This is based on joint work with Ilya Smirnov.

A question of Koll\'ar asks whether varieties with good algebraic properties (log-terminal singularities) have nice topological properties (finite local fundamental group). Motivated by this question, Carvajal-Rojas, Schwede, and Tucker proved a transformation rule for F-signature under finite maps with small ramification. Their result yields bounds for the size of the \'etale local fundamental group. In this talk, we will show the analogous result for differential signature, a numerical invariant that makes sense over fields of any characteristic. Our proof also yields a simplified approach to the aforementioned result. Speaker(s): Jack Jeffries (University of Michigan)