Wild ramification and the geometry of models over discrete valuation rings

One of the most striking discoveries in 20th century mathematics are the profound interactions between number theory and geometry. These are exemplified by the famous Weil Conjectures, an infuential list of proposals formulated by André Weil in 1949 that were eventually proved by Dwork, Grothendieck and Deligne in the period 1960-1974. The Belgian mathematician Pierre Deligne received a Fields medal (1978) and the Abel prize (2013) for his contributions, the highest honours in mathematics. The quest for a proof has been a major motivation for the development of modern algebraic geometry and, in particular, the theory of étale cohomology that serves as a bridge between arithmetic and geometric properties of algebraic varieties. The aim of this project is to understand the information contained in the étale cohomology of a degenerating family of algebraic varieties, especially in an arithmetic context. A major complication with respect to the purely geometric setting is the appearance of wild ramification, and this will be the focus of our research.