Dmitry Zakharov, Central Michigan University
"Maps to trees and loci in the moduli space of tropical curves"
Tropical
geometry
aims
to
find
combinatorial,
piecewise-linear
analogues
of
various
algebraic
and
geometric
objects.
One
particularly
well-developed
correspondence
is
the
one
between
algebraic
curves
and
metric
graphs,
which
are
also
called
tropical
curves.
There
exist
tropical
analogues
of
many
constructions
and
results
for
algebraic
curves,
such
as
meromorphic
functions,
divisors,
linear
equivalence,
the
Riemann--Roch
theorem,
Jacobians,
and
moduli
spaces.
Such
objects
can
be
studied
using
purely
combinatorial
methods,
and
results
about
them
can
then
be
used
to
understand
their
algebraic
analogues.
A
classical
problem
in
algebraic
geometry
is
the
study
of
loci
in
the
moduli
space
of
algebraic
curves
consisting
of
curves
admitting
linear
systems
of
a
particular
type.
A
major
difference
between
tropical
and
algebraic
curves
is
that
the
former
usually
have
a
much
larger
collection
of
principal
divisors
than
the
latter.
For
this
reason,
the
loci
of
tropical
curves
admitting
specific
linear
systems
have
unexpectedly
large
dimension
in
moduli.
I
will
talk
about
an
approach
to
this
problem
in
which,
instead
of
looking
at
tropical
curves
with
linear
systems,
we
look
at
tropical
curves
admitting
maps
to
trees
of
a
particular
type.
*M3-3103* Room Change