Sam Payne<br>

Tropicalization of the moduli space of curves
<br>
Tropical geometry allows a systematic study of algebraic curves
over
valued fields in terms of the marked dual graphs of special
fibers of
models of the curve over the valuation ring.  In the past
several
years, a number of researchers, including Caporaso, Gathmann,
Kozlov,
Mikhalkin, and their collaborators, have introduced and studied
moduli
spaces for these marked graphs, which are often called tropical
curves, and estabilshed various analogies to moduli spaces of
curves.
I will present work that explains and extends these analogies,
canonically and functorially, by applying a new generalized
tropicalization map for toroidal Deligne-Mumford stacks to the
moduli
space of stable curves.  Berkovich spaces appear in the
construction
of this new tropicalization map in a natural and elementary
way, but
no tropical or nonarchimedean analytic background is assumed.
This is
joint work with D. Abramovich and L. Caporaso.