Deformations of convex real projective manifolds and orbifolds

International audienceIn this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric structures on orbifolds, and Hilbert geometry. The main examples of finitely generated groups for us will be Fuchsian groups, 3-manifold groups and Coxeter groups

The maximal tubes under the deformations of a class of 3-dimensional hyperbolic cone-manifolds

Recently, Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-manifold is decreasing and that summed with the cone angle squared is increasing as we deform the cone-angles. We confirm this near 0 cone-angles for an infinite family of hyperbolic cone-manifolds obtained by Dehn surgeries along the Whitehead link complements. The basic method is based on explicit holonomy computations using the A-polynomials and finding the maximal tubes. One of the key tool is the Taylor expression of a geometric component of the zero set of the A-polynomial in terms of the cone-angles. We also show a sequence of Taylor expressions for Dehn surgered manifolds converges to one for the limit hyperbolic manifold.Comment: 27 pages, 10 figure

Green with Envy at Your Kid: The Effects of Two Different Types of Envy on Purchase Intention

Envy is an emotion that “arises when a person lacks another’s superior quality, achievement, or possession and either desires it or wishes that the other lacked it”. Envy has been classified into two types: benign envy and malicious envy. Benign envy emphasizes the brighter side of envy, which is related to moving-up motivation, while malicious envy represents the destructive side of envy, which motivates people to pull down. The purpose of this study is to address research gap by exploring how envy affects purchase intention among mom through experimental design. Participants were recruited by research company, total one hundred and twenty eight moms with 5-7 year old kids were randomized in each study. Envy is manipulated into two types depending on the deservingness of the situation. Participants were given a short scenario which described a friend who has more capital for raising her children. In purchasing economic capital related products, benign envy condition(M=2.57, SD=1.36) reported greater desire for purchasing products than malicious envy condition(M=2.35, SD=1.21)(F=5.392, p\u3c.05). In purchasing cultural capital related products, benign envy condition(M=4.00, SD=0.78) reported greater desire for purchasing products than malicious envy condition(M=3.98, SD=1.08)(F=.235,n.s). The findings confirm that difference depending on type of envy, and benign envy play an important role for Koreans in purchase intention. In result, envy has no influence in purchasing cultural capital related products. It reveals that in Korea, fashion is used as a strategy of cultural capital, and sense of advanced taste are likely to be interpreted as part of cultural capital

Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study the deformation space of real projective structures on a compact Coxeter $n$-orbifold $Q$ admitting a hyperbolic structure. Let $e_+(Q)$ be the number of ridges of order $\geq 3$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_+(Q) - n$ if $n=3$ and $Q$ is weakly orderable, i.e., the faces of $Q$ can be ordered so that each face contains at most $3$ edges of order $2$ in faces of higher indices, or $Q$ is based on a truncation polytope.Comment: 43 pages with 7 figures, to appear in Geometry & Topolog

Convex projective generalized Dehn filling

In dimension $d = 4, 5, 6, 7$, we find the first examples of complete finite volume hyperbolic $d$-dimensional manifolds $M$ with cusps such that an infinite number of orbifolds $M_m$ obtained by generalized Dehn fillings on $M$ admit a properly convex real projective structure. The manifolds $M$ are covering of hyperbolic Coxeter orbifolds and the orbifold fundamental groups $\Gamma_m$ of $M_m$ are Gromov hyperbolic relative to a collection of subgroups virtually isomorphic to $\mathbb{Z}^{d−2}

Design under uncertainty of carbon capture, utilization and storage infrastructure considering profit, environmental impact, and risk preference

This study presents a decision making tool for risk management of a carbon capture utilization and storage (CCUS) network under uncertainty among conflicting objectives. A two-phase-two-stage stochastic multi-objective optimization problem solving algorithm is formulated to balance environmental impact and various sources of uncertainty and corresponding risk by installing and operating a CCUS network. The algorithm allows decision makers to choose their own tolerance on risk. By conducting case studies that have different target profits for CCUS networks, the algorithm provides optimal results based on the decision maker's attitude to risk. To evaluate risks imposed by uncertain parameters, a concept of downside risk is introduced. By setting different target profit levels, the suggested tool enables decision makers to choose their own tolerance and preference for risk. In the model, the life cycle assessment is applied to evaluate all environmental contributions caused by installation and operations of the CCUS network. The model provides the trade-off relationship between total annual benefit with financial risk as well as corresponding environmental impact. The aim of this model to optimize CCUS supply chain networks is to provide an intuitive decision making algorithm to balance conflicting objectives within a single framework. This problem is formulated as a mixed integer linear program model. To illustrate the applicability of the model, four optimal CCUS network models for the various types of industrial complex of Korea in 2030 are presented. Results indicate that risk-averse cases with a low profit target are more reliable in stochastic uncertainty, and that risk-taking decision makers tend to invest more on capture facility and produce more product than do, risk-averse decision makers.11Nsciescopu