# Alexander Schmeding

- E-mailAlexander.Schmeding@uib.no
- Phone+47 405 39 912
- Visitor AddressRealfagbygget, Allégaten 41
- Postal AddressPostboks 78035020 Bergen

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:

- Lie theory for character groups
- a survey outlining some connections

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. - Geometric rough paths on infinite-dimensional spaces

**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 2020 I am teaching MAT331: Topics in Analysis (Infinite-dimensional geometry), this course is supported by the Trond Moen Stiftelse as a TMS gathering based course

**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.

Spring 2020, MAT102: Brukerkurs i matematikk II.

#### 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,

DOI: 10.3934/jgm.2016008.*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 group*s (with G. Bogfjellmo and R. Dahmen), in M. Barbero, K. Ebrahimi-Fard (Eds.): Discrete Mechanics, Geometric Integration and Lie-Butcher 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), Ann. Inst. Henri Poincare D 7 (2020), no. 3, 395-456, DOI: 10.4171/AIHPD/90*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, 144(2), 357-373 (2020), 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.*Algebra is geometry is algebra*, chapter (in press) for Ebrahimi-Fard (Ed.): Encyclopedia Book in Algebra and Geometry, J. Wiley-Elsevier, 2020*Extending Whitney’s extension theorem: nonlinear function spaces*, (with D.M. Roberts), to appear in Ann. Inst. Fourier (Grenoble) (2020).*Lie groupoids of mappings taking values in a Lie groupoid*(with H. Amiri and H. Glöckner), Archivum Mathematicum, vol. 56 (2020), issue 5, pp. 307-356, DOI: 10.5817/AM2020-5-307.

#### Preprints

*Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie group*s (with R. Dahmen and H. Glöckner), 32 pages, (2014).*Manifolds of absolutely continuous curves and the square root velocity framework,*(2015).*Incompressible Euler equations with stochastic forcing: a geometric approach*(with M. Maurelli and K. Modin), (2019).*Continuity of Chen-Fliess Series for Applications in System Identiﬁcation and Machine Learning*(with W.S. Gray and R. Dahmen), (2020).*Geometric rough paths on infinite-dimensional spaces*(with E. Grong and T. Nilssen), (2020).

Almost all of my preprints can be found on the arXiv in preprint form.

- 2021. Extending Whitney's extension theorem: nonlinear function spaces. Annales de l'Institut Fourier.

- 2020. Lie groups of controlled characters of combinatorial Hopf algebras. Annales de l’Institut Henri Poincaré D (AIHPD). 395-456.
- 2020. Lie groupoids of mappings taking values in a Lie groupoid. Archivum mathematicum. 307-356.

- 2019. The Lie group of vertical bisections of a regular Lie groupoid. Forum mathematicum. 479-489.
- 2019. Convergence of Lie group integrators. Numerische Mathematik. 357-373.
- 2019. A Differentiable Monoid of Smooth Maps on Lie Groupoids. Journal of Lie theory. 1167-1192.

- 2018. The geometry of characters of hopf algebras. Abel Symposia. 159-185.
- 2018. Shape analysis on homogeneous spaces: a generalised SRVT framework. Abel Symposia. 187-220.

- 2017. The Lie Group Structure of the Butcher Group. Foundations of Computational Mathematics. 127-159.
- 2017. Strong topologies for spaces of smooth maps with infinite-dimensional target. Expositiones mathematicae. 13-53.
- 2017. Shape analysis on lie groups and homogeneous spaces. Lecture Notes in Computer Science (LNCS). 49-56.

- 2016. The tame Butcher group. Journal of Lie theory. 1107-1144.
- 2016. Shape analysis on Lie groups with application in computer animation. Journal of Geometric Mechanics (JGM). 273-304.
- 2016. Functorial aspects of the reconstruction of Lie groupoids from their bisections. Journal of the Australian Mathematical Society. 253-276.
- 2016. Character groups of Hopf algebras as infinite-dimensional Lie groups. Annales de l'Institut Fourier. 2101-2155.
- 2016. (Re)constructing Lie groupoids from their bisections and applications to prequantisation. Differential geometry and its applications. 227-276.

- 2015. The diffeomorphism group of a non-compact orbifold. Dissertationes Mathematicae. 3-179.
- 2015. The Lie group of real analytic diffeomorphisms is not real analytic. Studia Mathematica. 141-172.
- 2015. The Lie group of bisections of a Lie groupoid. Annals of Global Analysis and Geometry. 87-123.
- 2015. Differentiable mappings on products with different degrees of differentiability in the two factors. Expositiones mathematicae. 184-222.

- 2021. Universal Zero Dynamics: The SISO Case. .

- 2015. The Lie group of bisections of a Lie groupoid.
- 2015. Linking Lie Groupoids and Infinite-Dimensional Lie Groups.
- 2015. Character groups of Hopf algebras as infinite-dimensional Lie groups.

- 2019. A geometric view on stochastic Euler equations.

- 2017. Shape analysis on Lie groups and beyond.
- 2017. Infinite-dimensional Lie groups for regularity structures of SPDEs.

- 2016. Shape analysis on Lie groups with applications in computer animation.
- 2016. Shape analysis on Lie groups with applications in computer animation.
- 2016. Shape analysis on Lie groups with applications in computer animation.
- 2016. Lie groups shape analysis and computer animation.
- 2016. Charakter Gruppen von Hopf-Algebren als unendlichdimensionale Lie Gruppen.

- 2014. The Lie group of real analytic diffeomorphisms is not real analytic.
- 2014. Differential geometry on orbifolds: A smooth path through singularities.

- 2019. Applications of inﬁnite-dimensional geometry and Lie theory Habilitationsschrift.

- 2017. Shape analysis on Lie groups and homogeneous manifolds. Proceedings of the GSI conference 2017.

- 2016. Shape Analysis on Lie Groups (and beyond) with Applications.

More information in national current research information system (CRIStin)

I am involved in the **CODYSMA** project headed by H. Munthe-Kaas.