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Added by
Michael R. Douglas
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Michael R. Douglas
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A very popular model in machine learning is the feedforward neural network (FFN). The FFN can approximate general functions and mitigate the curse of dimensionality. Here we introduce FFNs which represent sections of holomorphic line bundles on complex manifolds, and ask some questions about their approximating power. We also explain formal similarities between the standard approach to supervised learning and the problem of finding numerical Ricci flat Kahler metrics, which allow carrying some ideas between the two problems.
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This paper studies holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is a solution to a corresponding evolution problem. We then investigate the linearization problem: which semicocycles are cohomologous to constant semicocycles? In contrast with the case of commutative semicocycles, in the non-commutative case non-linearizable semicocycles are shown to exist. Simple conditions for linearizability are derived and are shown to be sharp.
Nirenberg and Spencer posed the question whether the germ of a compact complex submanifold in a complex manifold is determined by its infinitesimal neighborhood of finite order when the normal bundle is sufficiently positive. To study the problem for a larger class of submanifolds, including free rational curves, we reformulate the question in the setting of families of submanifolds and their infinitesimal neighborhoods. When the submanifolds have no nonzero vector fields, we prove that it is sufficient to consider only first-order neighborhoods to have an affirmative answer to the reformulated question. When the submanifolds do have nonzero vector fields, we obtain an affirmative answer to the question under the additional assumption that submanifolds have certain nice deformation properties, which is applicable to free rational curves. As an application, we obtain a stronger version of the Cartan-Fubini type extension theorem for Fano manifolds of Picard number 1. We also propose a potential application on hyperplane sections of projective K3 surfaces.
In this article we give an expository account of the holomorphic motion theorem based on work of M`a~ne-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have $|epsilon log epsilon|$ moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarzs lemma and integration over the holomorphic variable to produce Holder continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashis and Teichmullers metrics on the Teichmuller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions.
Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.
We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. We introduce Whitney sequences, a precise analogue (in higher dimensions) of the notion of finite Blaschke product (in one dimension). We show that terms in a Whitney sequence are homotopic to monomial mappings, and we establish an additional result about the target dimensions of such homotopies.
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