### Knotted singular surfaces in $S^4$

Alexandra Kjuchukova
All (closed oriented) smooth four-manifolds can be constructed as branched covers of the sphere. In this talk, I will consider a specific class of such covers: irregular 3-fold covers with non-smooth branching sets. This construction is particularly well-suited for explicitly constructing simply-connected manifolds. I'll give an overview of recent results in this area and discuss potential applications to understanding smooth structures and handle decompositions of simply-connected four-manifolds.
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