5 problems

Let $S$ be the surface in the preceding setup, let $C_i$ be one of the distinguished curves, and let $[\mathfrak Y_{i,(d)}]$ and $[\mathfrak Z_{i,\ell}]$ denote the corresponding d…

Let $\Delta^+$ be the positive roots of a finite-type ADE Lie algebra with a fixed total convex order, and let…

Fix integers $k\ge0$ and $N>k$. Let $g_1,\dots,g_N,h_1,\dots,h_N$ be generators of a group subject to the relations $g_Sh_S=h_Sg_S$ for all subsets $S$ with $|S|\le k$. Failure of…

Let $R=\mathcal{A}/I$ be a quotient ring, and let $g_S$ and $h_S$ denote the products associated with a subset $S$ in the definition of the Multiplicative Rule of $k$. Failure of t…

Let $R[[x,y]]$ be the ring of formal power series in $x$ and $y$ with coefficients in a … gi=1+alpha{i1}xui+alpha{i2}(xui)^2+alpha{i3}(xui)^3+cdots,qquad hi=1+beta{i1}yvi+beta{i2}(…