Olivetti Club

Nikhil SahooCornell University
Stay in the Loop!

Tuesday, May 3, 2022 - 4:35pm
Malott 406

This talk is about smooth closed curves in two and three dimensions.

Our main goal will be to investigate "total curvature" of such loops. For signed curvature in $\mathbb R^2\!,$ this yields the "rotation number" of our loop, about which we will mention some nifty results. Taking absolute curvature, we get a quantity that is floppier, but has interesting lower bounds.

Here, we take an interlude to discuss a length formula from geometric measure theory. This will give us a glimpse into the behavior of short curves on spheres and spaghetti thrown against a wall.

With these tools, we show that any loop in $\mathbb R^3$ has total curvature $\geq 2\pi$, with equality if and only if the curve is planar and convex (Fenchel). Finally, we show that any knotted loop in $\mathbb R^3$ has total curvature $\geq 4\pi$ (Fáry-Milnor).

Refreshments will be served in the lounge at 4:00 PM.