UNITE - Shortest Paths on Curved Spaces: Does the Best Route Exist?

Between two points on a surface, is there always a shortest route? On the Earth the answer is an arc of a great circle, which is why flights from Zürich to San Francisco pass over Greenland. On more complicated surfaces there is no formula, and in some spaces no shortest route exists at all: the best possible length can be approached ever more closely without any path ever reaching it. 

This talk is an introduction to the geometry of curved spaces built around that question. Through pictures and everyday examples, from a lifeguard running along a beach to the distortions of a world map, it explains how lengths and distances are measured on a curved space, and shows what can go wrong. It then presents, step by step, a proof that on a compact surface a shortest route always exists. 

The strategy, known as the direct method of the calculus of variations, is central to geometry and physics: rather than computing the best route, one catches it as the limit of better and better candidates.

11:00 - 12:00: Colloquium

12:00 - 12:30: Gong Show

12:30 - 15:00: Lunch & networking

Stay tuned for more details.