Shape spaces - theory and numerics

winter 2022/23

lecturer: Prof. Dr. Benedikt Wirth

Information on the lecture

time, location: Mo. 10 ct, weekly, M 3
Th. 10 ct, weekly, M 3
begin: Oct. 10, 2022
Learnweb: https://sso.uni-muenster.de/LearnWeb/learnweb2/course/view.php?id=62997
content: Shapes play a role in many application areas, e.g. the statistical distribution of organ shapes is examined in computational anatomy in order to detect illnesses from deviations (e.g. shape changes of the brain). For such applications one needs to provide the set of all shapes with an additional structure. This is typically a Hilbert manifold structure; some shape spaces also possess a Lie group structure. We will present so-called Riemannian and Hamiltonian approaches to shape spaces, the analysis of corresponding variational problems and PDEs as well as the numerical analysis for their numeric treatment. Some background in PDEs is required (be it PDEs or numerics for PDEs or geometric analysis); background in differential geometry may be helpful, but is not required.

The first half dozen lectures will be a crash course in standard differential geometry concepts from an applied point of view and a brief overview over different shape spaces; the rest of the lecture is then probably spent on the space of parameterized curves as one example shape space. We will consider differential geometric and analytical aspects of the shape space as well as corresponding discretization concepts for calculations in the shape space and their convergence.
prerequisites:  analysis, linear algebra, some pde background (e.g. pde or numerics for pde or geometric analysis)
examination: type 1: oral examinations (30 min) at the end of term + some type of homework assessment to be announced (e.g. presentation of solutions to two homework sheets)
type 2: oral examinations (20 min) at the end of term
literature:

Information on the tutorials

Learnweb: same as lecture
time, location: We. 8:00-10:00 or 10:00-12:00, weekly, SR 1B