arXiv subjects

Alexander Schmeding

Publications and source records attributed to Alexander Schmeding.

At least 19 recordsLinked to original sources

arXiv2026

Metric Geometry of the Signature Group for $p$-Variation Rough Paths

The signatures of $p$-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For $p \geq 1$, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot--Carath\'eodory geometries in the $p$-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Z\"ust from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.

math.MG↗

arXiv2026

Flows and Homotopies of Banach Lie algebroids

We establish flow and homotopy tools for Banach Lie algebroids that do not rely on compactly supported extensions. Under suitable hypotheses, we construct parameter-dependent local extensions of sections along algebroid paths, complete lifts and their fibrewise-linear evolutions, and a Bochner-integral variation-of-constants formula. We then prove that an admissible-path homotopy is equivalently a Lie algebroid morphism from the parameter square and a solution of the associated transport equation. This provides the extension, transport and infinitesimal variation theory required for a local integration program of Banach Lie algebroids to be developed in a sequel.

math.DG↗

arXiv2026

Optimization on Weak Riemannian Manifolds

Riemannian structures on infinite-dimensional manifolds arise naturally in shape analysis and shape optimization. These applications lead to optimization problems on manifolds which are not modeled on Banach spaces. The present article develops the basic framework for optimization via gradient descent on weak Riemannian manifolds leading to the notion of a Hesse manifold. Further, foundational properties for optimization are established for several classes of weak Riemannian manifolds connected to shape analysis and shape optimization.

math.OC↗

arXiv2025

Group Contractions via Infinite-Dimensional Lie Theory

Contractions are a procedure to construct a new Lie algebra out of a given one via a singular limit. Specifically, the \.{I}n\"on\"u--Wigner construction starts with a Lie algebra $\mathfrak{g}$ with Lie subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ and complement $\mathfrak{n}$. Then, the vectors in $\mathfrak{n}$ are rescaled by a formal parameter $\varepsilon \in \mathbb{R}_+$, which effectively turns the Lie bracket $[ \, \cdot \, , \cdot \, ]$ into an $\varepsilon$-dependent family $[ \, \cdot \, , \cdot \, ]_\varepsilon$. Notably, the limit $\varepsilon \to 0$ trivializes certain relations, such that the complement $\mathfrak{n}$ becomes an abelian ideal. In the present article, we are not only interested in the limiting Lie algebras and groups, but also in the corresponding power series expansions in $\varepsilon$ to understand their limiting behavior. Particularly, we are interested in the integration of the `power-series-expanded' Lie algebras to their corresponding Lie groups. To this end, we reformulate the above procedure using infinite-dimensional Lie algebras of analytic germs and then apply their integration theory. Our main results are a construction of the corresponding `Lie group expansions' in terms of quotients of groups of analytic germs and an explicit description of these groups in elementary terms. Applications of this procedure include the geometric Newtonian limit of General Relativity to Newton--Cartan gravity, where the Poincar\'e group is contracted to the Galilei group.

math-ph↗

arXiv2025

Cotangent Bundles as Coadjoint Orbits and Asymptotic Character Formulas

The present article presents geometric quantization on cotangent bundles as a special instance of Kirillov's orbit method. To this end, the cotangent bundle is realized as a coadjoint orbit of an infinite-dimensional Lie group constructed from the diffeomorphism group. We also develop two asymptotic character formulas by employing the tangent groupoid.

math.SG↗

arXiv2025

The Stacey-Roberts Lemma for Banach Manifolds

The Stacey-Roberts lemma states that a surjective submersion between finite-dimensional manifolds gives rise to a submersion on infinite-dimensional manifolds of smooth mappings by pushforward. This result is foundational for many constructions in infinite-dimensional differential geometry such as the construction of Lie groupoids of smooth mappings. We generalise the Stacey-Roberts lemma to Banach manifolds which admit smooth partitions of unity.The new approach also remedies an error in the original proof of the result for the purely finite-dimensional setting.

math.DG↗

arXiv2025

On the singularities of the exponential function of a semidirect product

We show that the Fréchet--Lie groups of the form $C^{\infty}(M)\rtimes \mathbb{R}$ resulting from smooth flows on compact manifolds $M$ fail to be locally exponential in several cases: when at least one non-periodic orbit is locally closed, or when the flow restricts to a linear one on an orbit closure diffeomorphic to a torus. As an application, we prove that the Bondi--Metzner--Sachs group of symmetries of an asymptotically flat spacetime is not locally exponential.

math.GR↗

arXiv2025

Boundary values of diffeomorphisms of simple polytopes, and controllability

We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup of face respecting diffeomorphisms and its Lie theoretic properties. We find a canonical Lie group structure for the quotient of the diffeomorphism by the subgroup Diff$^{\partial,id}(M)$ of maps that equal the identity on the boundary, turning the canonical quotient homomorphism Diff$(M)\rightarrow $Diff$(M)/$Diff$^{\partial,id}(M)$ into a smooth submersion. We also show that the identity component of the diffeomorphism group is generated by the exponential image, by proving general controllability results.

