Knot concordance and exotica

One well-known strategy for distinguishing smooth structures on closed 4-manifolds is to produce a knot K in the 3-sphere which is (smoothly) slice in one smooth filling W of the 3-sphere but not slice in some homeomorphic smooth filling W'. There are many techniques for distinguishing smooth structures on complicated closed 4- manifolds, but this strategy stands out for its potential to work for 4-manifolds W with very little algebraic topology. However, this strategy had never actually been used in practice, even for complicated W. I’ll discuss joint work with Manolescu and Marengon which gives the first application of this strategy. I’ll also discuss joint work with Manolescu which gives a systematic approach towards using this strategy to produce exotic definite closed 4-manifolds.