Discrete Riemann Surfaces and the Ising model

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality

Weyl Card Diagrams and New S-brane Solutions of Gravity

We construct a new card diagram which accurately draws Weyl spacetimes and represents their global spacetime structure, singularities, horizons and null infinity. As examples we systematically discuss properties of a variety of solutions including black holes as well as recent and new time-dependent gravity solutions which fall under the S-brane class. The new time-dependent Weyl solutions include S-dihole universes, infinite arrays and complexified multi-rod solutions. Among the interesting features of these new solutions is that they have near horizon scaling limits and describe the decay of unstable objects.Comment: 78 pages, 32 figures. v2 added referenc

Geometry, topology and dynamics of geodesic flows on noncompact polygonal surfaces

We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on $Z$-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a topologically typical $Z$-periodic surface with boundary are recurrent.Comment: 34 pages, 13 figures. To be published in V. V. Kozlov's Festschrif

Orthogonal structure on a quadratic curve

Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any quadratic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. Convergence of the Fourier orthogonal expansions is also studied in each case. As an application, we see that the resulting bases can be used to interpolate functions on the real line with singularities of the form $|x|$, $\sqrt{x^2+ \epsilon^2}$, or $1/x$, with exponential convergence