ELECTROSTATICS, HYPERBOLIC GEOMETRY AND WANDERING VECTORS

Hyperbolic Geometry

Wandering Vectors for Irrational Rotation Unitary Systems

An abstract characterization for those irrational rotation unitary systems with complete wandering subspaces is given. We prove that an irrational rotation unitary system has a complete wandering vector if and only if the von Neumann algebra generated by the unitary system is finite and shares a cyclic vector with its commutant. We solve a factorization problem of Dai and Larson negatively for ...

Geometry-dependent electrostatics near contact lines.

Long-ranged electrostatic interactions in electrolytes modify contact angles on charged substrates in a scale and geometry-dependent manner. For angles measured at scales smaller than the typical Debye screening length, the wetting geometry near the contact line must be explicitly considered. Using variational and asymptotic methods, we derive new transcendental equations for the contact angle ...

Algebraic surfaces and hyperbolic geometry

Many properties of a projective algebraic variety can be encoded by convex cones, such as the ample cone and the cone of curves. This is especially useful when these cones have only finitely many edges, as happens for Fano varieties. For a broader class of varieties which includes Calabi-Yau varieties and many rationally connected varieties, the Kawamata-Morrison cone conjecture predicts the st...

Holomorphic Dynamics and Hyperbolic Geometry

Rational maps are self-maps of the Riemann sphere of the form z → p(z)/q(z) where p(z) and q(z) are polynomials. Kleinian groups are discrete subgroups of PSL(2,C), acting as isometries of 3-dimensional hyperbolic space and as conformal automorphisms of its boundary, the Riemann sphere. Both theories experienced remarkable advances in the last two decades of the 20th century and are very active...