Philip L. Bowers

Quasi-Conformally Flat Mapping the Human Cerebellum

We present a novel approach to creating flat maps of the brain. It is impossible to flatten a curved surface in 3D space without metric and areal distortion; however, the Riemann Mapping Theorem implies that it is theoretically possible to preserve conformal (angular) information under flattening. Our approach attempts to preserve the conformal structure between the original cortical surface in 3-space and the flattened surface. We demonstrate this with data from the human cerebellum and we produce maps in the conventional Euclidean plane, as well as in the hyperbolic plane and on a sphere. Conformal mappings are uniquely determined once certain normalizations have been chosen, and this allows one to impose a coordinate system on the surface when flattening in the hyperbolic or spherical setting. Unlike existing methods, our approach does not require that cuts be introduced in the original surface. In addition, hyperbolic and spherical maps allow the map focus to be transformed interactively to correspond to any anatomical landmark.

Planar Conformal Mappings of Piecewise Flat Surfaces

There is a rich literature in the theory of circle packings on geometric surfaces that from the beginning has exposed intimate connections to the approximation of conformal mappings. Indeed, one of the first publications in the subject, Rodin and Sullivan’s 1987 paper [10], provides a proof of the convergence of a circle packing scheme proposed by Bill Thurston for approximating the Riemann mapping of an arbitrary proper simply-connected domain in ℂ to the unit disk. Bowers and Stephenson’s work in [4], which explains how to apply the Thurston scheme on nonplanar surfaces, may be viewed as a far reaching generalization of his scheme to the setting of arbitrary equilateral surfaces. Further, in [4] Bowers and Stephenson propose a method for uniformizing more general piecewise flat surfaces that necessitates a truly new ingredient, namely, that of inversive distance packings. This inversive distance scheme was introduced in a very preliminary way in [4] with some comments on the difficulty involved in proving that it produces convergence to a conformal map. Even with these difficulties, the scheme has been encoded in Stephenson’s packing software CirclePack and, though all the theoretical ingredients for proving convergence are not in place, it seems to work well in practice. This paper may be viewed as a commentary on and expansion of the discussion of [4]. Our purposes are threefold. First, we carefully describe the inversive distance scheme, which is given only cursory explanation in [4]; second, we give a careful analysis of the theoretical difficulties that require resolution before conformal convergence can be proved; third, we give a gallery of examples illustrating the power of the scheme. We should note here that there are special cases (e.g., tangency or overlapping packings) where the convergence is verified, and our discussion will give a proof of convergence in those cases.

Characterization of Hilbert space manifolds revisited

Abstract This exposition focuses not on manifolds modelled on Hilbert space but rather modelled on s =(0,1) ∞ , the product of countably many open intervals. The primary goal is to present an approach leading to the characterization of s -manifolds that relies exclusively on the geometry of s , contrasting with the original attack of Torunczyk in [21] that at times made essential use of the linear structure of Hilbert space. A secondary goal is to correct a misconception that has surfaced in several locations in characterizing manifolds in the non-locally compact setting; specifically, the usual notion of Z -set needs to be refined.

A \REGULAR" PENTAGONAL TILING OF THE PLANE

The paper introduces conformal tilings, wherein tiles have spec- ied conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a subdivision rule. Combinatorial symmetries imply rigid conformal symmetries, which in turn illustrate a new type of tiling self-similarity. In parallel with the conformal tilings, the paper develops discrete tilings based on circle packings. These faithfully reflect the key features of the theory and provide the tiling illustra- tions of the paper. Moreover, it is shown that under renement the discrete tiles converge to their true conformal shapes, shapes for which no other ap- proximation techniques are known. The paper concludes with some further examples which may contribute to the study of tilings and shinglings being carried forward by Cannon, Floyd, and Parry.

The set of circle packing points in the Teichmüller space of a surface of finite conformal type is dense

W. Thurston initiated interest in circle packings with his provocative suggestion at the International Symposium in Celebration of the Proof of the Bieberbach Conjecture (Purdue University, 1985) that a result of Andreev[2] had an interpretation in terms of circle packings that could be applied systematically to construct geometric approximations of classical conformal maps. Rodin and Sullivan [11] verified Thurstons conjecture in the setting of hexagonal packings, and more recently Stephenson [12] has announced a proof for more general combinatorics. Inspired by Thurstons work and motivated by the desire to discover and exploit discrete versions of classical results in complex variable theory, Beardon and Stephenson [4, 5] initiated a study of the geometry of circle packings, particularly in the hyperbolic setting. This topic is a recent example among many of the beautiful and sometimes unexpected interplay between Geometry, Topology, and Cornbinatorics that is evident in much of the topological research of the past decade, and that has its roots in the seminal work of the great geometrically-minded mathematicians – Riemann, Klein, Poincare – of the last century. A somewhat surprising example of this interplay concerns us here; namely, the fact that the combinatorial information encoded in a simplicial triangulation of a topological surface can determine its geometry.

