100 problems

Katz's Sato–Tate conjecture. For every $f\in\mathcal{C}([0,\pi])$,

Let $0\le s\le k$ be an integer. Let $f:\mathbb{P}^k\to\mathbb{P}^k$ be a holomorphic endomorphism of degree $d\geq 2$, and let $T$ be its Green $(1,1)$-current. For an analytic su…

Michel–Venkatesh adelic mixing conjecture. If $|D_i|\to\infty$ and $q_i\to\infty$ as $i\to\infty$, then

Gouvêa's distribution conjecture. The set $X_T$ is equidistributed on $[0,1]$ as $T$ tends to infinity.

Let $S_n$ be the set of permutations of $\\{1,2,\ldots,n\\}$, and let $\operatorname{des}_2$, $\operatorname{cyc}$, $\operatorname{pex}$, $\operatorname{des}$, and…

Let $\mathcal{A}_n$ be the set of ascent sequences of length $n$, and let $\mathsf{asc}$, $\mathsf{rep}$, $\mathsf{zero}$, and $\mathsf{\max}$ denote the corresponding Euler–Stirlin…

For $i=1,2$, let $g_i=\sum_{n=1}^{\infty}b_i(n)q^n$ be primitive eigenforms of weight $2k_i$ and level $2N_i$, and define … Let $g_1,g_2$ be distinct non-CM primitive forms of weig…

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a sequence and let $\ell\in {\mathbb N}$. A sequence is good for mean convergence along $\ell$-term arithmetic progressions if the corre…

Let $d\to\fty$ along ${\mathbb D}^\flat$, let ${\sf R}_d^\Delta([\mathfrak s])$ be the shifted diagonal Heegner packet, let $\mu$ be the quotient measure on ${\sf S}^2$, and let…

Equidistribution conjecture. For every generic $H$, one has

Fix $\rho > 0$. Let $G=\mathbf{G}(\mathbb{R})$ be an $\mathbb{R}$-split real algebraic group, let $\Gamma < G$ be an arithmetic lattice, and let $H$ be a maximal $\mathbb{R}$-split…

Let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module of generic characteristic, let $g\geq 1$, and let $\{x_k\}\subset\mathbb{G}_a^g(K^{\operatorname{sep}})$ be a strict sequence…

Dynamical Siegel conjecture. For any nonpreperiodic point $\beta\in\mathbb{P}^1_{\mathfrak{o}_K}(\overline{K})$, there are at most finitely many preperiodic points of $\varphi$ in…

Let $m\ge2$ be an integer, and let $f(X)\in\mathbb Z[X]$ represent infinitely many primes. A congruence class modulo $m$ is allowable if every integer $r$ in it satisfies…

Let $G$) be a connected semisimple Lie group with finite center, let $\Gamma$ be a lattice in $G$, and let $Y$ be a compact homogeneous space of $G$. Let $K$ be a maximal compact s…

Four-pattern involution equidistribution conjecture. The patterns

Let $\mathcal P$ be the family … For $\Pi\in\mathcal P$, let $\mathfrak S_n(\Pi)$ denote the permutations of length $n$ avoiding every pattern in $\Pi$, and let…

A mesh pattern of length $2$ is a pair $(\tau,R)$, with $\tau$ a permutation of length $2$ and $R\subseteq[0,2]\times[0,2]$ the set of shaded boxes. Two patterns are equivalent whe…

Let $p_1$ and $p_2$ be mesh patterns of length $2$. They are equivalent when $s_{n,k}(p_1)=s_{n,k}(p_2)$ for all $n,k\geq0$, where $s_{n,k}(p)$ counts $n$-permutations containing e…

For fixed $m\ge3$, consider the map … from uniformly chosen $n\in\mathbb Z_{2^k}$ to $\mathbb Z_{2^k}$. Asymptotic equidistribution conjecture. As $k\to\infty$, this map approaches…

Let $T$ denote the odd-to-odd Syracuse map, let $n_0$ be odd, and define $a_i=T^i(n_0)\bmod 2^M$. Let $\mu_{\mathrm{orb},N}=\frac{1}{N}\sum_{i=1}^N\delta_{a_i}$ and let $\mu_U$ be…

For each positive integer $n$, let $p(n)$ denote the partition function, and let $\{r\}$ denote the fractional part of a real number $r$. The equidistribution conjecture. For each…

For a positive integer $A$, let … where the set is counted as a multiset when cardinalities are computed. Equidistribution conjecture. For any interval $I\subset[0,1]$, … for some…

Let $p>q\geq 1$ be coprime integers and define $T_{p/q}(x)=\left\lceil\frac{p}{q}x\right\rceil$. The rational-base equidistribution conjecture. For every $n\in\mathbb N_{>0}$ and e…

Let $\mathcal{M}_n(P_1)$ be the set of Stoimenow matchings avoiding $P_1$, let $\mathcal{P}_n{\bf (3+1)}$ be the corresponding class of $(3+1)$-avoiding Fishburn posets, and let…