Tropical curves and why they are useful

Tropical geometry is a piecewise-linear analog of algebraic geometry in which some computations are ``easier" to carry out. Tropical geomery is useful as combinatorial properties of the tropical variety reflect much of the geometry of the original algebraic variety (as dimension, degree and irreducibility). In this talk, we focus mostly on tropical plane curves. In particular, we will see how to count algebraic plane curves using tropical geometry (and thus compute Gromov-Witten invariants of P^2 using combinatorics).