Circle packings in surfaces of finite type: an in situ approach with applications to moduli

ROBERT BRINKS [lo, 1 l] used circle packings to parametrize the deformation space of a Kleinian group and applied his results to prove that the “circle packing points” (closed Riemann surfaces that can be filled by circle packings) form a dense subset of moduli space. In [9], the authors combined techniques of Brooks, Thurston [28], and BeardonStephenson [4, 5) to extend Brooks’ result to surfaces of finite conformal type, closed surfaces with a finite number of punctures. The methods there involve the in situ manipulation of circle packings and rely heavily on certain canonical Brooks’ packings of quadrilaterals as developed in [lo], along with canonical infinite packings of cusp regions. The latter are rigid, but the Brooks’ packings act like shock absorbers, permitting the small adjustments to modulus that lead to circle packing points. Our purposes here are threefold: first, to develop more fully and systematically the in situ approach to circle packings in hyperbolic surfaces begun in [9]; second, to define canonical packings of annuli that have a flexibility reminiscent of Brooks’ packings of quadrilaterals; and thiid, to apply these in an extension of Brooks’ result to surfaces of finite topological type, closed surfaces having a finite number of punctures and a finite number of half-annular ends. Our main result is the following.

Limitation topologies on function spaces

Four competing definitions for limitation topologies on the set of continuous functions C(X, Y) are compared.

The upper Perron method for labelled complexes with applications to circle packings

The construction of geometric surfaces via labelled complexes was introduced by Thurston[ 16 , chapter 13], and subsequent applications and developments have appeared in [ 1, 3, 4, 5, 14, 15 ]. The basic idea of using labelled complexes to produce geometric structures is that the vertices of a simplicial triangulation of a surface can be labelled with positive real numbers that collectively determine a metric of constant curvature ±1 or 0, with possible singularities at vertices, by using the label values to identify 2-simplices of the triangulation with geometric triangles. Beardon and Stephenson[ 1 ] developed a particularly simple method for producing non-singular surfaces via labelled complexes that is modelled after the classical Perron method for producing harmonic functions, and they applied their method in [ 2 ] to construct a fairly comprehensive theory of circle packings in general Riemann surfaces. This Perron method was developed more fully by Stephenson and the author in [ 3, 4 ] and applied to the study of circle packing points in moduli space. At about the same time and independently of Beardon, Stephenson, and Bowers, Carter and Rodin [ 5 ] and Doyle [ 8 ] developed the method for flat surfaces and Minda and Rodin [ 14 ] developed the method for finite type surfaces. Minda and Rodin [ 14 ] applied their development to give partial solutions to the labelled complex version of the classical Schwarz-Picard problem that concerns the construction of singular hyperbolic metrics on surfaces with prescribed singularities. In this paper, we modify the aforementioned approaches and examine the upper Perron method for producing non-singular geometric surfaces. This upper method has several advantages over the Perron method as developed previously and provides a complete solution to the labelled complex version of the Schwarz-Picard problem.

Dense embeddings of nowhere locally compact separable metric spaces

Abstract Nowhere locally compact separable metric spaces are characterized (up to homeomorphism) as precisely the dense subsets of the separable Hilbert space and those of dimension at most n are characterized (up to homeomorphism) as precisely the dense subsets of the n-dimensional Menger-Nobeling space.

Fixed points in boundaries of negatively curved groups

An important feature of the natural action of a negatively curved (= word hyperbolic) group on its boundary is that the fixed points of hyperbolic (= infinite order) elements form a dense subset of the boundary. Gromov states this in his influential treatise Hyperbolic Groups [4, 8.2 D] and his explanation involves Stallings theorem on groups with infinitely many ends. This short note provides an elementary proof of this fact. The proof relies on a simple enhancement of the Pumping Lemma from the theory of finite state automata and on the fact that the geodesic words with respect to a fixed generating set for a negatively curved group form a regular language. For background material on negatively curved groups see [1] and on automata and regular languages on groups see [2]. For a different proof see [5].