Alexander SchmedingNord University | HIBO · FLU (Levanger Norway)
Alexander Schmeding
Dr. rer. nat.
About
73
Publications
3,648
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374
Citations
Citations since 2017
Introduction
My main research interests are infinite-dimensional (differential) geometry, global analysis and Lie theory.
In particular, this encompasses topologies on spaces of (smooth) functions and diffeomorphism groups.
Recently, I have become interested in aspects of higher category theory (in the form of Lie groupoids) and their connections to infinite-dimensional geometry. Further, Lie groups inspired by applications in numerical analysis and stochastic partial differential equations motivate my research.
(Almost) all of my preprints can be found on the arXiv in preprint form:
https://arxiv.org/a/schmeding_a_1.html
Additional affiliations
September 2019 - December 2020
September 2017 - August 2019
Position
- Wissenschaftlicher Mitarbeiter
Description
- Research and teaching in pure and applied mathematics
October 2013 - August 2017
Position
- Researcher
Description
- Research in pure and applied mathematics, teaching connected to the EU RISE MSCA project "Challenges in the preservation of structures", see https://www.ntnu.edu/imf/chips
Education
November 2019 - February 2021
April 2010 - August 2013
October 2005 - March 2011
Publications
Publications (73)
Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. Shapes are in fact unparametrized curves, evolving on a vector space, on a Lie group or on a manifold. On...
This paper is about the relation of the geometry of Lie groupoids over a
fixed compact manifold and the geometry of their (infinite-dimensional)
bisection Lie groups. In the first part of the paper we investigate the
relation of the bisections to a given Lie groupoid, where the second part is
about the construction of Lie groupoids from candidates...
In this article character groups of Hopf algebras are studied from the
perspective of infinite-dimensional Lie theory. For a graded and connected Hopf
algebra we construct an infinite-dimensional Lie group structure on the
character group. This structure turns the character group into a
Baker--Campbell--Hausdorff--Lie group which is regular in the...
Shape analysis methods have in the past few years become very popular, both
for theoretical exploration as well as from an application point of view.
Originally developed for planar curves, these methods have been expanded to
higher dimensional curves, surfaces, activities, character motions and many
other objects.
In this paper, we develop a frame...
The Butcher group is a powerful tool to analyse integration methods for
ordinary differential equations, in particular Runge--Kutta methods. In the
present paper, we complement the algebraic treatment of the Butcher group with
a natural infinite-dimensional Lie group structure. This structure turns the
Butcher group into a real analytic Baker--Camp...
The Nordic countries recently introduced programming as a new topic in school mathematics. Due to this reform, mathematics textbooks now include programming tasks. We propose in this article a framework for the analysis of programming tasks in mathematical textbooks. It seeks to classify programming tasks on the intersection of computational thinki...
In this short note, we identify the unit component of the Newman-Unti (NU) group in the fine very strong topology. In previous work, this component has been endowed with an infinite-dimensional Lie group structure, while the full NU-group does not support such a structure. The aim of the current comment is to answer a technical question which arriv...
Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Topics covered i...
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For α∈(1/3,1/2), we give criteria for when we can approximate Banach space-valued weakly geometric α-rough paths by signatures of curves of bounded variation, given some tuning of the Hölder parameter. We show that t...
In this chapter, we shall give an introduction to Euler–Arnold theory for partial differential equations (PDEs). The main idea of this theory is to reinterpret certain PDEs as smooth ordinary differential equations (ODEs) on infinite-dimensional manifolds. One advantage of this idea is that the usual solution theory for ODEs can be used to establis...
Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Topics covered i...
In this chapter, we study in detail the (weak) L^2-metric on spaces of smooth mappings. Its importance stems from the fact that this metric and its siblings, the Sobolev H^s -metrics are prevalent in shape analysis. It will be essential for us that geodesics with respect to the L^2-metric can explicitely be computed. Let us clarify what we mean her...
In this appendix, we give a short introduction to differential forms on infinite-dimensional manifolds. The main difference between the finite dimensional (or Banach) and our setting, is that it is in general impossible to interprete differential forms as (smooth) sections into certain bundles of linear forms. The reason for this is again that the...
Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Topics covered i...
This section contains some auxiliary results on topological vector spaces and locally convex spaces in particular. Note that for some of the results in this appendix, it is essential that we only consider Hausdorff topological vector spaces (which is the standing assumption in the present book). In some more specialised section, we will discuss the...
