Conrad Plaut - Academia.edu (original) (raw)
Papers by Conrad Plaut
Journal of Geometry, 2002
. We show how to construct the group using any sequence of Hadamard matrices. This construction... more . We show how to construct the group using any sequence of Hadamard matrices. This construction is nicely compatible with the classical Haar and Rademacher functions. We then show that every n-dimensional Euclidean lattice is isometrically isomorphic to a n-slice of . Finally we prove a similar embedding theorem for integral and p-rational lattices into the-module of all continuous integer-valued functions on the group of p-adic integers.
Journal of Topology and Analysis, 2016
We give various applications of essential circles (introduced in [16]) in a compact geodesic spac... more We give various applications of essential circles (introduced in [16]) in a compact geodesic space X. Essential circles completely determine the homotopy critical spectrum of X, which we show is precisely 2 3 the covering spectrum of Sormani-Wei. We use finite collections of essential circles to define "circle covers", which extend and contain as special cases the δcovers of Sormani and Wei (equivalently the ε-covers of [16]); the constructions are metric adaptations of those utilized by Berestovskii-Plaut in the construction of entourage covers of uniform spaces. We show that, unlike δ-and ε-covers, circle covers are in a sense closed with respect to Gromov-Hausdorff convergence, and we prove a finiteness theorem concerning their deck groups that does not hold for covering maps in general. This allows us to completely understand the structure of Gromov-Hausdorff limits of δ-covers. Also, we use essential circles to strengthen a theorem of E. Cartan by finding a new (even for compact Riemannian manifolds) finite set of generators of the fundamental group of a semilocally simply connected compact geodesic space. We conjecture that there is always a generating set of this sort having minimal cardinality among all generating sets.
We present a covering group theory (with a generalized notion of cover) for the category of compa... more We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism : G → G in this category. The abelian topological group K 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems.
Fundamenta Mathematicae, 2014
Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we de... more Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3 2 . The latter two spectra are known to be discrete for compact geodesic spaces, and correspond to the values at which certain special covering maps, called δ-covers (Sormani-Wei) or ε-covers (Plaut-Wilkins), change equivalence type. In this paper we initiate the study of these ideas for non-geodesic spaces, motivated by the need to understand the extent to which the accompanying covering maps are topological invariants. We show that discreteness of the critical spectrum for general metric spaces can fail in several ways, which we classify. The "newcomer" critical values for compact, non-geodesic spaces are completely determined by the homotopy critical values and refinement critical values, the latter of which can, in many cases, be removed by changing the metric in a bi-Lipschitz way.
Topology and its Applications, 2001
We develop a covering group theory for a large category of "coverable" topological groups, with a... more We develop a covering group theory for a large category of "coverable" topological groups, with a generalized notion of "cover". Coverable groups include, for example, all metrizable, connected, locally connected groups, and even many totally disconnected groups. Our covering group theory produces a categorial notion of fundamental group, which, in contrast to traditional theory, is naturally a (prodiscrete) topological group. Central to our work is a link between the fundamental group and global extension properties of local group homomorphisms. We provide methods for computing the fundamental group of inverse limits and dense subgroups or completions of coverable groups. Our theory includes as special cases the traditional theory of Poincaré, as well as alternative theories due to Chevalley, Tits, and Hoffmann-Morris. We include a number of examples and open problems.
Topology and its Applications, 2001
We present a covering group theory (with a generalized notion of cover) for the category of compa... more We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism : G → G in this category. The abelian topological group K 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems.
Topology and its Applications, 2007
Geometriae Dedicata, 2001
The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spac... more The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (¢nite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.
Journal of Pure and Applied Algebra, 2001
We present a covering group theory (with a generalized notion of cover) for the category of compa... more We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism : G → G in this category. The abelian topological group K 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems.
Journal of Geometric Analysis, 1999
ABSTRACT We prove that every locally connected quotient G/H of a locally compact, connected, firs... more ABSTRACT We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces.
