We showcase tropical geometry as a tool for geometric counting problems.
A nice feature of tropical geometry is that many techniques can be
applied simultaneously over various ground fields, e.g. for complex or
real counting problems. Our prime example will be the count of bitangent
liens to a smooth plane quartic. Already Plücker knew that a smooth
complex plane quartic curve has exactly 28 bitangents. Bitangents of
quartic curves are related to a variety of mathematical problems. They
appear in one of Arnold's trinities, together with lines in a cubic
surface and 120 tritangent planes of a sextic space curve. In this talk,
we review known results about counts of bitangents under variation of
the ground field. Special focus will be on counting in the tropical
world, and its relations to real and arithmetic counts. We end with new
results concerning the arithmetic multiplicity of tropical bitangent
classes, based on joint work with Sam Payne and Kris Shaw.
