calculus means the differentiation and integration of functions and differential forms, and leads naturally to the notion of de rham cohomology. classical hodge theory provides harmonic representatives for de rham cohomology classes. its more recent nonabelian version associates a higgs bundle to a representation $ ho$ of the fundamental group and higgs classes to $ ho$-twisted cohomology classes. calculus and de rham cohomology make sense algebraically and in any characteristic, but hodge theory is profoundly analytic. nevertheless i will describe recent attempts to construct an analog of nonabelian hodge theory in characteristic $p$ (joint work in progress with v. vologodsky).