# Mathematics Colloquia and Seminars

The theory of knotted surfaces in four-manifolds (the natural analogue of knot theory to dimension four) is one of the richest and least-explored domains of low-dimensional topology. In this talk, I'll outline some of the most intriguing open problems in this area, and I'll discuss a new approach to four-dimensional knot theory that is inspired by the theory of trisections. Particular focus will be placed on the study of complex curves in the simplest complex four-manifolds: $\mathbb{CP}^2$ and $S^\times S^2$. In this setting, the theory of bridge trisections has produced surprisingly beautiful pictures, which intriguing implications to the study of exotic smooth structures on (complex) four-manifolds.