In this chapter, we consider spaces of differentiable mappings as infinite-dimensional spaces. These spaces will then serve as the model spaces for manifolds of mappings, i.e. manifolds of differentiable mappings between manifolds. The resulting manifolds will allow us to construct essential examples in later chapters, such as the diffeomorphism gr...
Introducing foundational concepts in infinite-dimensional differential geometry beyond Banach manifolds, this text is based on Bastiani calculus. It focuses on two main areas of infinite-dimensional geometry: infinite-dimensional Lie groups and weak Riemannian geometry, exploring their connections to manifolds of (smooth) mappings. Topics covered i...
In this chapter, one aim is to study spaces of mappings taking their values in a Lie group. It will turn out that these spaces carry again a natural Lie group structure. However, before we prove this, the definition and basic properties of (infinite dimensional) Lie groups and their associated Lie algebras are recalled. Infinite-dimensional Lie the...
This appendix sketches the construction of a canonical manifold of mappings structure for smooth mappings between (finite-dimensional) manifolds. Before we begin, let us consider for a moment the locally convex space of smooth functions from a manifold with values in a locally convex space. The topology and vector space structure allow us to compar...
In this chapter, we will discuss Riemannian metrics on infinite-dimensional spaces. Particular emphasis will be placed on the new challenges which arise on infinite-dimensional spaces. One new feature is that Riemannian metrics comes in several flavours on infinite-dimensional spaces. These are not present in the finite dimensional setting. The str...
In this chapter, we will discuss the (infinite-dimensional) geometric framework for rough paths and their signature. Rough path theory originated in the 1990s with the work of T. Lyons. It seeks to establish a theory of integrals and differential equations driven by rough signals. For example, one is interested in controlled ordinary differential e...
In this chapter, we will highlight the interesting connection between finite and infinite dimensional differential geometry. To this end, we shall consider Lie groupoids, which can be understood as elements from higher geometry. The moniker higher geometry stems from the fact that in the language of category theory, these objects form higher catego...
It is well known that multidimensional calculus, aka Fréchet calculus, carries over to the realm of Banach spaces and Banach manifolds. Banach spaces are often not sufficient for our purposes. To generalise derivatives, we will, as a minimum, need vector spaces with an amenable topology (which need not be induced by a norm). This chapter presents f...
We report on an exploratory study in which we used self-assessment and peer assessment in a mathematics class for pre-service middle-school teachers.
In this short note we identify the unit component of the Newman--Unti (NU) group in the fine very strong topology. In previous work, this component has been endowed with an infinite-dimensional Lie group structure, while the full NU-group does not support such a structure.
We study the Newman--Unti (NU) group from the viewpoint of infinite-dimensional geometry. The NU group is a topological group in a natural coarse topology, but it does not become a manifold and hence a Lie group in this topology. To obtain a manifold structure we consider a finer Whitney-type topology. This turns the unit component of the NU group...
In shape analysis, one of the fundamental problems is to align curves or surfaces before computing a (geodesic) distance between these shapes. To find the optimal reparametrization realizing this alignment is a computationally demanding task which leads to an optimization problem on the diffeomorphism group. In this paper, we construct approximatio...
We prove a topological decomposition of the space of meromorphic germs at zero in several variables with prescribed linear poles as a sum of spaces of holomorphic and polar germs. Evaluating the resulting holomorphic projection at zero gives rise to a continuous evaluator (at zero) on the space of meromorphic germs in several variables. Our constru...
Formal power series products appear in nonlinear control theory when systems modeled by Chen–Fliess series are interconnected to form new systems. In fields like adaptive control and learning systems, the coefficients of these formal power series are estimated sequentially with real-time data. The main goal is to prove the continuity and analyticit...
We study the Lie group structure of asymptotic symmetry groups in General Relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group...
We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric cas...
Given smooth manifolds $$M_1,\ldots , M_n$$ M 1 , … , M n (which may have a boundary or corners), a smooth manifold N modeled on locally convex spaces and $$\alpha \in ({{\mathbb {N}}}_0\cup \{\infty \})^n$$ α ∈ ( N 0 ∪ { ∞ } ) n , we consider the set $$C^\alpha (M_1\times \cdots \times M_n,N)$$ C α ( M 1 × ⋯ × M n , N ) of all mappings $$f:M_1\tim...
