Mathematics Problem Archive

Suppose $G$ and $H$ are Cayley, or vertex-transitive, expanders on the same number of vertices and can be matched with asymptotically vanishing edge d...

If $G$ is a finite vertex-transitive expander, must every almost automorphism of $G$ be close to an automorphism?...

For a locally convergent sequence $(G_n)$ of bounded-degree integer-labeled graphs, does the normalized rank modulo $p$ of the adjacency matrix conver...

Let $(G_n)$ be a locally convergent graph sequence and let $\mu_n$ be the probability distribution of the roots of the chromatic polynomial of $G_n$. ...

Let $G$ be a $d$-regular Cayley graph with spectral measure $\mu$. Is $\mu$ a weak limit of spectral measures of finite $d$-regular graphs?...

Let $\Gamma$ have property (T), and let $(G_n)$ be a sofic approximation of a Cayley graph of $\Gamma$. Can $(G_n)$ be changed by an asymptotically va...

Can free spanning forests be used to prove basic properties of the first $L^2$ Betti number, such as multiplicativity upon passage to a finite-index s...

Does every infinite Cayley graph $G$ admit a $G$-invariant proper coloring with $\chi(G)$ colors?...

Let $X$ be the space of $k$-regular Cayley graphs with the local-convergence topology and let $T\subset X$ be the closed subset of transient graphs. I...

Let $G$ be a bounded-degree expander, or strongly ergodic, graphing that weakly contains a finite graph $H$. Does $G$ factor onto $H$?...

Does every higher-rank semisimple real lattice have rank gradient zero?...

Does every residually finite property-(T) group have rank gradient zero?...

Can the Euclidean minimum spanning tree (MST) of $n$ points in $\mathbb{R}^d$ be computed in time close to the lower bound of $\Omega(n \log n)$?...

Is the Yao-Yao Graph a $t$-spanner for constant $t$? A geometric graph is a $t$-spanner (or just a spanner) if, for every pair of nodes, the shortest ...

Can one decide in deterministic polynomial time whether an integer $n$ is squarefree?...

Let $C_{10}$ find $x$ satisfying $x^e\equiv a\pmod n$ under $\gcd(e,\varphi(n))=\gcd(a,n)=1$. Is complete integer factorization randomized polynomial-...

Given a prime $p$ and degree $d$, can an irreducible polynomial of degree $d$ over $\mathbb{F}_p$ be constructed in deterministic polynomial time?...

Is complete integer factorization deterministically polynomial-time reducible to discrete logarithms modulo composites?...

Is complete integer factorization deterministically polynomial-time reducible to computing $\varphi(n)$?...

Given $a,b$ and a prime $p$ with nonsingular curve $y^2=x^3+ax+b$, can a point on the curve modulo $p$ be found in deterministic polynomial time?...

Does there exist a constant $c$ for which one can find in deterministic polynomial time a nonzero lattice vector of length at most $n^c$ times the min...

Given $f\in\mathbb{Q}[x]$, can the degree of its splitting field, equivalently the order of its Galois group, be computed in deterministic polynomial ...

Given $d\in\mathbb{N}$, can the class number $h(-d)$ of binary quadratic forms of discriminant $-d$ be computed in deterministic polynomial time?...

Can $\gcd(a,b)$ be computed in the parallel complexity class $NC$?...

Can two positive integers $a,b$ be multiplied using $O(\log(ab))$ bit operations?...

Determine the least positive integer $n_1$ and the greatest negative integer $n_2$ for which a $D(n_i)$-quintuple exists....

Does there exist a rational Diophantine septuple, that is, seven nonzero rational numbers whose pairwise products plus $1$ are rational squares?...

For which rational numbers $q$ do there exist infinitely many rational $D(q)$-quintuples?...

Let $k$ be a positive integer. The equation $$ x^p-y^q=k, $$ where the unknowns $x$, $y$, $p$ and $q$ take integer values, all $\ge 2$, has only finit...

There exists a positive integer $k$ such that, for $m$ and $n$ positive integers, the conditions $$ R(m+i)=R(n+i)\quad (i=0,\ldots,k-1) $$ imply $m=n$...

If $a$ and $b$ are two positive integers with $a<b$ and $R(a)=R(b)$ then there is a prime $p$ with $a< p< b$....

Let $(\varepsilon_n)_{n\ge 0}$ be a sequence of elements in $\{0,1\}$. Assume that the real number $$ \sum_{n\ge 0}\varepsilon_n 3^{-n} $$ is irration...

As a $\mathbb{Q}$-algebra, $\mathfrak{Z}$ is the direct sum of $\mathfrak{Z}_p$ for $p\ge 0$....

For $p\ge 3$ we have $$ d_p=d_{p-2}+d_{p-3} $$ with $d_0=1$, $d_1=0$, $d_2=1$....

$\overline{G}$ is a universal odd distribution with values in groups where multiplication by $2$ is invertible....

Let $\tau\in\mathbb{C}$ have positive imaginary part. Assume that $\tau$ is not quadratic. Set $q=e^{2i\pi\tau}$. Then at least $4$ of the $5$ numbers...

For each integer $n\ge 1$ there exists a positive constant $c(n)$ such that, for any algebraic subvariety $V$ of $\mathbb{G}_m^n$ which is defined ove...

There exists a positive constant $c$, depending only on $A$ and $\mathcal{L}$, such that for any $P\in A(\overline{\mathbb{Q}})$ which has infinite or...

For any positive integer $n$ and any real number $\theta$ which is either transcendental or else is algebraic of degree $>n$, there exists a positive ...

Assume that $K=\mathbb{Q}$ and that $V(\mathbb{Q})$ is Zariski dense; is $Z$ a union of connected components of $V(\mathbb{R})$?...

Let $A$ be a simple Abelian variety of dimension $g$ over a number field $K$ embedded in $\mathbb{R}$. Denote by $\ell$ the rank over $\mathbb{Z}$ of ...

Let $n \ge 2$. Denote by $\mathrm{ZH}_n$ the set of real numbers $\xi$ with the following property: there exists $c_1(\xi) >0$ and $c_2(\xi)>0$ such t...

Let $(n_i)_{i\ge 0}$ be an increasing sequence of positive integers. Assume there is a prime number $p$ such that the power series $$ \sum_{i\ge 0}z^{...

Given an elliptic curve $E/\mathbb{F}_q$ and $P\in E(\mathbb{F}_q)$, characterize maps or isogenies $\psi:E\to E'$ for which $\psi(P)$ can be evaluate...

Find bilinear structures that are useful for cryptographic constructions....

Find orders $\mathcal{O}$ whose Picard groups contain $\mathbb{Z}/\ell$, admit compact element representations, and allow group composition in $O(\log...

Make the proposed realization of existing finite-field discrete-logarithm systems inside class groups explicit for practical systems....

Determine whether Weil descent compromises the security of the proposed class-group realization of finite-field discrete-logarithm systems....

Explicitly construct global algebras or Brauer classes with prescribed local data, especially when the local splitting fields are dihedral....

Determine whether natural de Rham cup-product pairings can be used to improve zeta-function computations....