6 problems

Let $(R,\mathfrak{m})$ be a regular local ring, and let $\mathfrak{p}$ and $\mathfrak{q}$ be prime ideals in $R$ such that … Here $\mathfrak{p}^{(n)}$ denotes the $n$th symbolic po…

Let $w,\tau\in W/W_{P_d}$. Let $R^+$ and $R_{P_d}^+$ denote the positive roots of the ambient root system and of the parabolic subsystem, respectively. Define $S'_{w,\tau}$ to be t…

Bhatt–Hochster–Ma conjecture. The ring $R$ has a lim Cohen–Macaulay sequence.

Consider a nonzero polynomial of the form … where each $f_{ij} \in \mathbb{C}[X]$ has at most $t$ monomials. Multiplicity $\tau$-conjecture. The multiplicity of any nonzero complex…

Let $v,w\in S_{n+2}$, and let ${\rm mult}_{e_v}(X_w)$ denote the multiplicity of the Schubert variety $X_w$ at the torus-fixed point $e_v$. If $C_n=\frac{1}{n+1}{2n\choose n}$ is t…

Serre's vanishing conjecture. The intersection multiplicity satisfies