Alexander Schmeding

About

Mathematician working on infinite-dimensional differential geometry, Lie theory and their application. Beyond research in pure and applied mathematics I am also interested in mathematics didactics (especially connected to higher education and teacher education programs).

I am responsible for the coordination of the EVU activities at IMF.

Background

I obtained my Doktorgrad (the german equivalent of PhD) in infinite-dimensional Lie theory at the University of Paderborn, Germany in 2013. Before, I completed both a diploma in mathematics (nowadays equivalent to a master) and a teacher training program (mathematics and history) at the university of Paderborn. After that I held postdoctoral positions at NTNU, TU Berlin and was substitute associate professor at the University of Bergen. Before joining NTNU I was associate professor at Nord university in Levanger were I worked several years in teacher education.

Research

My research centers around infinite-dimensional differential geometry, Lie theory and their applications. Infinite-dimensional Lie groups arise naturally for example in the theory of hydrodynamics. The flow of a fluid particle (under periodic boundary conditions) can be modelled as a curve taking values in the group of volume preserving diffeomorphism on the torus. The resulting curve turns out to be connected to Riemannian geometry on the infinite-dimensional Lie group of volume preserving diffeomorphisms. Hence infinite-dimensional geometric tools and insights can be used to study flows in hydrodynamics. This example shows one of the subtle connections between finite and infinite-dimensional geometry I am interested in.

Beyond research in geometry, I am also interested in questions connected to the didactics of mathematics related to higher education at the university level and teacher education.

Research interests (mathematics)

Lie groups of diffeomorphisms
Lie groupoids and their representations
manifolds of mappings
(weak) Riemannian geometry on infinite-dimensional spaces
Interactions of differential geometry with rough path theory and Hopf algebras
Applications of the above (e.g. Geometric hydrodynamics and shape analysis)

Research interests (didactics of mathematics)

programming in mathematics textbooks for norwegian schools
peer- and self-evaluation activities in higher education

CODYSMA project

The project CODYSMA: Computational Dynamics and Stochastics on Manifolds is a five year project which started January 1st, 2020 supported by the Norwegian Research Council through the program FRIPRO.

Lie groups and manifolds of non-smooth mappings

We propose to construct and study Lie groups of paths of bounded variation with values in a Lie group. The driving motivation comes from the theory of rough paths. Rough path theory provides a fresh look at differential equations driven by rough signals.

Publications

An Introduction to Infinite-Dimensional Differential Geometry

Introducing foundational concepts in infinite-dimensional differential geometry, this book focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry. Available as Open Access.

Errata for "An Introduction to infinite-dimensional Differential Geometry"

Unfortunately, no publication without errors. Here is a list of all known errata for the book on infinite-dimensional geometry by CUP.

Almost all of my mathematical publications are available on the arXiv. I wrote a book containing an introduction to many concepts in infinite-dimensional geometry I am interested in. It is available in open access at the publishers website (check link below).

My publications in didactics in mathematics can be found as preprints on Researchgate.

2024

Fleischmann, Silvie Yael Girlani; Julien, Antoine Laurent Christophe; Schmeding, Alexander. (2024) Peer assessment in an undergraduate geometry course: Fostering proof competency in teacher students.
Academic chapter/article/Conference paper
Gray, W. Steven; Ebrahimi-Fard, Kurusch; Schmeding, Alexander. (2024) Decompositions of nonlinear input–output systems to zero the output. Systems & control letters (Print)
Academic article
Dahmen, Rafael; Paycha, Sylvie; Schmeding, Alexander. (2024) A topological splitting of the space of meromorphic germs in several variables and continuous evaluators. Complex Analysis and its Synergies
Academic article
Grong, Erlend; Schmeding, Alexander. (2024) Controllability and diffeomorphism groups on manifolds with boundary. Forum mathematicum
Academic article
Gray, W Steven; Pham, Natalie; Schmeding, Alexander. (2024) Distributional Generating Series in Nonlinear Control Theory. IEEE conference proceedings
Academic chapter/article/Conference paper

