Alexander Petrov, a mathematician at MIT, presents this talk at the Institute for Advanced Study's workshop on Hodge theory and o-minimality. He starts from the familiar picture: an algebraic variety over a number field F comes with etale cohomology groups, an etale fundamental group, and higher etale homotopy groups, each carrying an action of the absolute Galois group of F, and the cohomology and fundamental group cases obey known constraints such as being de Rham at places above p and having Frobenius eigenvalues that are Weil numbers. Petrov's subject is where this breaks down. He shows that higher etale homotopy groups often contain subrepresentations that are not de Rham at p, tracing the phenomenon to the gap between the cohomology of an arithmetic group and its profinite completion, and discusses how this behavior splits depending on whether the arithmetic group's reductive group is of Hodge type. The work is joint with Lue Pan and George Pappas. The talk runs 67 minutes and assumes graduate-level familiarity with arithmetic geometry.