389 problems

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 $ai, aj, ak, al in A$, … For a real number $x > 0$ let ……

Call a finite set $S \subset \mathbb{R}^2$ in general position if no three of its points are collinear, and say that $n$ points of $S$ are in convex position if they are precisely…

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…

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…

For an integer $m$ with $m\neq 0$, let $\operatorname{rad}(m)$ denote the product of the distinct prime factors of $m$ (with $\operatorname{rad}(pm 1)=1$). For integers…

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…

Let $F$ be a nonarchimedean local field with ring of integers $\mathcal{O}_F$, and let $G$ be an unramified group over $F$, i.e. a quasi-split connected reductive $F$-group that sp…

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 $k$ be a field, let $\ell$ be a prime number invertible in $k$, and fix a separable closure $k_{\mathrm{sep}}$ of $k$ with absolute Galois group…

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 $(x_1, x_2, x_3, \ldots)$ be a sequence with $x_i \in \{+1, -1\}$ for every positive integer $i$, and let $C$ be an integer. Then there exist positive integers $k$ and $d$ such…

Ganea's conjecture is a now disproved claim in algebraic topology. It states that

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…

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…

In geometric topology, the spherical space form conjecture states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere.

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 Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of -Betti numbers.

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 differential geometry, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S3. The conjecture was featured by the Australian…

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

In computational complexity, the sensitivity theorem, proved by Hao Huang in 2019, states that the sensitivity of a Boolean function $f\colon \{0,1\}^{n}\to \{0,1\}$ is at least th…