Introduction to Whitney Towers

These introductory notes on Whitney towers in 4-manifolds, as developed in collaboration with Jim Conant and Peter Teichner, are an expansion of three expository lectures given at the Winter Braids X conference February 2020 in Pisa, Italy. Topics presented include local manipulations of surfaces in 4-space, fundamental definitions related to Whitney towers and their associated trees, geometric Jacobi identities, the classification of order n twisted Whitney towers in the 4-ball and higher-order Arf invariants, and low-order Whitney towers on 2-spheres in 4-manifolds and related invariants.