The video https://www.youtube.com/watch?v=wZvFKcQ_3Rc&t=8s mentioned something called the Index Theory. I can't find it on wikipedia. Where could I find more about the theory?
Here index is just the winding number of a curve in a vector field.
Also, the video mentioned that "In a vector field, every closed orbit must eclipse fixed points that have indices sum to 1".
My questions are: Is there some analogous results in complex analysis? Is there some connection between topology and complex integrals?
Since $\frac{1}{2\pi i}\int_\gamma \frac{1}{z-w}$ is the winding number of $\gamma$ around $w$. I guess many complex analysis results are consequences of applying topology to study differential equations. I would be grateful if someone could explain further the underlying topology in complex analysis.