Following the work of Linial—Meshulam, Gromov, and others, we will discuss higher-dimensional expansion from the viewpoint of isoperimetric inequalities for coboundary operators (signed incidence matrices between (k-1)-dimensional and k-dimensional faces of a complex) in cohomology. 

The resulting notion of expansion depends on the choice of coefficients as well as on the choice of a norm on the cochains.

For instance, for real coefficients and the L_2-norm, we recover the spectral definition of higher-dimensional expansion in terms of high-dimensional Laplacians. For Z/2-coefficients and the Hamming norm, we get combinatorial coboundary expansion.

We will also discuss applications (insofar as time permits), such as Gromov’s topological overlap theorem.