| Abstract: We discuss a few fascinating conjectures and open
problems on arithmetic and geometric cyclotomy. These fall into three
categories (given in no certain order): (1) representations of
differentials and integral bases, (2) equidistribution of roots of
$L$-functions, and (3) circle packing. We'll give attention to a few
recent results towards these, particularly the classification of
separable fields, methods for basis construction, a solution to the
``Hurwitz problem'' for differentials in characteristic $p > 0$,
\emph{almost} genus stability, and definitions of curvature. We will
also discuss a few other major open problems on normal bases and
modules over a PID, which have connections to lifting. This talk, for an audience with general knowledge of algebraic number
theory, will be given with a view towards the similarities and
differences between the classical cyclotomic fields and function field
cyclotomy. |