We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface S of genus g , we get a geometric lower bound on 2g 2.S/ W 2g 2.S/ > 1=4C 0.S/ > 0; where 0.S/ is an explicit constant which depends only on the systole of S . Mathematics Subject Classification (2010). Primary 30F; Secondary 35P05, 35P15.

We have observed, in metamaterial with hyperbolic dispersion (an array of silver nanowires in alumina membrane), a sixfold reduction of the emission lifetime of dye deposited onto the metamaterial's surface. This serves as evidence of an anomalously high density of photonic states in hyperbolic metamaterials, demonstrates the feasibility of an earlier-predicted single-photon gun, and paves the ...

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...

Hyperbolic materials enable numerous surprising applications that include far-field subwavelength imaging, nanolithography, and emission engineering. The wavevector of a plane wave in these media follows the surface of a hyperboloid in contrast to an ellipsoid for conventional anisotropic dielectric. The consequences of hyperbolic dispersion were first studied in the 50's pertaining to the prob...

Hyperbolic metamaterials (HMMs) represent a novel class of fascinating anisotropic plasmonic materials, supporting highly confined bulk plasmon polaritons in addition to the surface plasmon polaritons. However, it is very challenging to tailor and excite those modes at optical frequencies using prism coupling technique because of the intrinsic difficulties to engineer non-traditional optical pr...

We show that a general small deformation of the union of two general cones in P of degree ≥ 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in P of any given degree d ≥ 8. It was shown by Clemens [2] that a very general surface Xd of degree d ≥ 5 in P 3 has no rational curves; G. Xu [11] showed that Xd also has no elliptic curves (and in fact has no curves of genu...

We establish basic local existence as well as a stability result concerning small perturbations of the Catenoid minimal surface in R3 under hyperbolic vanishing mean curvature flow.

This talk re-examines some of the evolving interaction between hyperbolic geometry and low-dimensional topology, focussing in particular on some contributions of Murray Macbeath to surface topology.

Inside of a hyperbolic medium, the principal components of the permittivity tensor have opposite signs causing the medium to exhibit a 'metallicbr' type of response to light wave sin one direction, and a 'dielectric' response in the other. Our study shows that inside hyperbolic media, volume plasmon polaritons (VPPs) propagate along the characteristic planes, forming distinct, directionally dep...

We show that a general small deformation of the union of two general cones in P of degree ≥ 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in P of any given degree d ≥ 8. It was shown by Clemens [2] that a very general surface Xd of degree d ≥ 5 in P 3 has no rational curves; G. Xu [11] showed that Xd also has no elliptic curves (and in fact has no curves of genu...