2023

Gray, W. Steven; Ebrahimi-Fard, Kurusch; Schmeding, Alexander. (2023) Zeroing the Output of Nonlinear Systems Without Relative Degree. IEEE conference proceedings
Academic chapter/article/Conference paper
Jensen, Mari Solberg; Julien, Antoine Laurent Christophe; Rafiepour, Abolfazl; Schmeding, Alexander. (2023) An analytical framework for programming tasks in mathematics textbooks. European Society for Research in Mathematics Education
Academic chapter/article/Conference paper
Jensen, Mari Solberg; Schmeding, Alexander. (2023) Jupyter notatbok i programmering. Tangenten - Tidsskrift for matematikkundervisning
Article in business/trade/industry journal
Schmeding, Alexander. (2023) On the unit component of the Newman-Unti group. Classical and Quantum Gravity
Academic article
Celledoni, Elena; Glöckner, Helge; Riseth, Jørgen Nilsen; Schmeding, Alexander. (2023) Deep neural networks on diffeomorphism groups for optimal shape reparametrization. BIT Numerical Mathematics
Academic article
Maurelli, Mario; Modin, Klas; Schmeding, Alexander. (2023) Incompressible Euler equations with stochastic forcing: A geometric approach. Stochastic Processes and their Applications
Academic article

2022

Prinz, David Nicolas; Schmeding, Alexander. (2022) Lie theory for asymptotic symmetries in general relativity: The BMS Group. Classical and Quantum Gravity
Academic article
Glöckner, Helge; Schmeding, Alexander. (2022) Manifolds of mappings on Cartesian products. Annals of Global Analysis and Geometry
Academic article
Gray, W. Steven; Palmstrøm, Mathias; Schmeding, Alexander. (2022) Continuity of Formal Power Series Products in Nonlinear Control Theory. Foundations of Computational Mathematics
Academic article
Julien, Antoine Laurent Christophe; Romijn, Elisabeth Inge; Schmeding, Alexander. (2022) Peer and self-assessment to improve mathematics competence in pre-service middle-school teachers. European Society for Research in Mathematics Education
Academic chapter/article/Conference paper
Prinz, David Nicolas; Schmeding, Alexander. (2022) Lie theory for asymptotic symmetries in general relativity: The NU group. Classical and Quantum Gravity
Academic article
Schmeding, Alexander. (2022) An Introduction to Infinite-Dimensional Differential Geometry. Cambridge University Press Cambridge Studies in Advanced Mathematics
Academic monograph
Grong, Erlend; Nilssen, Torstein; Schmeding, Alexander. (2022) Geometric rough paths on infinite dimensional spaces. Journal of Differential Equations
Academic article

2021

Dahmen, Rafael; Gray, W. Steven; Schmeding, Alexander. (2021) Continuity of Chen-Fliess Series for Applications in System Identification and Machine Learning. IFAC-PapersOnLine
Academic article
Roberts, David M.; Schmeding, Alexander. (2021) Extending Whitney's extension theorem: nonlinear function spaces. Annales de l'Institut Fourier
Academic article
Schmeding, Alexander. (2021) Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application. John Wiley & Sons
Academic chapter/article/Conference paper
Schmeding, Alexander. (2021) Applications of inﬁnite-dimensional geometry and Lie theory Habilitationsschrift. Technische Universität Berlin
Doctoral dissertation
Ebrahimi-Fard, Kurusch; Gray, W. Steven; Schmeding, Alexander. (2021) Universal Zero Dynamics: The SISO Case. IEEE (Institute of Electrical and Electronics Engineers) IEEE (Institute of Electrical and Electronics Engineers)
Report

2020

Schmeding, Alexander; Amiri, Habib; Glöckner, Helge. (2020) Lie groupoids of mappings taking values in a Lie groupoid. Archivum mathematicum
Academic article
Schmeding, Alexander; Dahmen, Rafael. (2020) Lie groups of controlled characters of combinatorial Hopf algebras. Annales de l’Institut Henri Poincaré D (AIHPD)
Academic article

