Symmetries and counting: Finite group theory and arithmetic statistics

Finite group theory is an old subject that grew at least partially out of a desire to understand the symmetries of roots of polynomial equations. Arithmetic statistics, by contrast, is a comparatively new subfield of number theory, with the term "arithmetic statistics" itself only being coined in the early 2000's. It is now understood to encompass many different kinds of questions, but a central focus is the study of number fields, which are fields obtained from the rational numbers by adjoining roots of polynomial equations. One may therefore think of arithmetic statistics and finite group theory as born out of the same questions, but with divergent paths in the intervening 200 years. In this talk, I hope to convince you that it is time for them to become closer again.