Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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Boileau-Orevkov on the intersection of a complex curve with 3-sphere

Motivation: Feel free to skip. Let $\Sigma$ be a complex curve in $\mathbb{C}^2$. If $\Sigma$ is transverse to a 3-sphere $S^3 \subset \mathbb{C}^2$ of radius $r$, then $\Sigma \cap S^3$ is a link, ...