Manifolds of mappings on Cartesian products

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Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence of Lie groups G_n (all of which may be infinite-dimensional). We study the question when G is the direct limit of the G_n's in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtained for G the group Diff_c(M) of compactly supported smooth diffeomorphisms of a sigma-compact smooth manifold M, and for test function groups C^infty_c(M,H) of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.Comment: extended preprint version, 66 page

Deep learning of diffeomorphisms for optimal reparametrizations of shapes

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 approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms to solve the approximation problem. We propose a practical algorithm implemented in PyTorch which is applicable both to unparametrized curves and surfaces. We derive universal approximation results and obtain bounds for the Lipschitz constant of the obtained compositions of diffeomorphisms.Comment: 26 pages, 11 figures. Submitted to SIAM Journal of Scientific Computin

Overview of (pro-)Lie group structures on Hopf algebra character groups

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-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples: The Butcher group from numerical analysis and character groups which arise from the Connes--Kreimer theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on "New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai

Minimum Information about a Biosynthetic Gene cluster

A wide variety of enzymatic pathways that produce specialized metabolites in bacteria, fungi and plants are known to be encoded in biosynthetic gene clusters. Information about these clusters, pathways and metabolites is currently dispersed throughout the literature, making it difficult to exploit. To facilitate consistent and systematic deposition and retrieval of data on biosynthetic gene clusters, we propose the Minimum Information about a Biosynthetic Gene cluster (MIBiG) data standard.Chemistry and Chemical Biolog

Infinite-Dimensional Lie Groups Without Completeness Restrictions

We show that the differential calculus of smooth mappings between open subsets of locally convex spaces developed in [Hamilton, R., The inverse function theorem of Nash and Moser , Bull. Amer. Math. Soc. 7 (1982), 65-222] in the Fréchet case and in [Milnor, J., Remarks on infinite-dimensional Lie groups, in: DeWitt, B., and R. Stora (Eds.), \Relativity, Groups and Topology II," North-Holland, 1983] in the case of sequentially complete spaces, carries over without essential changes to the case of general, not necessarily sequentially complete, locally convex spaces. Our studies were dictated by the needs of infinite-dimensional Lie theory in the context of the existence problem of universal complexifications. We explain why satisfactory results in this area can only be obtained if the requirement of sequential completeness is abandoned

Discontinuous non-linear mappings on locally convex direct limits

We show that the self-map f: C ∞ c (R) → C ∞ c (R), f(γ): = γ ◦ γ − γ(0) of the space of real-valued test functions on the line is discontinuous, although its restriction to the space C ∞ K (R) of functions supported in K is smooth (and hence continuous), for each compact subset K ⊆ R. More generally, we construct mappings with analogous pathological properties on spaces of compactly supported smooth sections in vector bundles over non-compact bases. The results are useful in infinite-dimensional Lie theory, where they can be used to analyze the precise direct limit properties of test function groups and groups of compactly supported diffeomorphisms. Subject classification: 46F05, 46T20; 46A13, 46M40, 22E6

Implicit functions from topological vector spaces to Banach spaces

We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.Comment: 36 page