16 problems

Let $n\geq 1$. Let $F^+$ be a totally real field, and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_2({\mathbf A}_{F^+})$ without CM, with $\pi_\infty$ essenti…

Let $n\geq 1$. Let $F^+$ be a totally real field, and let $(\pi,\chi)$ be a RAESDC automorphic representation of $\mathrm{GL}_2({\mathbf A}_{F^+})$ without CM. Symmetric power func…

Lattice-isomorphism conjecture. These lattices are isomorphic, compatibly with the identification predicted by the periodicity conjecture for the corresponding Grothendieck-group c…

Periodicity conjecture. Under this isomorphism,

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

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 $k$ 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 $\mathcal{T}$ 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 $C$ 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…