My main research interests are infinite-dimensional (differential) geometry, global analysis and Lie theory. Lately, I have become interested in applications of infinite-dimensional geometry to the theory of rough paths and stochastic (partial) differential equations.
Rough path theory is in itself already quite geometric and can be formulated with ease in a Banach space setting. However, then geometric, combinatorial and topological questions come into play.
My research interests and expertise encompass in particular topologies on spaces of (smooth) functions, diffeomorphism groups and higher categories in differential geometry (in the form of Lie groupoids) and their connections to infinite-dimensional geometry.
Before joining UiB, I have held positions at TU Berlin and NTNU Trondheim (for example in the EU-project CHiPS). There applications of infinite-dimensional geometry and numerical analysis were key elements of my research. If you want to know more, (almost) all of my preprints can be found on the arXiv in preprint form.
Here are some key research areas I am interested in (together with some of my contributions):
Infinite-dimensional Geometry, rough paths and stochastic analysis
Rough paths are connected to character groups of certain Hopf algebras (see here for an explanation).These groups turn out to be infinite-dimensional Lie groups:
Applications of infinite-dimensional geometry to stochastic analysis
Shape Analysis (on spaces with ambient geometry)
cf. the survey ''Shape analysis on Lie groups and homogeneous spaces''
Geometry on manifolds of Differentiable mappings
- Mapping Groupoids
- Topology on spaces of differentiable mappings
- Whitney Extension theorem on manifolds
- Numerical Integrators on manifolds
Lie groupoids vs. Infinite-dimensional Lie groups
Linking Lie groupoids and infinite-dimensional Lie groups.
In Fall 2019 I am teaching MAT214: Complex analysis.
There are no fixed office hours. Feel free to drop by or make an appointment by Email.
Bachelor-/Master or other projects
You are a student interested in taking a Bachelor- or Master project with me? Then please feel free to get in touch! Usually I have several ideas for such projects from (infinite-dimensional) differential geometry and its applications. Here is a list of possible projects. However, own suggestions for projects are also welcome.
In the past I had the pleasure of supervising some Bachelor projects at TU Berlin (together with P. Friz in stochastic analysis). Further, I was involved in the StudForsk program at NTNU Trondheim. The project there was completed with a joint publication. See:
Hjelle, S.: Strong topologies for spaces of smooth maps with infinite-dimensional target, Expo. Math. 35 no. 1 (2017), S. 13-53 (cf. ArXiv)
- 2017. The Lie Group Structure of the Butcher Group. Foundations of Computational Mathematics. 17: 127-159. doi: 10.1007/s10208-015-9285-5
- 2017. Shape analysis on lie groups and homogeneous spaces. Lecture Notes in Computer Science. 10589 LNCS: 49-56. doi: 10.1007/978-3-319-68445-1_6
- 2017. Shape analysis on Lie groups and homogeneous manifolds. Proceedings of the GSI conference 2017.
- 2017. Strong topologies for spaces of smooth maps with infinite-dimensional target. Expositiones mathematicae. 35: 13-53. Published 2016-08-03. doi: 10.1016/j.exmath.2016.07.004
- 2016. Character groups of Hopf algebras as infinite-dimensional Lie groups. Annales de l'Institut Fourier. 66: 2101-2155.
- 2016. The tame Butcher group. Journal of Lie theory. 26: 1107-1144.
- 2016. Shape analysis on Lie groups with application in computer animation. Journal of Geometric Mechanics (JGM). 8: 273-304. doi: 10.3934/jgm.2016008
- 2016. (Re)constructing Lie groupoids from their bisections and applications to prequantisation. Differential geometry and its applications. 49: 227-276. doi: 10.1016/j.difgeo.2016.07.009
- 2016. Functorial aspects of the reconstruction of Lie groupoids from their bisections. Journal of the Australian Mathematical Society. 101. 253-276. doi: 10.1017/S1446788716000021
- 2015. The Lie group of real analytic diffeomorphisms is not real analytic. Studia Mathematica. 229: 141-172. doi: 10.4064/sm8130-12-2015
- 2015. The diffeomorphism group of a non-compact orbifold. Dissertationes Mathematicae. 507: 3-179. doi: 10.4064/dm507-0-1
- 2015. Differentiable mappings on products with different degrees of differentiability in the two factors. Expositiones mathematicae. 33: 184-222. doi: 10.1016/j.exmath.2014.07.002
- 2015. The Lie group of bisections of a Lie groupoid. Annals of Global Analysis and Geometry. 48: 87-123. doi: 10.1007/s10455-015-9459-z
Almost all of my preprints can be found on the arXiv in preprint form.