66 problems

Let $K$ be a field, and let $f\in K[x]$ be a non-constant monic univariate polynomial of degree $n$. For $1\leq i\leq n-1$, write $H_i(f)$ for its $i$-th Hasse derivative, and call…

Let $k$ be a perfect field of characteristic $p\geq 7$, and let $(X,\emptyset)$ be a log canonical pair over $k$ of dimension at most $p-2$. Logarithmic Extension Theorem conjectur…

Let $A$ be the algebra of regular functions on a quasihomogeneous isolated surface singularity $X$ defined by $Q(x,y,z)=0$ over a field of characteristic $p$, with the notation…

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, let $E/\mathbb{F}_q(T)$ be an elliptic curve, and let $P,Q\in E(\mathbb{F}_q(T))$ be nontorsion points. Positive common-…

Let $k$ be a field, let $W$ be a finite-dimensional $k$-representation of a linearly reductive group $G$, and let $R=S(W)^G$ be the ring of invariants in the symmetric algebra…

Reductive local structure conjecture. Under these hypotheses, such an étale morphism exists.

Let $K$ be a field of characteristic $p>0$, and let $f_k\in K\{x_1,\ldots,x_n\}$ be a sequence of isolated singularities whose Milnor numbers tend to infinity as $k\to\infty$. The…

Involution compatibility conjecture. For every $\mathfrak{s}\in\mathcal{I}$,

Infinite collision conjecture. The set $C(\alpha_1,\alpha_2;\beta)$ is infinite. The conjecture reverses the expectation stated immediately beforehand, which predicted finiteness w…

Let $X$ be a smooth complete curve over a field $k$, let $G$ be a reductive group, and let $\Shv_{\Nilp}(\Bun_G)$ be the category of $\ell$-adic sheaves on $\Bun_G$ with nilpotent…

Contact-inequivalence conjecture. For general $a,b,c,d ain F$, these singularities are not contact equivalent.

Let $p$ be an odd prime. Let $X'$ be a smooth curve of genus $3$ over $\overline{\mathbf F}_p$, and let $\pi':Y'\to X'$ be a double cover branched at two points. By the Riemann–Hur…

Positive-characteristic dynamical Mordell–Lang conjecture. The return set $R$ is a union of finitely many infinite arithmetic progressions together with finitely many sets of the f…

Same-degree intersection-of-orbits conjecture. At least one of the following holds: (1) there exists $m\in{\mathbb N}$ such that $f^m=g^m$; or (2) there exist linear polynomials…

Intersection-of-orbits conjecture. At least one of the following holds: (1) there exist $m,n\in{\mathbb N}$ such that $f^m=g^n$; or (2) there exist linear polynomials…

Let $\overline{K}$ be an algebraically closed field of characteristic $p$, and let the paper's singularity-classification algorithm be denoted by Algorithm 1. Positive-characterist…

Strict boundedness conjecture. The following assertions hold:

Let $k$ be a field of characteristic $p>0$, and let $\mathrm{Cr}_n(k)=\mathrm{Bir}_k(\mathbb{P}^n_k)$ be the Cremona group of birational automorphisms of projective $n$-space. An e…

Let $x\in X$ be an $n$-dimensional klt singularity in characteristic $0$, and for primes $p\gg0$ let $x_p\in X_p$ be its reduction modulo $p$. Let $s(x_p,X_p)$ denote the F-signatu…

Proposed structure. The quotient and its character and Hilbert polynomial are

Let the MZV's in positive characteristic be the multiple zeta values over the coefficient field used in the source, viewed as a set carrying the relevant shuffle product. Shuffle H…

Fix an integer $d\geq 2$ and a prime power $q$. Let $\left[\mathcal{L}_d\right]$ be the space of homothety classes of lattices, let $\tilde{\mathcal{K}}^d$ be the ambient vector sp…

Dynamical Mordell–Lang conjecture. The set $S$ is a finite union of arithmetic progressions together with finitely many sets of the form

Let $A$ be an Artinian Gorenstein algebra of codimension three, with socle degree $j_A$, over an infinite field $\mathsf{k}$ of characteristic $p>j_A$. For a general enough linear…

Principal-plinth-ideal conjecture.