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To understand the topology of a 4-dimensional manifold, one often investigates the embedded surfaces it contains. In this talk, I will describe an approach to studying surfaces in 4-manifolds using "corks". These are contractible pieces of 4-manifolds that are known to "localize" exotic smooth structures. For applications, I’ll begin with some badly behaved 2-spheres in 4-space that answer a question of Matsumoto, based on joint work with Piccirillo. Then I’ll use a twist on these ideas to construct smoothly (indeed, holomorphically) embedded surfaces in the 4-ball that are "exotically knotted", i.e., isotopic through ambient homeomorphisms but not diffeomorphisms.

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