Title:  Continued fractions and latice trigonometry.

Abstract:

In this talk we introduce lattice trigonometric functions of lattice
angles in lattice geometry. We show some properties of lattice
trigonometry and compare them with properties of standard (Euclidean)
trigonometry.  In particular we introduce a necessary and sufficient
condition for three angles to be the angles of some lattice triangle
in terms of lattice tangents. This condition is a version of the
Euclidean condition: three angles are the angles of some triangle iff
their sum equals Pi.  If we have enough time we say a few words about
connections with theory of complex projective toric varieties.