Description
In this course we sharpen and combine our tools from linear algebra and calculus to address geometric questions such as:
What is the shortest path from A to B in a curved space? What do we mean by ‘curved’ anyway? How can we understand vector fields qualitatively? And what does the graph of a holomorphic function look like in R^4?
A natural setting for addressing such questions are manifolds: spaces that locally look like R^n.
We will clean up the the vector calculus of R^n and lift it to the context of manifolds using differential forms. This includes a vast generalization of the fundamental theorem of calculus: the general Stokes theorem.
In the setting of manifolds we will be able to prove the famous Gauss-Bonnet theorem:
The integral of the curvature of a surface equals the total index of any vector field on the surface and this in turn equals the Euler characteristic.
This illustrates the beautiful interaction of geometry, differential equations and topology that manifolds are all about.
Audience
This course is aimed at a broad audience of 3rd year bachelor students, particularly all students with an interest in geometry and analysis. By working concretely in R^n we aim to strike a balance between learning how to prove theorems and how to do computations in curved space.
Examination
Written exam plus weekly homework
Prerequisites
Linear Algebra 1,2, Analysis 2,3, Complex Analysis, Algebra 1. Topology is not a prerequisite.
Link
Course homepage
Literature
Course notes available from the homepage.