Abstract: A natural algebraic and number-theoretic question to ask is how many number fields of given Galois group there are up to a certain size X, for an appropriate definition of "size". When size is measured by the discriminant of a number field, there are asymptotic results in the cubic case due to Davenport-Heilbronn and in the quartic and quintic case due to Manjul Bhargava. We will survey their work, and then see how these results can be used as input to prove a different, but related, question arising from predictions made by the Katz-Sarnak philosophy, on distribution of zeros of families of L-functions.

This talk will be accessible to graduate students.