Geodesics and optimization problems (Spring 2020)

A postman is trying to deliver a parcel from the factory to your house. He found that along different paths, the cost of the delivery is different. How to find a path that has the minimal cost? One way to approach this problem is to put a price function on the space where the postman moves so that certain regions are more `costly' then others. Such a price function is called a Riemannian metric in geometry and the plane would be what is called a Riemannian manifold, that is a manifold with a metric.

The minimal cost path is called a {\it geodesic} connecting the factory and your house. In Riemannian geometry, geodesics are viewed as a generalization of the straight lines in the plane. Computing the length of geodesics helps us understand the geometry of the curved objects and has important applications in optimization problems.

In this project, we will study the basic theory of Riemannian manifold and geodesics and see how it can be applied to practical optimization problems. We will learn several different techniques to compute the length of geodesics on Riemannian manifolds such as the plane and more generally curved surfaces.

People:

• Chad Berner (undergrad)

• Domenico D’Alessandro (Faculty)

• Nicklas Day (undergrad)

• Christopher Kunz (undergrad)

• Zhifei Zhu (Postdoc)

Pre-requisites:

• Multivariable calculus (Math 265), linear algebra (Math 317)

• Experience with proofs (Math 201)

• Ordinary differential equations and topology is desirable, but not necessary