math.GR↗

arXiv2024

Decompositions of Nonlinear Input-Output Systems to Zero the Output

Consider an input-output system where the output is the tracking error given some desired reference signal. It is natural to consider under what conditions the problem has an exact solution, that is, the tracking error is exactly the zero function. If the system has a well defined relative degree and the zero function is in the range of the input-output map, then it is well known that the system is locally left invertible, and thus, the problem has a unique exact solution. A system will fail to have relative degree when more than one exact solution exists. The general goal of this paper is to describe a decomposition of an input-output system having a Chen-Fliess series representation into a parallel product of subsystems in order to identify possible solutions to the problem of zeroing the output. For computational purposes, the focus is on systems whose generating series are polynomials. It is shown that the shuffle algebra on the set of generating polynomials is a unique factorization domain so that any polynomial can be uniquely factored modulo a permutation into its irreducible elements for the purpose of identifying the subsystems in a parallel product decomposition. This is achieved using the fact that this shuffle algebra is isomorphic to the symmetric algebra over the vector space spanned by Lyndon words. A specific algorithm for factoring generating polynomials into its irreducible factors is presented based on the Chen-Fox-Lyndon factorization of words.

eess.SY↗

arXiv2024

Controllability and diffeomorphism groups on manifolds with boundary

In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.

math.DG↗

arXiv2024

Manifolds of continuous BV-functions and vector measure regularity of Banach-Lie groups

We construct a smooth Banach manifold BV$([a,b], M)$ whose elements are suitably-defined functions $f:[a,b] \rightarrow M$ of bounded variation with values in a smooth Banach manifold $M$ which admits a local addition. If the target manifold is a Banach-Lie group $G$, with Lie algebra $\mathfrak{g}$, we obtain a Banach-Lie group BV$([a,b], G)$ with Lie algebra BV$([a, b], \mathfrak{g})$. Strengthening known regularity properties of Banach-Lie groups, we construct a smooth evolution map from a Banach space of $\mathfrak{g}$-valued vector measures on $[0,1]$ to BV$([0,1],G)$.

math.FA↗

arXiv2024

A differentiable monoid of smooth maps on Lie groupoids

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 conditions, i.e. the source map of the Lie groupoid is proper, one also obtains a differentiable structure on the monoid and can identify the bisection group as a Lie subgroup of its group of units. Finally, relations between groupoids associated to the underlying Lie groupoid and subgroups of the monoid are obtained. The key tool driving the investigation is a generalisation of a result by A. Stacey which we establish in the present article. This result, called the Stacey-Roberts Lemma, asserts that pushforwards of submersions yield submersions between the infinite-dimensional manifolds of mappings.

math.GR↗

arXiv2023

The Ebin-Marsden toolbox for stochastic PDEs: stochastic Euler equations

The Ebin-Marsden theory is a powerful geometric framework for many PDEs from fluid dynamics. In this paper we provide a toolbox to apply the Ebin-Marsden approach to stochastic PDEs, combining tools from infinite-dimensional geometry and stochastic analysis. We showcase our approach in the context of incompressible Euler equation for an ideal fluid with additive noise. Among our main results there are: (i) local well-posedness of maximal solutions by using the Ebin-Marsden framework; (ii) a stochastic version of the celebrated no-loss-no-gain theorem.

math.PR↗

arXiv2023

A topological splitting of the space of meromorphic germs in several variables and continuous evaluators

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 constructions are carried out in the framework of Silva spaces and use an inner product on the underlying space of variables. They generalise to several variables, the topological direct decomposition of meromorphic germs at zero as sums of holomorphic and polar germs previously derived by the first and third author and provide a topological refinement of a known algebraic decomposition of such spaces previously derived by the second author and collaborators.

math.CV↗

arXiv2023

Deep neural networks on diffeomorphism groups for optimal shape reparameterization

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.

math.OC↗

arXiv2023

On the unit component of the Newman-Unti group

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.

math.GR↗

arXiv2022

Lie Theory for Asymptotic Symmetries in General Relativity: The NU Group

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 into an infinite-dimensional Lie group. We then study the Lie theoretic properties of this group. Surprisingly, the group operations of the full NU group become discontinuous, whence the NU group does not support a Lie group structure. The NU group contains the Bondi--Metzner--Sachs (BMS) group as a subgroup, whose Lie group structure was constructed in a previous article. It is well known that the NU Lie algebra splits into a direct sum of Lie ideals of the Lie algebras of the BMS group and conformal rescalings of scri. However, the lack of a Lie group structure on the NU group implies that the BMS group cannot be embedded as a Lie subgroup therein.

gr-qc↗

arXiv2022

An introduction to infinite-dimensional differential geometry

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 infinite-dimensional geometry: 1. infinite-dimensional Lie groups, and 2. weak Riemannian geometry. Both topics are developed and connected to manifolds of (smooth) mappings. These manifolds are studied in detail to construct important examples such as diffeomorphism groups, loop groups and Riemannian metrics for shape analysis. Manifolds of mappings are prime examples for surprising connections between finite and infinite-dimensional geometry. However, also pathologies occurring in infinite-dimensions will be highlighted in many examples. The geometric techniques developed will then be showcased in modern applications of geometry such as geometric hydrodynamics, higher geometry in the guise of Lie groupoids and rough path theory.

math.DG↗