Advances in Mathematics, 2013
We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diam... more We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g1, ..., g k of length at most 2D and relators of the form gigm = gj . In particular, we obtain an explicit bound for the number k of generators in terms of the number "short loops" at every point and the number of balls required to cover a given semi-locally simply connected geodesic space. As a consequence we obtain a fundamental group finiteness theorem (new even for Riemannian manifolds) that implies the fundamental group finiteness theorems of Anderson and Shen-Wei. Our theorem requires no curvature bounds, nor lower bounds on volume or 1-systole. We use the method of discrete homotopies introduced by the first author and V. N. Berestovskii. Central to the proof is the notion of the "homotopy critical spectrum" that is closely related to the covering and length spectra. Discrete methods also allow us to strengthen and simplify the proofs of some results of Sormani-Wei about the covering spectrum.
Journal of Mathematical Biology, 2007
Endothelial cell adhesion and barrier function play a critical role in many biological and pathop... more Endothelial cell adhesion and barrier function play a critical role in many biological and pathophysiological processes. The decomposition of endothelial cell adhesion and barrier function into cell-cell and cell-matrix components using frequency dependent cellular micro-impedance measurements has, therefore, received widespread application. Few if any studies, however, have examined the precision of these model parameters. This study presents a parameter sensitivity analysis of a representative cellular barrier function model using a concise geometric formulation that includes instrumental data acquisition settings. Both model state dependence and instrumental noise distributions are accounted for within the framework of Riemannian manifold theory. Experimentally acquired microimpedance measurements of attached endothelial cells define the model state domain, while experimentally measured noise statistics define the data space Riemannian metric based on the Fisher information matrix. The results of this analysis show that the sensitivity of cell-cell and cell-matrix impedance components are highly model state dependent and several 123 722 A. E. English et al.
Journal of Geometry, 2002
. We show how to construct the group using any sequence of Hadamard matrices. This construction... more . We show how to construct the group using any sequence of Hadamard matrices. This construction is nicely compatible with the classical Haar and Rademacher functions. We then show that every n-dimensional Euclidean lattice is isometrically isomorphic to a n-slice of . Finally we prove a similar embedding theorem for integral and p-rational lattices into the-module of all continuous integer-valued functions on the group of p-adic integers.
Journal of Topology and Analysis, 2016
We give various applications of essential circles (introduced in [16]) in a compact geodesic spac... more We give various applications of essential circles (introduced in [16]) in a compact geodesic space X. Essential circles completely determine the homotopy critical spectrum of X, which we show is precisely 2 3 the covering spectrum of Sormani-Wei. We use finite collections of essential circles to define "circle covers", which extend and contain as special cases the δcovers of Sormani and Wei (equivalently the ε-covers of [16]); the constructions are metric adaptations of those utilized by Berestovskii-Plaut in the construction of entourage covers of uniform spaces. We show that, unlike δ-and ε-covers, circle covers are in a sense closed with respect to Gromov-Hausdorff convergence, and we prove a finiteness theorem concerning their deck groups that does not hold for covering maps in general. This allows us to completely understand the structure of Gromov-Hausdorff limits of δ-covers. Also, we use essential circles to strengthen a theorem of E. Cartan by finding a new (even for compact Riemannian manifolds) finite set of generators of the fundamental group of a semilocally simply connected compact geodesic space. We conjecture that there is always a generating set of this sort having minimal cardinality among all generating sets.
We present a covering group theory (with a generalized notion of cover) for the category of compa... more We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism : G → G in this category. The abelian topological group K 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems.
