In my talk, I will start by reviewing how various properties
of characteristic zero singularities can be understood topologically
by ways of the Riemann-Hilbert correspondence. After that, I will
explain how similar ideas can be applied in the study of mixed
characteristic singularities. This is based on a joint work (in
progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl
Schwede, Kevin Tucker, and Joe Waldron.