In logarithmic geometry, one can consider moduli spaces of stable maps which satisfy some specified tangency conditions. One example is the moduli space M(x) of relative stable maps to P^1 which satisfy some ramification data x. We will begin with a result of Kannan on how invariants of M(x) depend on x and then see how this can be generalised for maps to toric surfaces. We will then move on to introduce expanded stable maps, covering how to produce a map to a target expansion from a polyhedral subdivision of a toric fan. Finally, we will discuss how to tackle the analogue of Kannan's result in this setting using tropicalisation, and if time permits, the results so far. Any necessary toric or tropical background will be explained!