The present document is the draft of a book which presents an introduction to infinite-dimensional differential geometry beyond Banach manifolds. As is well known the usual calculus breaks down in this setting. Hence, we replace it by the more general Bastiani calculus which is built using directional derivatives. We then focus on two main areas of...
This chapter examines the interaction of algebra and geometry in the guise of Hopf algebras and certain associated character groups. The geometry mirrors the algebra in that equation becomes a Lie group anti-homomorphism. Furthermore, the geometric structure allows us to give intrinsic geometric meaning of certain constructions in numerical analysi...
We study the Newman--Unti (NU) group from the viewpoint of infinite-dimensional geometry. The NU group is a topological group in a natural coarse topology, but it does not become a manifold and hence a Lie group in this topology. To obtain a manifold structure we consider a finer Whitney-type topology. This turns the unit component of the NU group...
Given smooth manifolds $M_1,\ldots, M_n$ (which may have a boundary or corners), a smooth manifold $N$ modeled on locally convex spaces and $\alpha\in({\mathbb N}_0\cup\{\infty\})^n$, we consider the set $C^\alpha(M_1\times\cdots\times M_n,N)$ of all mappings $f\colon M_1\times\cdots\times M_n\to N$ which are $C^\alpha$ in the sense of Alzaareer. S...
We study the Lie group structure of asymptotic symmetry groups in General Relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group...
Formal power series products appear in nonlinear control theory when systems modeled by Chen-Fliess series are interconnected to form new systems. In fields like adaptive control and learning systems, the coefficients of these formal power series are estimated sequentially with real-time data. The main goal of the present article is to prove the co...
Similar to ordinary differential equations, rough paths and rough differential equations can be formulated in a Banach space setting. For $\alpha\in (1/3,1/2)$, we give criteria for when we can approximate Banach space-valued weakly geometric $\alpha$-rough paths by signatures of curves of bounded variation, given some tuning of the H\"older parame...
Model continuity plays an important role in applications like system identification, adaptive control, and machine learning. This paper provides sufficient conditions under which input-output systems represented by locally convergent Chen-Fliess series are jointly continuous with respect to their generating series and as operators mapping a ball in...
We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie–Butcher theory of Lie group integrators leads to global error estimates.
In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e., a gauge groupoid, the vertical bisections coinc...
We consider stochastic versions of Euler--Arnold equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden. For the Euler equation on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution (in the stochastic sense) in spaces of Sobolev mappings (of high...
The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self-maps. Then, representations of the Lie groupoids giv...
In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e. a gauge groupoid, the vertical bisections coinci...
Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that...
Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)...
We relate two notions of local error for integration schemes on Riemannian homogeneous spaces, and show how to derive global error estimates from such local bounds. In doing so, we prove for the first time that the Lie-Butcher theory of Lie group integrators leads to global error estimates.
The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of associated Lie groups. The groups envisaged here are the bisection group and a group of smooth groupoid self maps. It is known that in the topological cat...
This article shows that there is a continuous extension operator for compactly-supported smooth sections of vector bundles on possibly non-compact smooth manifolds, where the closed set to which sections are restricted can have outward polynomial cusps. These function spaces are only locally convex in general and so cannot use existing theory for F...
Character groups of Hopf algebras appear in a variety of mathematical contexts. For example, they arise in non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. A Hopf algebra is a structure that is simultaneously a (unital, associ...
Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. One computationally efficient approach to shape analysis is based on the Square Root Velocity Transform (...
Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)...
In this paper we are concerned with the approach to shape analysis based on the so called Square Root Velocity Transform (SRVT). We propose a generalisation of the SRVT from Euclidean spaces to shape spaces of curves on Lie groups and on homogeneous manifolds. The main idea behind our approach is to exploit the geometry of the natural Lie group act...
In this paper we are concerned with the approach to shape analysis based on the so called Square Root Velocity Transform (SRVT). We propose a generalisation of the SRVT from Euclidean spaces to shape spaces of curves on Lie groups and on homogeneous manifolds. The main idea behind our approach is to exploit the geometry of the natural Lie group act...
In this article we investigate a monoid of smooth mappings on the space of arrows of a Lie groupoid and its group of units. The group of units turns out to be an infinite-dimensional Lie group which is regular in the sense of Milnor. Furthermore, this group is closely connected to the group of bisections of the Lie groupoid. Under suitable conditio...