2019

Schmeding, Alexander. (2019) The Lie group of vertical bisections of a regular Lie groupoid. Forum mathematicum
Academic article
Amiri, Habib; Schmeding, Alexander. (2019) A Differentiable Monoid of Smooth Maps on Lie Groupoids. Journal of Lie theory
Academic article
Curry, Charles Henry Alexander; Schmeding, Alexander. (2019) Convergence of Lie group integrators. Numerische Mathematik
Academic article

2018

Celledoni, Elena; Eidnes, Sølve; Schmeding, Alexander. (2018) Shape analysis on homogeneous spaces: a generalised SRVT framework. Abel Symposia
Academic article
Bogfjellmo, Geir; Schmeding, Alexander. (2018) The geometry of characters of hopf algebras. Abel Symposia
Academic article

2017

Celledoni, Elena; Eidnes, Sølve; Eslitzbichler, Markus; Schmeding, Alexander. (2017) Shape analysis on Lie groups and homogeneous manifolds. Proceedings of the GSI conference 2017
Abstract
Schmeding, Alexander; Hjelle, Eivind Otto. (2017) Strong topologies for spaces of smooth maps with infinite-dimensional target. Expositiones mathematicae
Academic article
Celledoni, Elena; Eidnes, Sølve; Eslitzbichler, Markus; Schmeding, Alexander. (2017) Shape analysis on lie groups and homogeneous spaces. Lecture Notes in Computer Science (LNCS)
Academic article
Bogfjellmo, Geir; Schmeding, Alexander. (2017) The Lie Group Structure of the Butcher Group. Foundations of Computational Mathematics
Academic article

2016

Bogfjellmo, Geir; Dahmen, Rafael; Schmeding, Alexander. (2016) Character groups of Hopf algebras as infinite-dimensional Lie groups. Annales de l'Institut Fourier
Academic article
Schmeding, Alexander; Wockel, Christoph. (2016) Functorial aspects of the reconstruction of Lie groupoids from their bisections. Journal of the Australian Mathematical Society
Academic article
Bogfjellmo, Geir; Schmeding, Alexander. (2016) The tame Butcher group. Journal of Lie theory
Academic article
Celledoni, Elena; Eslitzbichler, Markus; Schmeding, Alexander. (2016) Shape analysis on Lie groups with application in computer animation. Journal of Geometric Mechanics (JGM)
Academic article
Schmeding, Alexander; Wockel, Christoph. (2016) (Re)constructing Lie groupoids from their bisections and applications to prequantisation. Differential geometry and its applications
Academic article

2015

Dahmen, Rafael; Schmeding, Alexander. (2015) The Lie group of real analytic diffeomorphisms is not real analytic. Studia Mathematica
Academic article
Schmeding, Alexander. (2015) The diffeomorphism group of a non-compact orbifold. Dissertationes Mathematicae
Academic article
Schmeding, Alexander; Alzaareer, Hamza. (2015) Differentiable mappings on products with different degrees of differentiability in the two factors. Expositiones mathematicae
Academic article
Schmeding, Alexander; Wockel, Christoph. (2015) The Lie group of bisections of a Lie groupoid. Annals of Global Analysis and Geometry
Academic article

