Webrelaunch 2020

Differential Geometry (Summer Semester 2024)

Course material will be made available on the Ilias page.

Lecture: Tuesday 15:45-17:15 20.30 SR 2.067
Wednesday 9:45-11:15 20.30 SR 2.058
Problem class: Thursday 17:30-19:00 20.30 SR 3.069

Differential Geometry is one of the most research intensive areas of mathematics in the late twentieth and early twenty-first century. Going back to the seminal studies by Gauss, it has not only a long history, but its modern form given by Riemann can also trace its development back well over a hundred years. Its current prominence stems in part from its position at the crossroads of many active mathematical research fields, such as geometric analysis, topology, discrete and metric geometry, analysis, partial differential equations, Lie and geometric group theory, stochastics on manifolds, etc., and looking beyond the confines of mathematics, the subject continues to influence and be influenced by theoretical physics and has a multitude of further applications as, e.g., in engineering, robotics, computer vision, and machine learning. In particular, being the mathematical backbone of Einstein’s theory of general relativity, Differential Geometry lies at the core of any modern location technologies like the global positioning system GPS, and is nowadays also a fundamental tool in robotics assisted surgery, manifold learning and protein design.
The course itself will cover fundamental concepts, methods and results of differential geometry and analysis on manifolds, together with glimpses into some of their above-mentioned applications. In particular, it will treat smooth manifolds, tensors and Riemannian metrics, linear connections, covariant derivatives and parallel translation, geodesics and curvature.

Analysis I-III, Linear Algebra I,II, basic concepts of topology as, e.g., studied in the KIT course 'Elementare Geometrie'.