Character groups of Hopf algebras appear in a variety of mathematical contexts such as non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. In these applications, several species of "series expansions" can then be described as cha...
A classical result in Riemannian geometry states that the absolutely continuous curves into a (finite-dimensional) Riemannian manifold form an infinite-dimensional manifold. In the present paper this construction and related results are generalised to absolutely continuous curves with values in a strong Riemannian manifolds. As an application we co...
The Butcher group is a powerful tool to analyse integration methods for
ordinary differential equations, in particular Runge--Kutta methods. Recently,
a natural Lie group structure has been constructed for this group.
Unfortunately, the associated topology is too coarse for some applications in
numerical analysis. In the present paper, we propose t...
In this article groups of controlled characters of a combinatorial Hopf algebra are considered from the perspective of infinite-dimensional Lie theory. A character of a combinatorial Hopf algebra is controlled in our sense if it satisfies certain growth bounds, e.g. exponential growth. These groups appear in the guise of groups of locally convergen...
In this article we study two "strong" topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space. Namely, we construct Whitney type topologies for these spaces and a certain refinement corresponding to Michor's $\mathcal{FD}$-topology. Then we establis...
To a Lie groupoid over a compact base, the associated group of bisection is
an (infinite-dimensional) Lie group. Moreover, under certain circumstances one
can reconstruct the Lie groupoid from its Lie group of bisections. In the
present article we consider functorial aspects of these construction
principles. The first observation is that this proce...
We endow the diffeomorphism group DiffOrb(Q, U) of a paracompact (reduced) orbifold with the structure of an infinite-dimensional Lie group modeled on the space of compactly supported sections of the tangent orbibundle. For a second countable orbifold, we prove that DiffOrb(Q, U) is C0-regular, and thus regular in the sense of Milnor. Furthermore,...
We construct an infinite dimensional real analytic manifold structure for the
space of real analytic mappings from a compact manifold to a locally convex
manifold. Here a map is real analytic if it extends to a holomorphic map on
some neighbourhood of the complexification of its domain. As is well known the
construction turns the group of real anal...
Let M be a real analytic manifold modeled on a locally convex space and K be
a non-empty compact subset of M. We show that if an open neighborhood of K in M
admits a complexification which is a regular topological space, then the germ
of the latter (as a complex manifold) is uniquely determined. If M is regular
and the complexified modeling space o...
In this article we endow the group of bisections of a Lie groupoid with
compact base with a natural locally convex Lie group structure. Moreover, we
develop thoroughly the connection to the algebra of sections of the associated
Lie algebroid and show for a large class of Lie groupoids that their groups of
bisections are regular in the sense of Miln...
Orbifolds are a generalization of manifolds. They arise naturally in different areas of mathematics and physics, e.g.:
- Spaces of symplectic reduction are orbifolds,
- Orbifolds may be used to construct a conformal field theory model.
In "The diffeomorphism group of a non-compact orbifold", we considered the diffeomorphism group of a paracompac...
We develop differential calculus of C^{r,s}-mappings on products of locally
convex spaces and prove exponential laws for such mappings. As an application,
we consider differential equations in Banach spaces depending on a parameter in
a locally convex space. Under suitable assumptions, the associated flows are
mappings of class C^{r,s}.
We reconsider a classical theorem by Bican and El Bashir, which guarantees
the existence of non-trivial relatively pure submodules in a module category
over a ring with unit. Our aim is to generalize the theorem to module
categories over rings with several objects. As an application we then consider
the special case of alpha-pure objects in such mo...
Projects
Projects (2)
This is an ongoing book project for a book titled
"An introduction to infinite-dimensional geometry" (to be published by Cambridge University press).
The aim is to write an introduction to infinite-dimensional differential geometry. As this is a field too vast to completely cover in one book, we will concentrate on infinite-dimensional Lie groups, (weak) Riemannian geometry and applications of these topics.
Explore techniques for shape spaces in the presence of additional geometric information. Typically we are interested in methods for manifolds with geometric structures which can be exploited for shape analysis.
In the past we have developed methods for Lie groups and homogeneous spaces. Here the group action of the Lie group and tools from Lie theory played a key role in the development and analysis of numerical algorithms.
