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Alexander Schmedings bilde

Alexander Schmeding

Førsteamanuensis, I have left UiB and can for now be found at Nord university
  • E-postAlexander.Schmeding@uib.no
  • Telefon+47 405 39 912
  • Besøksadresse
    Realfagbygget, Allégaten 41
  • Postadresse
    Postboks 7803
    5020 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:

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

Lie groupoids vs. Infinite-dimensional Lie groups

Linking Lie groupoids and infinite-dimensional Lie groups.

Høst 2020: MAT331: Topics in Analysis (Infinite-dimensional geometry), også støttet av Trond Moen Stiftelse som TMS gathering based course

Kontortid: Tirsdag kl. 13.00-14.00, eller skriv en epost for en tidsavtale!

På min engelske side om undervisning kan du finne mer informasjon om bachelor og masterprosjekter.

Undervisning i tidligere semester

Høst 2019: MAT214: Complex analysis.
Vår 2020: MAT102: Brukerkurs i matematikk II

Publications in refereed journals and books

  1. A construction of relatively pure submodules, Communications in Algebra Vol. 42, Issue 1 (2014) pp. 228-237.
  2. Differentiable mappings on products with different degrees of differentiability in the two factors (with H. Alzaareer), Expo. Math. 33 (2015), pp. 184-222.
  3. Orbifold diffeomorphism groups, in: Kielanowski, P. et al. (Eds.), Geometric Methods in Physics XXXII Workshop Bialowieza 2013, (2014) pp. 153-162.
  4. 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.
  5. The Lie group of bisections of a Lie groupoid (with C. Wockel), Ann. Global Anal. Geom. Vol 48, Issue 1 (2015), pp. 87-123.
  6. The Lie group structure of the Butcher group (with G. Bogfjellmo), 33 pages, (2015), Found. Comput. Math., DOI: 10.1007/s10208-015-9285-5.
  7. 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.
  8. 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.
  9. (Re)constructing Lie groupoids from their bisections and applications to prequantisation (with C. Wockel), Dierential Geom. Appl. 49 (2016), pp. 227-276.
  10. 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.
  11. The tame Butcher group (with G. Bogfjellmo), J. Lie theory 26 (2016), No. 4, pp. 1107-1144. 
  12. 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.
  13. 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.
  14. 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 Lie-Butcher Series, Springer Proceedings in Mathematics & Statistics Vol. 267 (2018) pp. 284-314.
  15. 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 
  16. 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)
  17. 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).
  18. 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
  19. 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.
  20. A differentiable monoid of smooth maps on Lie groupoids (with H. Amiri), Journal of Lie theory, Vol. 29, No. 4, pp. 1167-1192 (2019)
  21. Convergence of Lie group integrators, (with C. Curry), Numerische Mathematik, 144(2), 357-373 (2020), DOI: 10.1007/s00211-019-01083-1.
  22. The Lie group of vertical bisections of a regular Lie groupoid, Forum Mathematicum (2019), DOI: 10.1515/forum-2019-0128.
  23. Algebra is geometry is algebra, chapter (in press) for Ebrahimi-Fard (Ed.): Encyclopedia Book in Algebra and Geometry, J. Wiley-Elsevier, 2020
  24. Extending Whitney’s extension theorem: nonlinear function spaces, (with D.M. Roberts), to appear in Ann. Inst. Fourier (Grenoble) 2020
  25. 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

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

Sjekk min side i arXiv.

