A $C^k$-seeley-extension-theorem for Bastiani’s differential calculus - Hanusch, Maximilian (Canadian Journal of Mathematics)
Keywords: extension of differentiable maps

The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups - Hanusch, Maximilian (Annals of Global Analysis and Geometry)
Keywords: Lax equation; generalized Baker-Campbell-Dynkin-Hausdorff formula; regularity of Lie groups

Decompositions of Analytic 1-Manifolds - Hanusch, Maximilian (Indagationes Mathematicae.)
Keywords: Lie group actions and analytic 1-submanifolds

Deep neural networks on diffeomorphism groups for optimal shape reparametrization - Celledoni, Elena; Glöckner, Helge; Riseth, Jørgen; Schmeding, Alexander (BIT Numerical Mathematics)

Aspects of control theory on infinite-dimensional Lie groups and G-manifolds - Glöckner, Helge; Hilgert, Joachim (Journal of Differential Equations)
Keywords: 22E65; 28B05; 34A12; 34H05; 46E30; 46E40

Diffeomorphism groups of convex polytopes - Glöckner, Helge (Journal of Convex Analysis)
Let $E$ be a finite-dimensional real vector space and $M\subseteq E$ be a convex polytope with non-empty interior. We turn the group of all $C^∞$-diffeomorphisms of $M$ into a regular Lie group.

Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groups - Glöckner, Helge; Tárrega, Luis (Journal of Lie Theory)

Non-Lie subgroups in Lie groups over local fields of positive characteristic - Glöckner, Helge (p-Adic Numbers, Ultrametric Analysis, and Applications)
Keywords: 20Exx; 22Exx; 32Cxx

Manifolds of mappings on Cartesian products - Glöckner, Helge; Schmeding, Alexander (Annals of Global Analysis and Geometry)
Keywords: 58D15; 22E65; 26E15; 26E20; 46E40; 46T20; 58A05

Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces - Glöckner, Helge (Axioms)
We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter kR-spaces and locally convex spaces E such that E×E is a kR-space.

Birkhoff decompositions for loop groups with coefficient algebras - Glöckner, Helge (arXiv:2206.11711)
Starting with a finite-dimensional complex Lie algebra, we extend scalars using suitable commutative topological algebras. We study Birkhoff decompositions for the corresponding loop groups. Some results remain valid for loop groups with valued in complex Banach-Lie groups.

Regularity of Lie groups - Hanusch, Maximilian (Communications in Analysis and Geometry)
Keywords: regularity of Lie groups

Analysis 1 und 2 Skript/Buch - Hanusch, Maximilian

Decompositions of locally compact contraction groups, series and extensions - Glöckner, Helge; Willis, George A. (Journal of Algebra)
A locally compact contraction group is a pair (G,α), where G is a locally compact group and α:G→G an automorphism such that αn(x)→e pointwise as n→∞. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups (G,α) which are central extensions0→Fp((t))→G→Fp((t))→0 of the additive group of the field of formal Laurent series over Fp=Z/pZ by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.
Keywords: Contraction group; Torsion group; Extension; Cocycle; Section; Equivariant cohomology; Abelian group; Nilpotent group; Isomorphism types

Locally pro-p contraction groups are nilpotent - Glöckner, Helge; Willis, George A. (J. Reine Angew. Math.)
Keywords: 22D05; 22A05; 20E18

Direct limits of regular Lie groups - Glöckner, Helge (Mathematische Nachrichten)

