Abstract: We present a proof of Reisner's criterion, which characterizes when the Stanley-Reisner ring of a simplicial complex is Cohen-Macaulay in terms of the homology of the links of in the complex.

Abstract: A central question in homotopy theory is how to do mathematics in a homotopy-invariant way. Broadly speaking, there are two approaches: homotopical algebra, where one studies relative categories (categories with a distinguished class of maps) and higher category theory, where one studies objects, such as quasicategories, encoding higher dimensional morphisms.

The goal of this talk is to give a brief introduction to ∞-category theory and to explain how combinatorial model categories and presentable ∞-categories capture the same homotopy theory, generalizing (locally) presentable categories.