Welcome to Differential Geometry – as you need it in Science and Engineering

**Description: **Geometry underpins many branches of science and engineering, for example:

- if you wish to describe the motionof an airplane or of a robot arm
- if you look closer at the design of the GPS navigation system in a car
- if you analyse the control of a telecommunication satellite
- if you want to exploit the options and understand the constraints you meet in customary programmes in computer graphics, computer vision or computer games
- if you study the configurations of a mechanical or molecular system, or problems in relativity theory or in elementary particle physics.

It is quite common that the underlying configuration space of a system is not some Euclidean space Rn, but a manifold that only locally can be described by a fixed set of coordinates. The number of coordinates required corresponds to the degrees of freedom of the system. Familiar examples of manifolds are spheres - in arbitrary dimensions - and tori - in 2 dimensions visualized as a donut. Other typical examples occur as the space of orthogonal matrices – the configurations of a mechanical system with a fixed point; in dimension 3, these are related to the so-called Euler angles. Similarly, projective space is the space of directions of a 1-dimensional rod fixed at a point. How can one describe such a manifold, how do coordinates change? What corresponds to velocity vectors? What is the counterpart of a differential equation on such a gadget, under which conditions can it be solved? What types of symmetries are there on (some of) these manifolds, and how to exploit them? …

In mathematical terms, subjects covered by the course will include

- Manifolds: concept, definition and examples
- Tangent vector and tangent bundle
- Differentiable maps between manifolds and their differentials
- Inverse and implicit function theorem as a tool to uncover and to analyse manifolds
- Vector fields, flows, Lie brackets

Other possible topics to be chosen from and depending on the audience:

- Differential forms and Stokes’ Theorem
- Distributions and the Frobenius Theorem
- Riemannian metrics
- Curvature concepts

The lectures will incorporate examples of the use of notions and results from differential geometry in engineering and science; the choice of examples will depend on the audience.

Prerequisites:

Solid background in mathematical techniques as obtained through engineering studies at Aalborg University, is expected.

Key literature: L. W. Tu, An Introduction to manifolds, Second Edition, 2011, Springer, New York.

** Organizer: **Associate professor Lisbeth Fajstrup, fajstrup@math.aau.dk

**Lecturers: **Associate professor Lisbeth Fajstrup, fajstrup@math.aau.dk, Professor Rafael Wisniewski, raf@es.aau.dk and Professor Martin Raussen, raussen@math.aau.dk

**ECTS: **4

**Time: **31 October and 2, 7, 9, 12, 14 November 2018

**Place: **Fibigerstræde 11, room 115

** Zip code:** 9220

**City: **Aalborg Øst** **

**Number of seats: **40

**Deadline: **10 October 2018

**Important information concerning PhD courses **We have over some time experienced problems with no-show for both project and general courses. It has now reached a point where we are forced to take action. Therefore, the Doctoral School has decided to introduce a no-show fee of DKK 5,000 for each course where the student does not show up. Cancellations are accepted no later than 2 weeks before start of the course. Registered illness is of course an acceptable reason for not showing up on those days. Furthermore, all courses open for registration approximately three months before start. This can hopefully also provide new students a chance to register for courses during the year. We look forward to your registrations.** **

- Teacher: Lisbeth Fajstrup
- Teacher: Martin Raussen
- Teacher: Rafal Wisniewski