Mathematics Problem Archive

Manin conjecture: if K-rational points on Fano variety are Zariski-dense subset, then the distribution of points of height: $H(x)\leq B$ in any Zarisk...

For an abelian variety or suitable motive over a number field, are its normalized Frobenius conjugacy classes equidistributed in the associated compac...

Vojta's conjecture: points on non-singular algebraic variety over algebraic number field that not satisfy certain height inequality are contained in s...

Fix $n\geq3$. If coprime nonzero integers $a_1,\ldots,a_n$ have sum zero and no proper subsum zero, is it true that for every $\varepsilon>0$ there is...

Szpiro's conjecture: for any $\varepsilon > 0$, there is some constant $C(\varepsilon)$ such that, for any elliptic curve $E$ defined over $\mathbb{Q}...

Erdős–Moser problem: is $1^1 + 2^1 = 3^1$ the only solution to the Erdős–Moser equation?...

Which integers can be written as the sum of three perfect cubes?...

For coprime integers $1\leq a<d$, is the least prime $p(a,d)$ congruent to $a\pmod d$ always less than $d^2$?...

Are there infinitely many palindromic primes to every base?...

In the global Gan–Gross–Prasad setting, is nonvanishing of the relevant $H$-period on an irreducible cuspidal automorphic representation equivalent to...

For every number field $K$ and every prime $\ell$, does the Iwasawa invariant $\mu_\ell(K)$ vanish?...

For every odd prime $p$ and every positive integer $t$, does there exist a $p$-rational number field $K$ with $\operatorname{Gal}(K/\mathbb Q)\cong(\m...

For a finite abelian extension of number fields, does the Brumer–Stark element have the predicted ideal-class annihilation and principalization proper...

Let $N$ particles move as independent Brownian motions in a bounded connected open set $D\subset\mathbb{R}^d$. Whenever a particle hits the complement...

For the city-growth model, prove: (a) if $\alpha>1$, the eventual number of cities $M(\infty)$ is finite almost surely; (b) if $\beta<2\alpha$, the la...

Construct a scale-invariant random spatial network whose law is mathematically natural, for example with an explicit formula for the distribution of $...

Construct a scale-invariant random spatial network that is visually realistic, in the sense of not looking very different from a real-world road netwo...

Prove or disprove each of the proposed implications between the SIRSN properties numbered (16), (20), (49), (50), and (51): (16)$\Rightarrow$(20), uni...

For the measure-theoretic limit transition densities $p_\infty(x,y,t)$ arising from sequences of finite reversible Markov chains, construct a natural ...

For the null-recurrent renewal process of Problem 1.1, is there a non-decreasing function $\phi$ such that $D_t/\phi(t)$ converges in distribution to ...

For the null-recurrent renewal process of Problems 1.1–1.2, assuming their answers are positive, does $(D_t/\phi(t),U_t)$ converge in distribution to ...

Let $(a_n)_{n\in\mathbb{Z}}$ be a locally finite configuration of points on $\mathbb{R}$. For the force law $F(x,y)=|x-y|^{-2}$, call the configuratio...

Let $(G_n)$ be a bounded-degree expander family converging locally to an infinite graph $G$. Prove that the finite-graph percolation thresholds $p_c(G...

Prove that Bernoulli percolation has at most one infinite cluster on every bounded-degree graph roughly isometric to $\mathbb{Z}^d$....

For bounded-degree graphs, prove that the property $p_c<p_u$ is invariant under rough isometry....

In one-dimensional long-range percolation with edge probabilities proportional to $\beta|i-j|^{-2}$, study the distance exponent $\theta(\beta)$ defin...

Prove that every bounded-degree graph roughly isometric to a nonamenable Cayley graph is non-Liouville....

Let $D_n$ be the source-to-sink first-passage distance in the recursively substituted hierarchical graph whose distances satisfy $D_n\stackrel d=D_{n-...

For the hierarchical first-passage distances $D_n$ satisfying $D_n\stackrel d=D_{n-1}+\min(D'_{n-1},D''_{n-1})$, determine concentration around the me...

For i.i.d. first-passage percolation on $\mathbb{Z}^2$, prove or disprove that boundary fluctuations have a Tracy–Widom limit and that the variance of...

Prove that natural i.i.d. first-passage-percolation models on $\mathbb{Z}^d$, including exponential edge lengths, almost surely contain no two-sided i...

Determine the almost-sure asymptotic growth rate of the effective resistance from the root to graph-distance $r$ in the uniform infinite planar triang...

Let $G$ be a distributional local limit of finite planar graphs. Prove that $p_c^{\mathrm{site}}(G)\ge1/2$ almost surely and that there is no infinite...

On the $n\times n$ grid with a Gaussian free field with no boundary conditions, give every vertex length equal to the exponential of the field. If $\g...

For every $p\ge4$, determine the joint law of the matrix of pairwise distances among $p$ independent points sampled from the volume measure of the Bro...

Let $G$ be a connected quasi-transitive graph and let $p\in(0,1)$. If Bernoulli percolation has more than one infinite cluster almost surely, prove th...

Which hyperbolic $3$–manifolds admit taut foliations? Give an effective procedure to decide if a hyperbolic $3$–manifold admits a taut foliation. For ...

Let $M$ be atoroidal. Is there a natural refinement of the polyhedral structure of the unit ball of the Thurston norm to a polyhedron $\mathscr{P}_\ma...

Suppose $\mathscr{F}$ is a taut foliation of $M$. Characterize the space of metrics on $M$ for which $\mathscr{F}$ can be isotoped to consist of minim...

Which boundary slopes are realized by essential laminations carried by a fixed taut ideal triangulation? Give an algorithm....

Do branched surfaces without sink disks carry automatic laminations?...

Suppose $M$ is irreducible. Suppose further that $\pi_1(M)$ admits a nontrivial action on $\mathbb{R}$. When does $M$ admit a taut foliation with a tr...

Suppose $M$ is atoroidal and admits a taut foliation. Must it admit an $\mathbb{R}$–covered foliation?...

What possibilities are there for universal circles $S^1_\mathrm{univ}$ for a fixed manifold? For a fixed foliation? For what taut foliations is there ...

What is the most general class of knots to which the techniques of Delman–Roberts (in constructing persistent laminations) can be extended?...

Do leaves of $\widetilde{\Lambda}$ for $\Lambda$ an essential lamination have the continuous extension property? More generally, what is the relations...

Suppose $M$ an atoroidal $3$–manifold admits an essential lamination. Does it admit a (necessarily genuine) lamination with quasi–geodesic leaves?...

Is there some notion of a Godbillon–Vey invariant for a lamination?...

Is there a geometric notion for a $3$–manifold analogous to LERFness for foliations? What properties could a manifold have so that immersed essential ...

Let $\mathscr{F}$ be a taut foliation of $M$. Can leaves of $\mathscr{F}$ be approximated by compact essential surfaces? That is, given a leaf $\lambd...