Extremal set theory asks questions such as "what is the largest family of pairwise intersecting k-element subsets of an n-element set?" Many questions of this type can be modeled with exterior algebras, and admit extensions such as "what is the largest self-annihilating subspace of k-forms in an n-variable exterior algebra?" The shifting technique in extremal set theory corresponds to algebraic geometric limits of certain curves in a Grassmannian.

In this talk, I will overview these connections, and go on to discuss recent joint work with Bulavka and Gandini. We find a maximum subspace of k-forms in an n-variable exterior algebra that is self-annihilating but not annihilated by any 1-form. A difficulty is that "not annihilated by any 1-form" is not a closed condition in the Grassmannian. This work extends the Hilton-Milner theorem of extremal set theory to the exterior algebra setting, and answers questions of Scott and Wilmer and of my own.