Reimagining universal covers and fundamental groups in algebraic and arithmetic geometry

Tuesday, February 12, 2008 -
4:30pm to 6:30pm
In topology, the notions of the fundamental group and the universal cover are inextricably intertwined. In algebraic geometry, the traditional development of the ├ętale fundamental group is somewhat different, reflecting the perceived lack of a good universal cover. However, I will describe how the usual notions from topology carry over directly to the algebraic and arithmetic setting without change, rectifying imperfections in the ├ętale fundamental group. One key example is the absolute Galois group scheme, which contains more information than the traditional absolute Galois group, in a choice-free manner, and has a rich arithmetic structure. Its geometric fiber is the classical absolute Galois group as a topological group (the profinite topology is the Zariski topology, and comes from geometry). I will also discuss the example of abelian varieties and the Tate module. This is joint work with Kirsten Wickelgren.
Ravi Vakil
Stanford University
Event Location: 
Fine Hall 322