# Mathematics Colloquia and Seminars

An exceptional Dehn surgery on $K$, a hyperbolic knot in $S^3$, is one that results in a non-hyperbolic manifold. Some important examples are surgeries that result in a manifold that is Seifert-fibred or one that has cyclic or finite fundamental group. A Dehn surgery slope that results in a manifold that contains an essential surface is called a boundary slope. Dunfield showed that if $t$ is a cyclic surgery slope, then there must be a boundary slope $r$ with $|r-t| \leq 1$. In joint work with Masaharu Ishikawa and Koya Shimokawa, we show that finite and Seifert surgeries are also accompanied by nearby boundary slopes.