Vitenskapelig artikkel
  • Vis forfatter(e) 2020. Lie groups of controlled characters of combinatorial Hopf algebras. Annales de l’Institut Henri Poincaré D (AIHPD). 395-456.
  • Vis forfatter(e) 2020. Lie groupoids of mappings taking values in a Lie groupoid. Archivum mathematicum. 307-356.
  • Vis forfatter(e) 2019. The Lie group of vertical bisections of a regular Lie groupoid. Forum mathematicum. 479-489.
  • Vis forfatter(e) 2019. Convergence of Lie group integrators. Numerische Mathematik. 357-373.
  • Vis forfatter(e) 2019. A Differentiable Monoid of Smooth Maps on Lie Groupoids. Journal of Lie theory. 1167-1192.
  • Vis forfatter(e) 2018. The geometry of characters of hopf algebras. Abel Symposia. 159-185.
  • Vis forfatter(e) 2018. Shape analysis on homogeneous spaces: a generalised SRVT framework. Abel Symposia. 187-220.
  • Vis forfatter(e) 2017. The Lie Group Structure of the Butcher Group. Foundations of Computational Mathematics. 127-159.
  • Vis forfatter(e) 2017. Strong topologies for spaces of smooth maps with infinite-dimensional target. Expositiones mathematicae. 13-53.
  • Vis forfatter(e) 2017. Shape analysis on lie groups and homogeneous spaces. Lecture Notes in Computer Science (LNCS). 49-56.
  • Vis forfatter(e) 2016. The tame Butcher group. Journal of Lie theory. 1107-1144.
  • Vis forfatter(e) 2016. Shape analysis on Lie groups with application in computer animation. Journal of Geometric Mechanics (JGM). 273-304.
  • Vis forfatter(e) 2016. Functorial aspects of the reconstruction of Lie groupoids from their bisections. Journal of the Australian Mathematical Society. 253-276.
  • Vis forfatter(e) 2016. Character groups of Hopf algebras as infinite-dimensional Lie groups. Annales de l'Institut Fourier. 2101-2155.
  • Vis forfatter(e) 2016. (Re)constructing Lie groupoids from their bisections and applications to prequantisation. Differential geometry and its applications. 227-276.
  • Vis forfatter(e) 2015. The diffeomorphism group of a non-compact orbifold. Dissertationes Mathematicae. 3-179.
  • Vis forfatter(e) 2015. The Lie group of real analytic diffeomorphisms is not real analytic. Studia Mathematica. 141-172.
  • Vis forfatter(e) 2015. The Lie group of bisections of a Lie groupoid. Annals of Global Analysis and Geometry. 87-123.
  • Vis forfatter(e) 2015. Differentiable mappings on products with different degrees of differentiability in the two factors. Expositiones mathematicae. 184-222.
Faglig foredrag
  • Vis forfatter(e) 2015. The Lie group of bisections of a Lie groupoid.
  • Vis forfatter(e) 2015. Linking Lie Groupoids and Infinite-Dimensional Lie Groups.
  • Vis forfatter(e) 2015. Character groups of Hopf algebras as infinite-dimensional Lie groups.
Vitenskapelig foredrag
  • Vis forfatter(e) 2019. A geometric view on stochastic Euler equations.
  • Vis forfatter(e) 2017. Shape analysis on Lie groups and beyond.
  • Vis forfatter(e) 2017. Infinite-dimensional Lie groups for regularity structures of SPDEs.
  • Vis forfatter(e) 2016. Shape analysis on Lie groups with applications in computer animation.
  • Vis forfatter(e) 2016. Shape analysis on Lie groups with applications in computer animation.
  • Vis forfatter(e) 2016. Shape analysis on Lie groups with applications in computer animation.
  • Vis forfatter(e) 2016. Lie groups shape analysis and computer animation.
  • Vis forfatter(e) 2016. Charakter Gruppen von Hopf-Algebren als unendlichdimensionale Lie Gruppen.
  • Vis forfatter(e) 2014. The Lie group of real analytic diffeomorphisms is not real analytic.
  • Vis forfatter(e) 2014. Differential geometry on orbifolds: A smooth path through singularities.
Doktorgradsavhandling
  • Vis forfatter(e) 2019. Applications of infinite-dimensional geometry and Lie theory Habilitationsschrift.
Sammendrag/abstract
  • Vis forfatter(e) 2017. Shape analysis on Lie groups and homogeneous manifolds. Proceedings of the GSI conference 2017.
Poster
  • Vis forfatter(e) 2016. Shape Analysis on Lie Groups (and beyond) with Applications.

Se fullstendig oversikt over publikasjoner i CRIStin.

Jeg er involvert i CODYSMA prosjekten av H. Munthe-Kaas.