24 problems

Mason's rank conjecture. The rank of $N$ is

For a quasi-projective variety $X$ over $K$, let … where $\operatorname{Sym}^n(X)$ is the $n$-fold symmetric power of $X$ and $\operatorname{Sym}^0(X)=\operatorname{Spec}K$. Kapran…

Let $F$ be a number field, let $\mathbb A_F$ be its ring of adeles, and let $\pi\in\mathfrak{F}_{2}$ have central character $\omega_{\pi}$. For $m\geq 2$, let…

Let $R$ be a commutative ring, let $I$ be an ideal of $R$, let $V$ be an $R/I$-module, and let $P.(V)$ be a projective resolution of $V$ over $R$. Write $N$ for normalization,…

Let $\pi\in\mathfrak{F}_{2}$ be a cuspidal automorphic representation, and let $\mathrm{Sym}^{m}(\pi)$ denote its $m$th symmetric power lift. Let $\mathfrak{A}_{m+1}$ denote the re…

Let $X$ be a smooth variety over $k$, and let $m\geq 1$. The symmetric power $\operatorname{Sym}^m X=X^m/S_m$ is the quotient by the symmetric group. A variety has $\mathbb{D}$-sin…

Let $K_0(\operatorname{Var}_k^{\dim})$ be the graded Grothendieck ring of varieties, equipped with the symmetric-power operations $\operatorname{Sym}^m$ and the involution…

Let $F$ be a number field, let $\mathbb A_F$ be its ring of adeles, and let $\pi\in\mathfrak{F}_{2}$. For each place $v$ of $F$, let $\pi_v$ be the local component of $\pi$, and le…

Frobenius–Perron growth conjecture. For every $X\in\mathcal{C}$,

Let $k$ be a field of characteristic not $2$. Let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of $k$-varieties, let $\operatorname{GW}(k)$ be the Grothendieck–Witt ring, a…

Let $n$ be a positive integer. An element of $\operatorname{Sym}^n \mathbb{P}^1(\mathbb{Q})$ is a pair $(k,\alpha)$, where $k$ is an étale extension of degree $n$ and $\alpha\in k$…

Let $\mathcal{T}$ be a smooth proper dg category, and let $\operatorname{Sym}^n\mathcal{T}$ denote its $n$th symmetric power in the sense of Ganter–Kapranov. Write…

Let $M$ be a matroid on the ground set $E$, let $(M_1,\dots,M_d)$ be a symmetric quasi power of $M$, and suppose that $F\cdot\operatorname{Sym}_{d-1}(E)$ is a flat of $M_d$ for eve…

Let $M$ be a matroid with ground set $E$, let $(M_1,\dots,M_d)$ be a symmetric quasi power of $M$, and let $\sF(M)$ denote the set of flats of $M$. For each $F\in\sF(M)$, write…

Let $\mathcal{E}$ be an $\alpha$-twisted locally free sheaf on $\operatorname{Spec}K$ of rank $n$, and let $\mathbb{P}\mathcal{E}$ denote its associated twisted projective space. T…

Let $C$ be a smooth complex projective curve of genus $g\geq 2$, let $C_k$ denote its $k$-th symmetric power, and let $K_C$ be the canonical bundle. Say that a line bundle $L$ on…

Let $\mathit\Pi$ be the cohomological cuspidal automorphic representation under consideration, with central character $\omega_\mathit\Pi$, Whittaker vector $f_\mathit\Pi$, period…

Let $L(n)$ denote the $n$-th layer, defined by … Let $S^0$ be the sphere spectrum and $H\mathbb{Z}$ the Eilenberg–MacLane spectrum of the integers. The spectra $L(n)$ are spacelike…

Let $C$ be a curve and let $D'\geq D$ be effective divisors on $C$. For the motivic Białynicki–Birula decompositions, write $C^{(\underline m)}$ for the relevant product of symmetr…

Let $k$ be a field, let $K_0(\operatorname{Var}_k)$ be the Grothendieck ring of varieties over $k$, and let $\mathbb{L}=[\mathbb{A}^1_k]$. Let $\hat K$ be the completion of…

Let $M_k(f,T)$ denote the symmetric-power $L$-function associated with the generalized Airy family, and let $p$ be the characteristic of the underlying finite field. Haessig's poly…

Symmetric-power factorization conjecture. The operator $L_{46}$ has a left-most operator of order six, and this operator is the fifth symmetric power of $L_E$.

Let ${\mathbb F}$ be an algebraically closed field of characteristic $\ell \geq 0$, let $V={\mathbb F}^{d}$, and let $\mathcal{G}$ be $GL(V)$, $GO(V)$, or $Sp(V)$, where in the lat…

Let $X$ be a topological space, let $A=C(X)$, and let $\boldsymbol{f}\blcolon A\to\mathbb{R}$ be a linear map. For all $k\geqslant p-q+1$, define the Hankel determinant equations b…