28 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 …

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 (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: 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.…

Conjecture. For every discrete group $G$ and every $x\in H^*(BG;\mathbb Q)$ the higher signature is an oriented homotopy invariant. That is, if $M'$ is another closed oriented smoo…

Conjecture. For every integer $n\ge1$ there is a prime strictly between $n^2$ and $(n+1)^2$: …

Conjecture (Elliott–Halberstam). The primes have level of distribution $\theta$ for every $\theta<1$. That is, for every fixed $\theta<1$ and every $A>0$ there is a constant…

Conjecture (Schanuel). Let $n\ge1$ and let $z_1,\dots,z_n\in\mathbb C$ be linearly independent over $\mathbb Q$. Then …

Conjecture (Bombieri–Lang). If $X$ is of general type, then $X(K)$ is not Zariski dense in $X$. That is, there is a proper Zariski-closed subset $Z\subsetneq X$ with…

Conjecture $T^i(X,\ell)$ (Tate). The map $c^i_\ell$ is surjective. In words, every Galois-invariant class in $H^{2i}_{\mathrm{\acute et}}(X_{\bar k},\mathbb Q_\ell(i))$ is a…

Let $k$ be a field and let $H^*$ be a Weil cohomology theory for smooth projective $k$-varieties with coefficients in a field $F$ of characteristic $0$. Examples include $\ell$-adi…

Lehmer's problem. Does there exist an absolute constant $c>0$ such that … for every nonzero algebraic integer $\alpha$ that is not a root of unity?

Conjecture. If $\dim H\ge2$, then every bounded linear operator $T$ on $H$ has a nontrivial closed invariant subspace.