Hyperbolicity of general deformations

We present two methods of constructing low degree Kobayashi hyperbolic hypersurfaces in the projective space: the projection method and the deformation method. The talk is based on joint works of the speaker with B. Shiffman and C. Ciliberto.Comment: This is the content of the author's talk given at the conference "Effective Aspects of Complex Hyperbolic Varieties", Aber Wrac'h, France, September 10-14, 200

Convolution equations on lattices: periodic solutions with values in a prime characteristic field

These notes are inspired by the theory of cellular automata. A linear cellular automaton on a lattice of finite rank or on a toric grid is a discrete dinamical system generated by a convolution operator with kernel concentrated in the nearest neighborhood of the origin. In the present paper we deal with general convolution operators. We propose an approach via harmonic analysis which works over a field of positive characteristic. It occurs that a standard spectral problem for a convolution operator is equivalent to counting points on an associate algebraic hypersurface in a torus according to the torsion orders of their coordinates.Comment: 30 pages, a new editio