K(pi, 1) Spaces in Algebraic Geometry

The theme of this dissertation is the study of fundamental groups and classifying spaces in the context of the étale topology of schemes. The main result is the existence of K(pi,1) neighborhoods in the case of semistable (or more generally log smooth) reduction, generalizing a result of Gerd Faltings. As an application to p-adic Hodge theory, we use the existence of these neighborhoods to compare the cohomology of the geometric generic fiber of a semistable scheme over a discrete valuation ring with the cohomology of the associated Faltings topos. The other results include comparison theorems for the cohomology and homotopy types of several types of Milnor fibers. We also prove an l-adic version of a formula of Ogus, describing the monodromy action on the complex of nearby cycles of a log smooth family in terms of the log structure.