Topics for projects - Ronny Bergmann

In my research I am interested in differential geometry, especially numerical aspects therein, and optimization. Given a Riemannian manifold \(\mathcal M\), for example the sphere, we are given the task to find a minimizer of a function \(f\colon \mathcal M \to \mathbb R\)

Informally, working on a Riemannian manifold, means that “locally” the space we are in still looks like some \(\mathbb R^n\), but globally it does not. We, however, still have a certain structure on this space, that allows us to consider “looking directions” (tangent vectors) and measure angles and lengths of these.

Examples are

• the already mentioned Sphere \(\mathbb S^n\) of unit vectors in \(\mathbb R^{n+1}\)
• the manifold of rotation matrices \(\mathrm{SO}(n)\)
• the Stiefel manifold \(\mathrm{St}(n,p)\) of \(p\) orthonormal vectors in \(\mathbb R^n\)
• the symmetric positive definite (spd) matrices \(\mathcal P(n)\)
• …and many others, see for example the “Basic manifolds” menu entry at Manifolds.jl

You can see several previous topics for example in my list of supervised master theses or the list of supervised specialisation projects. Most of these topics can include programming, where my personal favorite is Julia, using Manopt.jl and Manifolds.jl or more recently LieGroups.jl

You should have had the course TMA4180 Optimization I. Some knowledge on Differential Geometry and Riemannian manifolds would be beneficial, but is not necessary. We could for example do the Specialisation Course as a reading course towards Differential Geometry and/or Optimisation alongside the Specialisation Project.

If you are interested and would like to discuss ideas of topics, please send me an email.