  Here we introduce researchers in algebraic biology to the exciting new field
of cophylogenetics. Cophylogenetics is the study of concomitantly evolving
organisms (or genes), such as host and parasite species. Thus the natural
objects of study in cophylogenetics are tuples of related trees, instead of
individual trees. We review various research topics in algebraic statistics for
phylogenetics, and propose analogs for cophylogenetics. In particular we
propose spaces of cophylogenetic trees, cophylogenetic reconstruction, and
cophylogenetic invariants. We conclude with open problems.
