Galois Theory and Fundamental Groups --- Alex Perry
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I will discuss the analogy between the galois theory of fields and the theory of fundamental groups and covering spaces.  The goal is to explain how both theories fit naturally into a framework of categorical galois theory, which gives an equivalence between a category equipped with a "fiber functor" to the category of finite sets equipped with the action of a certain profinite group, called the fundamental group.  Finally, I will discuss the fundamental group of a scheme in terms of this machinery.  If the scheme is the spectrum of a field we get usual galois theory; if the scheme is a smooth variety over the complex numbers, we get the theory of finite covering spaces.