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
- A geometric view on stochastic Euler equations
Here is a talk at the Hausdorff Institute explaining the ideas.
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 Spring 2020 I am teaching MAT102: Brukerkurs i matematikk II.
Office hours (Spring 2020):
Tuesdays 13-14, 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)
Teaching previous semesters
Fall 2019, MAT214: Complex analysis.
Publications in refereed journals and books
- A construction of relatively pure submodules, Communications in Algebra Vol. 42, Issue 1 (2014) pp. 228-237.
- Differentiable mappings on products with different degrees of differentiability in the two factors (with H. Alzaareer), Expo. Math. 33 (2015), pp. 184-222.
- Orbifold diffeomorphism groups, in: Kielanowski, P. et al. (Eds.), Geometric Methods in Physics XXXII Workshop Bialowieza 2013, (2014) pp. 153-162.
- The diffeomorphism group of a non-compact orbifold, Ph.D. thesis Paderborn (2013), urn:nbn:de:hbz:466:2-12166, Published as: Dissertationes Math. (Rozprawy Mat.) 507 (2015), 179 pages.
- The Lie group of bisections of a Lie groupoid (with C. Wockel), Ann. Global Anal. Geom. Vol 48, Issue 1 (2015), pp. 87-123.
- The Lie group structure of the Butcher group (with G. Bogfjellmo), 33 pages, (2015), Found. Comput. Math., DOI: 10.1007/s10208-015-9285-5.
- The Lie group of real analytic diffeomorphisms is not real analytic (with R. Dahmen), 32 pages, Studia Math. 229(2) (2015), pp. 141-172, DOI: 10.4064/sm8130-12-2015.
- Character groups of Hopf algebras as infinite-dimensional Lie groups (with G. Bogfjellmo and R. Dahmen), Ann. Inst. Fourier (Grenoble), 66 no. 5 (2016), pp. 2101-2155.
- (Re)constructing Lie groupoids from their bisections and applications to prequantisation (with C. Wockel), Dierential Geom. Appl. 49 (2016), pp. 227-276.
- Functorial aspects of the reconstruction of Lie groupoids from their bisections (with C. Wockel), J. Aust. Math. Soc. 101 (2016), p. 253-276, DOI: 10.1017/S1446788716000021.
- The tame Butcher group (with G. Bogfjellmo), J. Lie theory 26 (2016), No. 4, pp. 1107-1144.
- Shape Analysis on Lie Groups with Applications in Computer Animation (with E. Celledoni and M. Eslitzbichler), J. Geom. Mech. 8, no. 3 (2016), pp. 273-304,
- Strong topologies for spaces of smooth maps with infinite-dimensional target (with E.O. Hjelle), Expo. Math. (2016), DOI: 10.1016/j.exmath.2016.07.004.
- Overview of (pro-)Lie group structures on Hopf algebra character groups (with G. Bogfjellmo and R. Dahmen), in M. Barbero, K. Ebrahimi-Fard (Eds.): Discrete Mechanics, Geometric Integration and LieButcher Series, Springer Proceedings in Mathematics & Statistics Vol. 267 (2018) pp. 284-314.
- Shape Analysis on Lie groups and homogeneous spaces (with E. Celledoni, S. Eidnes and M. Eslitzbichler) in: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science, vol 10589. Springer, Cham
- Shape analysis on homogeneous spaces (with E. Celledoni und S. Eidnes), In: Celledoni E.,et. al. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. (2019)
- The geometry of characters of Hopf algebras (with G. Bogfjellmo), In: Celledoni E., et. al. (eds) Computation and Combinatorics in Dynamics, Stochastics and Control. Abelsymposium 2016. Abel Symposia, vol 13. (2019).
- Lie Groups of controlled characters of combinatorial Hopf algebras (with R. Dahmen), 52 pages, to appear in Ann. Inst. Henri Poincare D, cf. arXiv:1609.02044.
- Linking Lie groupoid representations and representations of infinite-dimensional Lie groups, (with H. Amiri), Ann Glob Anal Geom (2019) 55 Issue 4, pp 749-775.
- A differentiable monoid of smooth maps on Lie groupoids (with H. Amiri), Journal of Lie theory, Vol. 29, No. 4, pp. 1167-1192 (2019)
- Convergence of Lie group integrators, (with C. Curry), Numerische Mathematik (2019), DOI: 10.1007/s00211-019-01083-1.
- The Lie group of vertical bisections of a regular Lie groupoid, Forum Mathematicum (2019), DOI: 10.1515/forum-2019-0128.
- Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie groups (with R. Dahmen and H. Glöckner), 32 pages, (2014), arXiv:1410.6468.
- Manifolds of absolutely continuous curves and the square root velocity framework, arXiv:1612.02604.
- Extending Whitney’s extension theorem: nonlinear function spaces, (with D.M. Roberts), arXiv:1801.04126.
- Lie groupoids of mappings taking values in a Lie groupoid (with H. Amiri and H. Glöckner), arXiv:1811.02888.
- Algebra is geometry is algebra, chapter (in preparation) for Ebrahimi-Fard (Ed.): Encyclopedia Book in Algebra and Geometry, J. Wiley-Elsevier, 2019
- Incompressible Euler equations with stochastic forcing: a geometric approach (with M. Maurelli and K. Modin), arXiv:1909.09982.
Almost all of my preprints can be found on the arXiv in preprint form.
- 2019. A Differentiable Monoid of Smooth Maps on Lie Groupoids. Journal of Lie theory. 29: 1167-1192.
- 2019. Convergence of Lie group integrators. Numerische Mathematik. 1-17. doi: 10.1007/s00211-019-01083-1
- 2019. The Lie group of vertical bisections of a regular Lie groupoid. Forum mathematicum. doi: https://doi.org/10.1515/forum-2019-0128
- 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. Functorial aspects of the reconstruction of Lie groupoids from their bisections. Journal of the Australian Mathematical Society. 101. 253-276. doi: 10.1017/S1446788716000021
- 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
- 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
- 2019. Applications of inﬁnite-dimensional geometry and Lie theory Habilitationsschrift. Technische Universität Berlin, Technische Universität Berlin. 305 pages.