6 problems

Let $B$ be a $k$-row reduced bumpless pipe dream such that row $k$ has no blanks (“weighty tiles”). Let $\phi(B)$ be the RC graph corresponding to $B$ under the natural bijection b…

Bumpless pipe dreams are related to RC graphs by a bijection: for a bumpless pipe dream $B$ with $n$ rows, let $cphi(B)$ denote the corresponding RC graph, and let $\mathcal{Z}$ be…

RBPD limit-shape and arctic-curve conjecture. As $n\to\infty$, the height function converges to a deterministic limit shape consisting of four frozen regions adjacent to the domain…

Connectivity conjecture. For every $n$, the set of reduced bumpless pipe dreams of size $n$ is connected under the $2\times2$ flips that preserve reducedness.

Let $\omega$ be a permutation, and let $P\in\operatorname{Pipes}(\omega)$. If $P$ is non-reduced, replace the redundant crossings with bump tiles. Up-elbow bound conjecture. Pipe…

Let $S_n$ be the symmetric group, let $\omega\in S_n$, and let $x_1^{i_1}x_2^{i_2}\ldots x_n^{i_n}$ be a monomial with nonzero coefficient and non-maximal degree in the Grothendiec…