Given a family of smooth projective varieties over the complex numbers, it is well-known (Ehresmann) that any two fibers are homeomorphic; in particular, they have the same (integral) cohomology. In my talk, I will discuss the analogous situation over a mixed characteristic base, i.e., when one specializes a smooth projective variety from characteristic 0 to characteristic p. In this case, it is unreasonable to expect a direct analog of Ehresmann's theorem: examples of such specializations where there is extra cohomology for the special fiber are ubiquitous (and will be discussed in the talk). Nevertheless, I'll explain why this is essentially the only problem: the cohomology of the special fiber always provides an upper bound for the cohomology of the general fiber (integrally).
(This is based on joint work with Matthew Morrow and Peter Scholze.)