### Justin Lyle

University of Kansas

Title: Cohen-Macaulay Rings of Finite $\operatorname{\mathsf{CM}}_+$-Representation Type.

Abstract: Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay local ring. We say $R$ has finite $\operatorname{\mathsf{CM}}_+$-representation type if $R$ admits only finitely many nonisomorphic indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum of $R$. We prove several necessary conditions for $R$ to have finite $\operatorname{\mathsf{CM}}_+$-representation type. In some cases, we prove Gorenstein local rings of finite $\operatorname{\mathsf{CM}}_+$-representation type must be hypersurfaces, and we provide a classification of these rings in dimension $1$; if $R$ is complete, equicharacteristic, and with some hypotheses on $k$, they are exactly the hypersurfaces of countable Cohen-Macaulay representation type.