388 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…

For a permutation $\sigma$ of $\{1,2,\dots,n\}$ and a permutation $\beta$ of $\{1,2,\dots,k\}$, say that $\sigma$ contains $\beta$ as a pattern if there are indices…

Let $\mathcal{F}$ be a finite family of distinct finite sets that is union-closed, i.e. … and suppose $\mathcal{F}\neq\varnothing$ and $\mathcal{F}\neq\{\varnothing\}$. For an elem…

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…

Let $n \ge 1$ and let $K \subset \mathbb{R}^n$ be a convex body, i.e. a bounded closed convex set with nonempty interior. Call a set of the form $sK + v = \{sx + v : x \in K\}$, wi…

For a set $A \subseteq \mathbb{N}$, call $A$ a Sidon set if all pairwise sums of its elements are distinct, i.e. if for all $a_i, a_j, a_k, a_l \in A$, … For a real number $x > 0$…

Let $n \ge 2$ be an integer and let … be a monic polynomial all of whose roots lie in the closed unit disk, i.e. $|r_{k}|\le 1$ for every $k \in \{1,\dots ,n\}$. Since…

Let $G$ be a finite simple undirected graph with vertex set $V(G)$ and edge set $E(G)$, and for $u\in V(G)$ let $\deg(u)$ denote the number of neighbours of $u$ in $G$. For an edge…

In mathematics, the Berry–Robbins problem asks whether there is a continuous map from configurations of n points in R3 to the flag manifold U(n)/Tn that is compatible with the acti…

In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard T…

Let $n\ge 1$ and let $A:\mathbb{R}^n\to M_n(\mathbb{C})$ be a measurable matrix-valued function for which there exist constants $\lambda>0$ and $\Lambda<\infty$ such that, for almo…

Let $F$ be a field with $\operatorname{char} F \neq 2$. Define the Milnor $K$-ring of $F$ as the graded ring … where the tensor algebra is taken over $\mathbb{Z}$ on the abelian gr…

Peter David Lax was a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics.

Let ${\cal M}$ be a closed hyperbolic $3$-manifold. A surface subgroup is the image of the fundamental group of a closed surface of genus at least $2$ under a map inducing an injec…

Let $n\ge 1$ and let $\mu$ be a centered Gaussian probability measure on $\mathbb{R}^n$, i.e. the distribution of a random vector $X=(X_1,\dots,X_n)$ having a multivariate normal l…

Let $X$ be a compact Hausdorff space. Call a linearly ordered set $(L,<)$ with its open-interval topology a linearly ordered topological space, and say that $X$ is a continuous ima…

A $3$-manifold is closed if it is compact and has empty boundary. A closed $3$-manifold $M$ is prime if whenever $M$ is homeomorphic to a connected sum $M_1 \# M_2$, one of…

Fix an integer $n\geq 1$. Let $\mathrm{Bord}_n^{\mathrm{fr}}$ denote the symmetric monoidal $(\infty,n)$-category of framed bordisms: its objects are $0$-dimensional framed manifol…

In mathematics, the Ehrenpreis conjecture of Leon Ehrenpreis states that for any K greater than 1, any two closed Riemann surfaces of genus at least 2 have finite-degree covers whi…

In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is…

Call a triple $(a,b,c)$ of positive integers a Pythagorean triple if $a^{2}+b^{2}=c^{2}$; such triples are not required to be primitive, i.e. $\gcd(a,b,c)$ may exceed $1$. For a ma…

For an undirected graph $G$, the degeneracy of $G$ is the minimum integer $p$ such that every subgraph of $G$ contains a vertex of degree at most $p$; a graph of degeneracy at most…

For all integers $x, y, a, b$ with $x > 0$, $y > 0$, $a > 1$, $b > 1$, the equation … holds only for $x = 3$, $a = 2$, $y = 2$, $b = 3$; that is, $3^{2} - 2^{3} = 1$ is the unique…

Fix a prime $\ell$ and let $G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. Let $F=\mathbb{F}_{\ell^{r}}$ be a finite f…

Call an integer $p$ prime if $p>1$ and its only positive divisors are $1$ and $p$; the primes in a representation are not required to be distinct. For every odd integer $n>5$ there…