# A Gentle Introduction to Crystalline Cohomology

Let X be a smooth affine algebraic variety over the field C of complex numbers (that is, a smooth submanifold of C^n which can be described as the solutionsto a system of polynomial equations). Grothendieck showed that the de Rham cohomology of X can be computed using only polynomial differential forms on X. This observation was the starting point for the theory of algebraic de Rham cohomology, which has proved to be a useful invariant for algebraic varieties over an arbitrary field k. In the case where k has positive characteristic, Berthelot and Grothendieck introduced a refinement of algebraic de Rham cohomology, known as crystalline cohomology. In this talk, I will review the theory of algebraic de Rham cohomology and sketch an elementary construction of crystalline cohomology, based on recent joint work with Bhargav Bhatt and Akhil Mathew.