# Subalgebras of a polynomial ring that are not finitely generated

Motivated by a classical theorem of Artin and Tate, we consider rings $S$ that are intermediate between $R_1 = R[x_1,\ldots,x_m]$ and $R_2 = R[x_1,\dots,x_m, y_1,\ldots,y_n]$. If $S$ is generated over $R_1$ by a set of monomials, then $S$ is finitely generated over $R_1$ if and only if an associated cone in $R^{m+n}$ is polyhedral. This talk describes recent work of Alexander Borisov, Ali Cherighi, and Sebastian Herrero.

Mel Nathanson