An important goal in studying the relations between unitary VOAs and conformal nets is to prove the equivalence of their ribbon categories. In this article, we prove this conjecture for many familiar examples. Our main idea is to construct new structures associated to conformal nets: the categorical extensions.

Mesh parameterization is a key problem in digital geometry rocessing. By cutting a surface along a set of edges (a seam), one can map an arbitrary topology surface mesh to a single chart. Unfortunately, high distortion occurs when protrusions of the surface (such as fingers of a hand and horses’ legs) are flattened into a plane. This paper presents a novel skeleton-based algorithm for computing a seam on a triangulated surface. The seam produced is a full component Steiner tree in a graph constructed from the original mesh. By generating the seam so that all extremal vertices are leaves of the seam, we can obtain good parametrization with low distortion.

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Let ($X, L$) be a polarized 3-fold over the complex number field. In [Fk3], we proved that$g(L)≥q(X)$if$h$^{0}($L$)≥2 and moreover we classified ($X, L$) with$h$^{0}($L$)≥3 and$g(L)$=$q(X)$, where$g(L)$is the sectional genus of ($X, L$) and$q$($X$)=dim$H$^{1}($O$_{$X$}) the irregularity of$X$. In this paper we will classify polarized 3-folds ($X, L$) with$h$^{0}($L$)≥4 and$g(L)$=$q(X)$+1 by the method of [Fk3].

The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in dimension 2) condition for the diffeomorphism to extend to a finite holomorphic map. The multiplicity of this map is a biholomorphic invariant that is precisely the Moser invariant of the image, when it is a Bishop surface with vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR singular images and we prove that the set of CR singular points must be large, and in the case of codimension 2, necessarily Levi-flat or complex. We also show that there exist real-analytic CR functions on such images that satisfy the tangential CR conditions at the singular points, yet fail to extend to holomorphic functions in a neighborhood. We provide many examples to illustrate the phenomena that arise.