Contraction groups and the big cell for endomorphisms of Lie groups over local fields - Glöckner, Helge (arXiv:2101.02981)
Let $G$ be a Lie group over a totally disconnected local field and $α$ be an analytic endomorphism of $G$. The contraction group of $α$ ist the set of all $x\in G$ such that $α^n(x)→ e$ as $n→∞$. Call sequence $(x_{-n})_{n≥ 0}$ in $G$ an $α$-regressive trajectory for $x\in G$ if $α(x_{-n})=x_{-n+1}$ for all $n≥ 1$ and $x_0=x$. The anti-contraction group of $α$ is the set of all $x\in G$ admitting an $α$-regressive trajectory $(x_{-n})_{n≥ 0}$ such that $x_{-n}→ e$ as $n→∞$. The Levi subgroup is the set of all $x\in G$ whose $α$-orbit is relatively compact, and such that $x$ admits an $α$-regressive trajectory $(x_{-n})_{n≥ 0}$ such that $\{x_{-n}\colon n≥ 0\}$ is relatively compact. The big cell associated to $α$ is the set $Ω$ of all all products $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and $z$ in the anti-contraction group. Let $π$ be the mapping from the cartesian product of the contraction group, Levi subgroup and anti-contraction group to $Ω$ which maps $(x,y,z)$ to $xyz$. We show: $Ω$ is open in $G$ and $π$ is étale for suitable immersed Lie subgroup structures on the three subgroups just mentioned. Moreover, we study group-theoretic properties of contraction groups and anti-contraction groups.

Symmetries of analytic curves - Hanusch, Maximilian (Differential Geometry and its Applications)
Keywords: Geometry and Topology; Analysis

Lie groupoids of mappings taking values in a Lie groupoid - Amiri, Habib; Glöckner, Helge; Schmeding, Alexander (Archivum Mathematicum)
Keywords: 22A22; 22E65; 22E67; 46T10; 47H30; 58D15; 58H05

Products of regular locally compact spaces are k_R-spaces - Glöckner, Helge; Masbough, Niku (Topology Proceedings)
Keywords: 54B10; 54D45; 54D50

Smoothing operators for vector-valued functions and extension operators - Glöckner, Helge (arXiv:2006.00254)
For suitable finite-dimensional smooth manifolds M (possibly with various kinds of boundary or corners), locally convex topological vector spaces F and non-negative integers k, we construct continuous linear operators S_n from the space of F-valued k times continuously differentiable functions on M to the corresponding space of smooth functions such that S_n(f) converges to f in C^k(M,F) as n tends to infinity, uniformly for f in compact subsets of C^k(M,F). We also study the existence of continuous linear right inverses for restriction maps from C^k(M,F) to C^k(L,F) if L is a closed subset of M, endowed with a C^k-manifold structure turning the inclusion map from L to M into a C^k-map. Moreover, we construct continuous linear right inverses for restriction operators between spaces of sections in vector bundles in many situations, and smooth local right inverses for restriction operators between manifolds of mappings. We also obtain smoothing results for sections in fibre bundles.

Lie groups of real analytic diffeomorphisms are L^1-regular - Glöckner, Helge (arXiv:2007.15611)
Let $M$ be a compact, real analytic manifold and $G$ be the Lie group of all real-analytic diffeomorphisms of $M$, which is modelled on the (DFS)-space ${\mathfrak g}$ of real-analytic vector fields on $M$. We study flows of time-dependent real-analytic vector fields on $M$ which are integrable functions in time, and their dependence on the time-dependent vector field. Notably, we show that the Lie group $G$ is $L^1$-regular in the sense that each $[\gamma]$ in $L^1([0,1],{\mathfrak g})$ has an evolution which is an absolutely continuous $G$-valued function on $[0,1]$ and smooth in $[\gamma]$. As tools for the proof, we develop several new results concerning $L^p$-regularity of infinite-dimensional Lie groups, for $1\leq p\leq \infty$, which will be useful also for the discussion of other classes of groups. Moreover, we obtain new results concerning the continuity and complex analyticity of non-linear mappings on open subsets of locally convex direct limits.

The regularity problem for Lie groups with asymptotic estimate Lie algebras - Hanusch, Maximilian (Indagationes Mathematicae)
Keywords: regularity of Lie groups

The Strong Trotter Property for Locally μ-convex Lie Groups - Hanusch, Maximilian (Journal of Lie Theory)
Keywords: Lie theory; strong Trotter property

Differentiability of the evolution map and Mackey continuity - Hanusch, Maximilian (Forum Mathematicum)
Keywords: regularity of Lie groups; differentiability of the evolution map