Differential Geometry for Engineers

Overview

This tutorial endeavours to introduce differential geometry simply, efficiently, rigorously and engagingly. A motivating example used throughout is optimisation on manifolds: how can gradient descent or a Newton method be performed on a manifold? An integral part of the tutorial is teaching how to do calculations involving manifolds, including how to compute derivatives, tangent spaces, curvature and so forth. At a higher level, the different ways of working with manifolds will be explained (e.g., extrinsic coordinates versus local coordinates), and ways for extending algorithms from Euclidean space to manifolds will be discussed.

Schedule

8:30-9:15 Motivation and Multi-dimensional Differential Calculus

9:15-10:00 Tangent Spaces and Directional Derivatives

10:00-10:30 Coffee Break

10:30-11:15 Concrete Manifolds, Coordinate Charts and Smooth Functions

11:15-12:00 Metrics, Parallel Transport and Connections

12:00-12:30 Gradients, Hessians and Worked Examples

Presenter

Jonathan Manton, University of Melbourne