https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

A large range of sophisticated brain image analysis tools have been developed by the neuroscience community, greatly advancing the field of human brain mapping. Here we introduce the Computational Anatomy Toolbox (CAT) - a powerful suite of tools for morphometric analyses with an intuitive graphical user interface, but also usable as a shell script. CAT is suitable for beginners, casual users, experts, and developers alike providing a comprehensive set of analysis options, workflows, and integrated pipelines. The available analysis streams – illustrated on an example dataset – allow for voxel-based, surface-based, as well as region-based morphometric analyses. Importantly, CAT includes various quality control options and covers the entire analysis workflow, from cross-sectional or longitudinal data processing, to the statistical analysis, and visualization of results. The overarching aim of this article is to provide a complete description of CAT, while, at the same time, offering a citable standard reference.

CAT – A Computational Anatomy Toolbox for the Analysis of Structural MRI Data

http://library.um.ac.id/free-contents/downloadpdf.php/buku/methods-of-modern-mathematical-physics-vol-i-functional-analysis-reed-michael-barry-simon-12066.pdf

Methods of modern mathematical physics (vol.) I : functional analysis / Reed Michael, Barry Simon)

Schwarz meets Schwann: Design and fabrication of biomorphic and durataxic tissue engineering scaffolds

It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a Legendrian immersion into the unit cotangent bundle. We give easily computable criteria for a singular point on a front to be a cuspidal edge or a swallowtail. Using this, we prove that generically flat fronts in hyperbolic 3-space admit only cuspidal edges and swallowtails. We also show that any complete flat front (provided it is not rotationally symmetric) has associated parallel surfaces whose singularities consist of only cuspidal edges and swallowtails.

/pdf/singularities-of-flat-fronts-in-hyperbolic-space-4jhu5fkxv2.pdf

Singularities of flat fronts in hyperbolic space

After Galvez, Martinez and Milan discovered a (Weierstrass-type) holomorphic representation formula for flat surfaces in hyperbolic 3 -space H 3 , the first, third and fourth authors here gave a framework for complete flat fronts with singularities in H 3 . In the present work we broaden the notion of completeness to weak completeness, and of front to p-front. As a front is a p-front and completeness implies weak completeness, the new framework and results here apply to a more general class of flat surfaces. This more general class contains the caustics of flat fronts --- shown also to be flat by Roitman (who gave a holomorphic representation formula for them) --- which are an important class of surfaces and are generally not complete but only weakly complete. Furthermore, although flat fronts have globally defined normals, caustics might not, making them flat fronts only locally, and hence only p-fronts. Using the new framework, we obtain characterizations for caustics.

/pdf/flat-fronts-in-hyperbolic-3-space-and-their-caustics-22yc2g3py8.pdf

Flat fronts in hyperbolic 3-space and their caustics

In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same symmetry properties exists for all given minimal surfaces satisfying certain conditions. The surfaces we construct in this paper are irreducible, and in the process of showing this, we also prove some results about the reducibility of surfaces. 
Furthermore, in the case that the surfaces are of genus 0, we are able to make some estimates on the range of the parameter for the one-parameter family.

/pdf/irreducible-constant-mean-curvature-1-surfaces-in-hyperbolic-2iwkdjeuf4.pdf

Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus

We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply-connected 3-dimensional space forms $\R^3$, $\bbS^3 $ and $\bbH^3$. Additionally, we compute the extended frame for any associated family of Delaunay surfaces.

Wayne Rossman

Papers

Singularities of flat fronts in hyperbolic space

Flat fronts in hyperbolic 3-space and their caustics

Irreducible constant mean curvature 1 surfaces in hyperbolic space with positive genus

Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms

Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods