We present a novel approach to creating at maps of the brain. It is impossible to atten 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 attening. Our approach attempts to preserve the con-formal structure between the original cortical surface in 3-sp...

Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling b...

We present a theoretical analysis showing that layered topological insulators, for example, Bi2Se3 are optically hyperbolic materials in the range of terahertz (THz) frequencies. As such, these topological insulators possess deeply subdiffractional, highly directional collective modes: hyperbolic phonon polaritons. We predict that in thin crystals the dispersion of these modes is split into dis...

A graph is of parabolic or hyperbolic type if the simple random walk on the vertices is, respectively, recurrent or transient. A plane triangulation graph is CP-parabolic or CP-hyperbolic if the maximal circle packing determined by the graph packs, respectively, the complex plane C or the Poincar e disk D. We examine the implications that (Gromov) negative curvature carries for determining type...

The aim of this short lecture course is to develop Selberg's trace formula for a compact hyperbolic surface M, and discuss some of its applications. The main motivation for our studies is quantum chaos: the Laplace-Beltrami operator −∆ on the surface M represents the quantum Hamiltonian of a particle, whose classical dynamics is governed by the (strongly chaotic) geodesic flow on the unit tange...

Let M be a closed hyperbolic three manifold. We show that the number of genus g surface subgroups of π1(M ) grows like g.

The theme of this survey is that subgroups of the mapping class group MCG of a finite type surface S can be studied via the geometric/dynamical properties of their action on the Thurston compactification of Teichmüller space T = T ∪PMF , just as discrete subgroups of the isometries of hyperbolic space Hn can be studied via their action on compactified hyperbolic space H n = Hn ∪ S ∞ . 2000 Math...

For a compact spacelike constant mean curvature surface with nonempty boundary in the threedimensional Lorentz–Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilical point, which was developed by Choe [‘Sufficient conditions for constant mean curvature surfaces to be round’, Math. Ann. 323(1) (2002), 143–156]. Using the concept of the rotation index a...

An explicit construction of a geodesic flow-invariant distribution lying in the discrete series of weight 2k isotopic component is found, using techniques from representation theory of SL2(R). It is found that the distribution represents an AC measure on the unit tangent bundle of the hyperbolic plane minus an explicit singular set. Finally, via an averaging argument, a geodesic flow-invariant ...

The papers [Gálvez et al. 2000, Kokubu et al. 2003, Kokubu et al. 2005] gave a method of constructing flat surfaces in the three-dimesnional hyperbolic space. Such surfaces have generically singularities, since any closed nonsigular flat surface is isometric to a horosphere or a hyperbolic cylinder. In the paper [Sasaki et al. 2006], we defined a map, called the hyperbolic Schwarz map, from the...