Ramsey's theorem states that given a partition of the n-element subsets of a countably infinite set S into finitely many pieces, there is an infinite subset H of S so that all of the n-element subsets of H belong to the same piece. There are multiple ways one can attempt to generalize this result. In one direction, one can ask about coloring the infinite subsets of S. Here one needs to put some definability constraints on the partition (for instance, demanding that each piece is Borel), but upon doing so, Ellentuck's theorem gives a very satisfactory positive result. In another direction, one can add more structure to the infinite set S and demand that the witness H share this structure. For instance, S might be the rationals, and H would then be a subset of S which is order-isomorphic to the rationals. Here we can no longer demand that the n-element subsets of H belong to one piece of the partition, but we can put an absolute bound on how many pieces are needed. It turns out that these two ways of generalizing Ramsey's theorem can be combined, and this is the subject of joint work with Natasha Dobrinen.

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.

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.