### Differential Geometry - as you need it in Science and Engineering (2016)

• 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 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:  Lisbeth Fajstrup, Associate Professor, e-mail: 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/4-106
4 and 11 November: Frederik Bajers Vej 7C/2-209
9 November: Frederik Bajers Vej 7B/2-109

• Zip code: 9220

• City: Aalborg

• Number of seats: 40