 Description: Geometry underpins many branches of science and engineering, for example:
 if you wish to describe the motion of 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 socalled Euler angles. Similarly, projective space is the space of directions of a 1dimensional 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: Lisbeth Fajstrup, Associate Professor, email: fajstrup@math.aau.dk and Martin Raussen, Associate Professor, email:raussen@math.aau.dk
 Lecturers: ssociate Professor, Lisbeth Fajstrup, Associate Professor
Martin Raussen and Associate Professor Rafael Wisniewski  ECTS: 4
 Time: 2, 4, 9, 11, 14, 16 November, 2016. All days: 8.45 15.45
 Place:
2, 14, and 16 November: Frederik Bajers Vej 7A/4106
4 and 11 November: Frederik Bajers Vej 7C/2209
9 November: Frederik Bajers Vej 7B/2109  Zip code: 9220
 City: Aalborg
 Number of seats: 40
 Deadline: 12 October, 2016

Important information concerning PhD courses:
We have over some time experienced problems with noshow 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 noshow 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 Hubert Raussen
 Teacher: Rafal Wisniewski