Journal publications

Gray, W. Steven; Ebrahimi-Fard, Kurusch; Schmeding, Alexander. (2024) Decompositions of nonlinear input–output systems to zero the output. Systems & control letters (Print)
Academic article
Dahmen, Rafael; Paycha, Sylvie; Schmeding, Alexander. (2024) A topological splitting of the space of meromorphic germs in several variables and continuous evaluators. Complex Analysis and its Synergies
Academic article
Grong, Erlend; Schmeding, Alexander. (2024) Controllability and diffeomorphism groups on manifolds with boundary. Forum mathematicum
Academic article
Jensen, Mari Solberg; Schmeding, Alexander. (2023) Jupyter notatbok i programmering. Tangenten - Tidsskrift for matematikkundervisning
Article in business/trade/industry journal
Schmeding, Alexander. (2023) On the unit component of the Newman-Unti group. Classical and Quantum Gravity
Academic article
Celledoni, Elena; Glöckner, Helge; Riseth, Jørgen Nilsen; Schmeding, Alexander. (2023) Deep neural networks on diffeomorphism groups for optimal shape reparametrization. BIT Numerical Mathematics
Academic article
Maurelli, Mario; Modin, Klas; Schmeding, Alexander. (2023) Incompressible Euler equations with stochastic forcing: A geometric approach. Stochastic Processes and their Applications
Academic article
Prinz, David Nicolas; Schmeding, Alexander. (2022) Lie theory for asymptotic symmetries in general relativity: The BMS Group. Classical and Quantum Gravity
Academic article
Glöckner, Helge; Schmeding, Alexander. (2022) Manifolds of mappings on Cartesian products. Annals of Global Analysis and Geometry
Academic article
Gray, W. Steven; Palmstrøm, Mathias; Schmeding, Alexander. (2022) Continuity of Formal Power Series Products in Nonlinear Control Theory. Foundations of Computational Mathematics
Academic article
Prinz, David Nicolas; Schmeding, Alexander. (2022) Lie theory for asymptotic symmetries in general relativity: The NU group. Classical and Quantum Gravity
Academic article
Grong, Erlend; Nilssen, Torstein; Schmeding, Alexander. (2022) Geometric rough paths on infinite dimensional spaces. Journal of Differential Equations
Academic article
Dahmen, Rafael; Gray, W. Steven; Schmeding, Alexander. (2021) Continuity of Chen-Fliess Series for Applications in System Identification and Machine Learning. IFAC-PapersOnLine
Academic article
Roberts, David M.; Schmeding, Alexander. (2021) Extending Whitney's extension theorem: nonlinear function spaces. Annales de l'Institut Fourier
Academic article
Schmeding, Alexander; Amiri, Habib; Glöckner, Helge. (2020) Lie groupoids of mappings taking values in a Lie groupoid. Archivum mathematicum
Academic article
Schmeding, Alexander; Dahmen, Rafael. (2020) Lie groups of controlled characters of combinatorial Hopf algebras. Annales de l’Institut Henri Poincaré D (AIHPD)
Academic article
Schmeding, Alexander. (2019) The Lie group of vertical bisections of a regular Lie groupoid. Forum mathematicum
Academic article
Amiri, Habib; Schmeding, Alexander. (2019) A Differentiable Monoid of Smooth Maps on Lie Groupoids. Journal of Lie theory
Academic article
Curry, Charles Henry Alexander; Schmeding, Alexander. (2019) Convergence of Lie group integrators. Numerische Mathematik
Academic article
Celledoni, Elena; Eidnes, Sølve; Schmeding, Alexander. (2018) Shape analysis on homogeneous spaces: a generalised SRVT framework. Abel Symposia
Academic article
Bogfjellmo, Geir; Schmeding, Alexander. (2018) The geometry of characters of hopf algebras. Abel Symposia
Academic article
Celledoni, Elena; Eidnes, Sølve; Eslitzbichler, Markus; Schmeding, Alexander. (2017) Shape analysis on Lie groups and homogeneous manifolds. Proceedings of the GSI conference 2017
Abstract
Schmeding, Alexander; Hjelle, Eivind Otto. (2017) Strong topologies for spaces of smooth maps with infinite-dimensional target. Expositiones mathematicae
Academic article
Celledoni, Elena; Eidnes, Sølve; Eslitzbichler, Markus; Schmeding, Alexander. (2017) Shape analysis on lie groups and homogeneous spaces. Lecture Notes in Computer Science (LNCS)
Academic article
Bogfjellmo, Geir; Schmeding, Alexander. (2017) The Lie Group Structure of the Butcher Group. Foundations of Computational Mathematics
Academic article
Bogfjellmo, Geir; Dahmen, Rafael; Schmeding, Alexander. (2016) Character groups of Hopf algebras as infinite-dimensional Lie groups. Annales de l'Institut Fourier
Academic article
Schmeding, Alexander; Wockel, Christoph. (2016) Functorial aspects of the reconstruction of Lie groupoids from their bisections. Journal of the Australian Mathematical Society
Academic article
Bogfjellmo, Geir; Schmeding, Alexander. (2016) The tame Butcher group. Journal of Lie theory
Academic article
Celledoni, Elena; Eslitzbichler, Markus; Schmeding, Alexander. (2016) Shape analysis on Lie groups with application in computer animation. Journal of Geometric Mechanics (JGM)
Academic article
Schmeding, Alexander; Wockel, Christoph. (2016) (Re)constructing Lie groupoids from their bisections and applications to prequantisation. Differential geometry and its applications
Academic article
Dahmen, Rafael; Schmeding, Alexander. (2015) The Lie group of real analytic diffeomorphisms is not real analytic. Studia Mathematica
Academic article
Schmeding, Alexander. (2015) The diffeomorphism group of a non-compact orbifold. Dissertationes Mathematicae
Academic article
Schmeding, Alexander; Alzaareer, Hamza. (2015) Differentiable mappings on products with different degrees of differentiability in the two factors. Expositiones mathematicae
Academic article
Schmeding, Alexander; Wockel, Christoph. (2015) The Lie group of bisections of a Lie groupoid. Annals of Global Analysis and Geometry
Academic article

