II. Your Present Moment in Spacetime

This essay extends the argument begun in "Why Quantum Mechanics Makes Sense," exploring the conditions under which a physical world can define and communicate information. I argue that like the structure of quantum physics, the principles of Special and General Relativity can be understood as reflecting the requirements of a universe in which things are observable and measurable. I interpret the peculiar hyperbolic structure of spacetime not as the static, four-dimensional geometry of an unobservable "block universe", but as the background metric of an evolving web of communicated information that we, along with all our measuring instruments and recording devices, actually experience in our local "here and now." Our relativistic universe is conceived as a parallel distributed processing system, in which a common objective reality is constantly being woven out of many kinds of facts determined separately in countless local measurement-contexts.

Combinatorial Ricci Flows with Applications to the Hyperbolization of Cusped 3-Manifolds

In this paper, we adopt combinatorial Ricci flow to study the existence of hyperbolic structure on cusped 3-manifolds. The long-time existence and the uniqueness for the extended combinatorial Ricci flow are proven for general pseudo 3-manifolds. We prove that the extended combinatorial Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature, and the flow converges exponentially fast in this case. For an ideally triangulated cusped 3-manifold admitting a complete hyperbolic metric, the flow provides an effective algorithm for finding the hyperbolic metric.

Geometrisation of purely hyperbolic representations in PSL2R$ {\mathrm{PSL}_2\mathbb{R}} $

Let S be a surface of genus g at least 2. A representation ρ : π 1 S → P S L 2 R $ \rho:\pi_1 S\to{\mathrm{PSL}_2\mathbb{R}} $ is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S. In this work we characterise them completely, giving necessary and sufficient conditions.

Surjectivity of certain adjoint operators and applications

This paper is an extension and generalization of some previous works, such as the study of M. Benalili and A. Lansari. Indeed, these authors, in their work about the finite co-dimension ideals of Lie algebras of vector fields, restricted their study to fields $$X_0$$ X 0 of the form $$X_0=\sum _{i=1}^{n}( \alpha _i \cdot x_i+\beta _i\cdot x_i^{1+m_i}) \frac{\partial }{\partial x_i}$$ X 0 = ∑ i = 1 n ( α i · x i + β i · x i 1 + m i ) ∂ ∂ x i , where $$\alpha _i, \beta _i $$ α i , β i are positive and $$m_i$$ m i are even natural integers. We will first study the sub-algebra U of the Lie-Fréchet space E, containing vector fields of the form $$Y_0 = X_0^+ + X_0^- + Z_0$$ Y 0 = X 0 + + X 0 - + Z 0 , such as $$ X_0\left( x,y\right) =A\left( x,y\right) =\left( A^{-}\left( x \right) ,A^{+}\left( y\right) \right) $$ X 0 x , y = A x , y = A - x , A + y , with $$A^-$$ A - (respectively, $$ A^+ $$ A + ) a symmetric matrix having eigenvalues $$ \lambda < 0$$ λ < 0 (respectively, $$\lambda >0 $$ λ > 0 ) and $$Z_0$$ Z 0 are germs infinitely flat at the origin. This sub-algebra admits a hyperbolic structure for the diffeomorphism $$\psi _{t*}=(exp\cdot tY_0)_*$$ ψ t ∗ = ( e x p · t Y 0 ) ∗ . In a second step, we will show that the infinitesimal generator $$ad_{-X}$$ a d - X is an epimorphism of this admissible Lie sub-algebra U. We then deduce, by our fundamental lemma, that $$U=E$$ U = E .

The relative hyperbolicity and manifold structure of certain right-angled Coxeter groups

In this paper, we study the manifold structure and the relative hyperbolic structure of right-angled Coxeter groups with planar nerves. We then apply our results to the quasi-isometry problem for this class of right-angled Coxeter groups.

TG-Hyperbolicity of virtual links

We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We prove that all virtual alternating links are tg-hyperbolic. We further extend tg-hyperbolicity to several classes of non-alternating virtual links. We then consider bounds on volumes of virtual links and include a table for volumes of the 116 nontrivial virtual knots of four or fewer crossings, all of which, with the exception of the trefoil knot, turn out to be tg-hyperbolic.

Teichmuller Space

This chapter deals with Teichmüller space Teich(S) of a surface S. It first defines Teichmüller space and a topology on Teich(S) before giving two heuristic counts of its dimension. It then describes explicit coordinates on Teich(Sɡ) coming from certain length and twist parameters for curves in a pair of pants decomposition of Sɡ; these are the Fenchel–Nielsen coordinates on Teich(Sɡ). The chapter also considers the Teichmüller space of the torus and concludes by proving the 9g – 9 theorem, which states that a hyperbolic structure on Sɡ is completely determined by the lengths assigned to 9g – 9 isotopy classes of simple closed curves in Sɡ.

On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings

We give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure.