In low-dimensional topology, there are many problems that can be studied with a combination of geometric, combinatorial, dynamical, and algebraic tools. In this talk, we'll focus on a particular concept from geometry and dynamics, a

*geodesic current*,*which is a generalization of a closed geodesic on a hyperbolic surface. We'll apply the theory of currents to attack problems in both group theory and in combinatorics, specifically:*(1) equipping the moduli space of graphs with a metric resembling the classical Weil-Petersson metric, and using its geometry to study the outer automorphism group of the free group (joint work with Matt Clay and Yo'av Rieck);

(2) counting mapping class group orbits of closed curves on a surface of negative Euler characteristic, and relating these counts to geometric features of the Teichmuller space (joint work with Juan Souto).