22 problems

Assume that the characteristic exponent $p$ is inverted and take the étale topology. For every smooth $k$-variety $X$, write $h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)$ for its zerot…

Let $X$ be a smooth real projective variety. Let $RT_tH_n(X)$ denote the topological filtration on $H_n(ReX;\mathbb Z_2)$, given by the image of $\Phi_{t,n}$, and let $RG_tH_n(X)$…

Let $\mathcal F_n$ and $\mathcal F_{n.5}$ denote the successive terms of the Cochran–Orr–Teichner filtration of the smooth knot concordance group, indexed by $n\in\mathbb N_0$. Coc…

Let $\hat {\cal B}_{m}^{\,u}$ be the diagram space equipped with the filtration whose quotient at $(f,o,e)$ is ${\cal G}_{f,o,e}^{\,}\,\hat {\cal B}_{m}^{\,u}$. Let $T(m,u)$ be the…

Let $X$ be an abelian surface, let $K_n(X)$ be the generalized Kummer variety, and let $z=\{z_0,\ldots,z_n\}\in K_n(X)$ with $z_m\in X$. For a point $z$, let $O_z$ denote its ratio…

Roé-Urbinati's conjecture. The valuation $v$ is b-divisorial if and only if

Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $I$ be an $\mathfrak{m}$-primary ideal, and let $\mathcal{A}$ be an $I$-good filtration. Vasconcelos' uniformity conjecture.…

Let $\{\mathfrak L_\alpha\}_{\alpha\in[0,1]}$ be the filtration of $\mathcal G_0$ defined in the paper. The family is called strict when its inclusions are strict as the parameter…

Let $\mathfrak G^{(0)}=\{\mathfrak G^{(0)}_\alpha\}_{\alpha\in[\alpha_*,1]}$ be the generalized filtration, and let $f\in\mathfrak G^{(0)}_\alpha$. The function $f$ generates a sem…

Boundary conjecture. The function $f$ belongs to $\partial\mathfrak H_\alpha$ whenever $\alpha\in(0,2/3]$. The source notes that several other boundary functions are known, while t…

Let $\mathfrak A=\{\mathfrak A_\alpha\}_{\alpha\le1}$ be the analytic filtration, and let $F_\alpha$ denote the function used in its definition. For each $\alpha\le1$, consider the…

Weight–stable-rank comparison conjecture.

Pour un corps $F$, soit $D_k\mathcal{C}(F)$ la filtration croissante par la profondeur, définie par $D_{-1}\mathcal{C}(F)=0$, $D_0\mathcal{C}(F)=\mathcal{C}_1(F)$ et la condition s…

Let $X$) be a separable Banach space. For integers $N,M\geq 0$, let $(f_{n,m}:0\leq n\leq N,0\leq m\leq M)$ be an $X$-valued martingale on $(\Omega,\mathcal{F},\mathbb{P})$ with re…

Consider a graph providing an adic realization of an automorphism with positive entropy, together with its associated limiting simplex. Positive-entropy adic graph conjecture. Such…

Let $c4$ and $c4'$ be finitely isomorphic ergodic filtrations. Assume that both filtrations are nonstandard and that both have virtual matrix distributions for the sequences of ass…

Let $\widetilde U=\widetilde U(\mathfrak g):=U(\mathfrak g)\otimes_{{\mathcal Z}(\mathfrak g)}S(\mathfrak h)$, and let … with…

Heaviside filtration conjecture. The graded representations $M_n$ and $D_n$ of $S_n$ are both isomorphic to the associated graded representation of the Heaviside filtration on…

Teh's conjecture.

Let $F$ be the ambient field over $k$, let $\Omega^j_{F|k}$ be the space of $j$-forms over $k$, and let $N_j$ denote the level filtration defined by subfields of transcendence degr…

Principal-filtration conjecture. If $\mu\in\mathfrak X_{+,H}$ satisfies the vanishing conditions of Theorem, then

Stable equality conjecture. For any $g\geq 0$ and all $j\geq 1$,