Fundamenta Mathematicae, 2014
Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we de... more Utilizing the discrete homotopy methods developed for uniform spaces by Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space, generalizing to the non-geodesic case the covering spectrum defined by Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs from the covering spectrum by a factor of 3 2 . The latter two spectra are known to be discrete for compact geodesic spaces, and correspond to the values at which certain special covering maps, called δ-covers (Sormani-Wei) or ε-covers (Plaut-Wilkins), change equivalence type. In this paper we initiate the study of these ideas for non-geodesic spaces, motivated by the need to understand the extent to which the accompanying covering maps are topological invariants. We show that discreteness of the critical spectrum for general metric spaces can fail in several ways, which we classify. The "newcomer" critical values for compact, non-geodesic spaces are completely determined by the homotopy critical values and refinement critical values, the latter of which can, in many cases, be removed by changing the metric in a bi-Lipschitz way.
Topology and its Applications, 2001
We develop a covering group theory for a large category of "coverable" topological groups, with a... more We develop a covering group theory for a large category of "coverable" topological groups, with a generalized notion of "cover". Coverable groups include, for example, all metrizable, connected, locally connected groups, and even many totally disconnected groups. Our covering group theory produces a categorial notion of fundamental group, which, in contrast to traditional theory, is naturally a (prodiscrete) topological group. Central to our work is a link between the fundamental group and global extension properties of local group homomorphisms. We provide methods for computing the fundamental group of inverse limits and dense subgroups or completions of coverable groups. Our theory includes as special cases the traditional theory of Poincaré, as well as alternative theories due to Chevalley, Tits, and Hoffmann-Morris. We include a number of examples and open problems.
Topology and its Applications, 2001
We present a covering group theory (with a generalized notion of cover) for the category of compa... more We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism : G → G in this category. The abelian topological group K 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems.
Topology and its Applications, 2007
Geometriae Dedicata, 2001
The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spac... more The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (¢nite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.
Journal of Pure and Applied Algebra, 2001
We present a covering group theory (with a generalized notion of cover) for the category of compa... more We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism : G → G in this category. The abelian topological group K 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems.
Journal of Geometric Analysis, 1999
ABSTRACT We prove that every locally connected quotient G/H of a locally compact, connected, firs... more ABSTRACT We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry of these spaces, in particular of non-negatively curved homogeneous spaces.
Advances in Mathematics, 2013
We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diam... more We generalize and strengthen the theorem of Gromov that every compact Riemannian manifold of diameter at most D has a set of generators g1, ..., g k of length at most 2D and relators of the form gigm = gj . In particular, we obtain an explicit bound for the number k of generators in terms of the number "short loops" at every point and the number of balls required to cover a given semi-locally simply connected geodesic space. As a consequence we obtain a fundamental group finiteness theorem (new even for Riemannian manifolds) that implies the fundamental group finiteness theorems of Anderson and Shen-Wei. Our theorem requires no curvature bounds, nor lower bounds on volume or 1-systole. We use the method of discrete homotopies introduced by the first author and V. N. Berestovskii. Central to the proof is the notion of the "homotopy critical spectrum" that is closely related to the covering and length spectra. Discrete methods also allow us to strengthen and simplify the proofs of some results of Sormani-Wei about the covering spectrum.
Journal of Mathematical Biology, 2007
Endothelial cell adhesion and barrier function play a critical role in many biological and pathop... more Endothelial cell adhesion and barrier function play a critical role in many biological and pathophysiological processes. The decomposition of endothelial cell adhesion and barrier function into cell-cell and cell-matrix components using frequency dependent cellular micro-impedance measurements has, therefore, received widespread application. Few if any studies, however, have examined the precision of these model parameters. This study presents a parameter sensitivity analysis of a representative cellular barrier function model using a concise geometric formulation that includes instrumental data acquisition settings. Both model state dependence and instrumental noise distributions are accounted for within the framework of Riemannian manifold theory. Experimentally acquired microimpedance measurements of attached endothelial cells define the model state domain, while experimentally measured noise statistics define the data space Riemannian metric based on the Fisher information matrix. The results of this analysis show that the sensitivity of cell-cell and cell-matrix impedance components are highly model state dependent and several 123 722 A. E. English et al.