linear com…

Birch and Swinnerton-Dyer conjecture. For every elliptic curve $E$ over $\mathbb Q$, …

Conjecture. For every integer $n\ge 4$, every Kakeya set $E\subseteq\mathbb R^n$ has Hausdorff dimension $n$: …

Problem. Determine whether …

Problem (Simon's Problem 1). Prove that for every $\nu\ge3$ there are $a<b$, with $b-a$ small, and an open interval $I\subset\mathbb R$ such that, almost surely, $H_\omega$ has pur…

Whole space. The data decay rapidly: for all multi-indices $\alpha$, all $m\ge0$ and all $K\ge0$, … A solution is admissible if $u,p\in C^\infty(\mathbb R^3\times[0,\infty))$ and t…

Riemann hypothesis. Every nontrivial zero of $\zeta$ has real part $\tfrac12$. Equivalently, for every $s\in\mathbb C$, …

Conjecture (spacetime Penrose inequality). Every asymptotically flat initial data set as above satisfies … and equality holds only if $(M,g,k)$ arises as a spacelike hypersurface,…

Conjecture (termination of flips). For every $n$, there is no infinite sequence … of $(K_{X_i}+B_i)$-flips over $S$ starting from a $\mathbb Q$-factorial klt pair of dimension $n$.

Conjecture. $\omega=2$. Equivalently, for every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that for every $n\ge1$ some arithmetic program multiplies $n\times n$ compl…

Conjecture (Fontaine–Mazur). Every irreducible geometric representation $\rho\colon G_K\to\mathrm{GL}_n(\overline{\mathbb Q}_p)$ comes from algebraic geometry.

Problem. Prove or disprove that one-way permutations exist.

Problem. Prove or disprove that one-way functions exist.

Problem (Hilbert's twelfth problem). For every number field $K$, give an explicit construction of $K^{\mathrm{ab}}$ of the same kind. That is, specify analytic functions attached t…

Conjecture (generalized Ramanujan conjecture for $\mathrm{GL}_n$). For every number field $F$, every $n\ge2$, every such $\pi$, and every place $v$ of $F$, archimedean or not,…

Conjecture (global functoriality). For every $F$, $H$, $G$ and $\rho$ as above, and every automorphic representation $\pi'$ of $H(\mathbb A_F)$, there is an automorphic representat…

Problem. Prove or disprove: for every finite group $G$ there is a finite Galois extension $L/\mathbb Q$ with …

Conjecture. There are infinitely many primes $p$ such that $p+2$ is also prime: …

Conjecture. For every admissible family, there are infinitely many positive integers $n$ such that $f_1(n),\dots,f_k(n)$ are all prime.

Conjecture. For every pair of real numbers $\alpha,\beta$, … where $n$ runs over the positive integers.

Conjecture (Zilber–Pink). For every such $X$ and every irreducible closed subvariety $V\subseteq X$, the set of maximal atypical subvarieties of $V$ is finite.

Conjecture (Masser–Oesterlé). For every real $\varepsilon>0$ there is a constant $K_\varepsilon>0$ such that every abc triple satisfies …

Conjecture (plane case). For every field $K$ of characteristic $0$ and all $p,q\in K[x,y]$: if $J_F=c$ for some constant $c\in K\setminus\{0\}$, then $F$ has a polynomial inverse.…