Books

Schmeding, Alexander. (2022) An Introduction to Infinite-Dimensional Differential Geometry. Cambridge University Press Cambridge Studies in Advanced Mathematics
Academic monograph

Part of book/report

Fleischmann, Silvie Yael Girlani; Julien, Antoine Laurent Christophe; Schmeding, Alexander. (2024) Peer assessment in an undergraduate geometry course: Fostering proof competency in teacher students.
Academic chapter/article/Conference paper
Gray, W Steven; Pham, Natalie; Schmeding, Alexander. (2024) Distributional Generating Series in Nonlinear Control Theory. IEEE conference proceedings
Academic chapter/article/Conference paper
Gray, W. Steven; Ebrahimi-Fard, Kurusch; Schmeding, Alexander. (2023) Zeroing the Output of Nonlinear Systems Without Relative Degree. IEEE conference proceedings
Academic chapter/article/Conference paper
Jensen, Mari Solberg; Julien, Antoine Laurent Christophe; Rafiepour, Abolfazl; Schmeding, Alexander. (2023) An analytical framework for programming tasks in mathematics textbooks. European Society for Research in Mathematics Education
Academic chapter/article/Conference paper
Julien, Antoine Laurent Christophe; Romijn, Elisabeth Inge; Schmeding, Alexander. (2022) Peer and self-assessment to improve mathematics competence in pre-service middle-school teachers. European Society for Research in Mathematics Education
Academic chapter/article/Conference paper
Schmeding, Alexander. (2021) Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application. John Wiley & Sons
Academic chapter/article/Conference paper

Report

Schmeding, Alexander. (2021) Applications of inﬁnite-dimensional geometry and Lie theory Habilitationsschrift. Technische Universität Berlin
Doctoral dissertation
Ebrahimi-Fard, Kurusch; Gray, W. Steven; Schmeding, Alexander. (2021) Universal Zero Dynamics: The SISO Case. IEEE (Institute of Electrical and Electronics Engineers) IEEE (Institute of Electrical and Electronics Engineers)
Report

Teaching

Spring 2024

TMA4115 - Matematikk 3

Fall 2024

MA3407 - Introduction to Lie theory

MA3001 - Introduction to Riemannian geometry

Supervision

If you are a student interested in writing a thesis or a project under my supervision, please contact me. I keep a list of suitable problems for such projects connected to my research interests (check Research page).

Outreach