10 problems

Depth criterion conjecture. One has

Wang's conjecture. One has

Guerrieri's conjecture. One has

Depth-preservation conjecture. The depths agree:

Let $W(C_m)$ be the graph obtained by attaching a whisker to each vertex of the cycle $C_m$, let $I(W(C_m))$ be its edge ideal in the corresponding polynomial ring…

Let $C_n$ be a cycle of length $n$, let $I(C_n)$ be its edge ideal in the corresponding polynomial ring $R=\mathbb{K}[x\mid x\in V(C_n)]$, and let $\nu(C_n)$ denote its matching nu…

Non-increasing normalized depth function conjecture. The function $g_I$ should be non-increasing for every squarefree monomial ideal $I$.

Let $R$ be a Noetherian polynomial ring over a field $k$, and let $I$ be a squarefree monomial ideal of $R$. Consider the function $t\mapsto\operatorname{depth} R/I^t$ for integers…

Let an arrangement consist of $n$ hyperplanes in general position in $\mathbb{R}^d$. Rousseeuw–Hubert conjecture. There exists a point such that every ray starting at that point in…

Let $G$ be a reductive $p$-adic group, let $R(G)$ be its category of smooth representations, and let $\Pi_r\in Z_G$ be the Bernstein projector onto the subcategory of representatio…