A number of questions in Galois theory can be phrased in the following way: how large (in various senses) can the Galois group G of an extension of the rational numbers be, if the extension is only allowed to ramify at a small set of primes? If we assume that G is abelian, class field theory provides a complete answer, but the question is open is almost every nonabelian case, since there is no known way to systematically and explicitly construct such extensions in full generality.

However, due to some recent advances in our understanding of various types of arithmetic families (with some heavy lifting due to the representation theory of reductive groups over local fields), it has become possible to apply certain "coarse" or "soft" methods in the theory of automorphic forms to attack problems like the above. For concreteness, we will focus on a specific question raised by R. Greenberg and show that such "slightly ramified" number fields, despite not being explicitly constructible by known methods, turn out to "exist in abundance" and allow us to find bounds on the sizes of such Galois groups. Speaker(s): Brian Hwang (Cornell University)