Abstract: Computations in hyperbolic spaces are difficult due to the group properties of these spaces which make it difficult to use these tools, in sharp contrast with the Euclidean situation.
The author devised a way to navigate in the tessellations of the hyperbolic plane and one of the four tessellations of the hyperbolic 3D space. This constitutes an actual GPS which allows to know the position of each tile with respect to the others and to go from one tile to another one. From this, true coordinates for the points of these spaces themselves can be devised which have a nice behaviour undeer shifts which preserve the tessellation.
In the talk, we sketchilly remind the Poincare''s disc model which we use in order to try to see something in these spaces. Then, we describe the coordinate system. In the main part of the talk, we present and discuss the wide range of its applications: this goes from cosmology to computer science itself, including the Internet. We shall look at already realized applications and at those waiting for realization as well.

Brief Biography of the Speaker:
MARGENSTERN Maurice, born on June, 6, 1947, Paris, France, married, 2 children, 1 grand-child, is full professor at the University of Metz, IUT of Metz, France, from 1995. Formerly, he was associated professor at the department of mathematics of the University Paris-Sud. He was the head of LITA (Laboratoire d'Informatique The'orique et Applique'e), from 2000 up to 2008 and an elected member of the scientific council of his university from 2000 up to 2004. He was also the head of the hiring committee of his university for computer science from 1998 up to 2004. Recently, he was promoted to the exceptional class for university professors by the National Council of Universities in France.
His scientific activity deals with the frontier between decidability and undecidability which is studied in various models of discrete computations. He has important results in Turing machines, in cellular automata and in (bio)molecular computing. He wrote 183 papers, among them 55 in well known international journals, 48 in international conferences with proceedings. He is a member of the Editorial Board of the Journal of Universal Computer Science, of the Journal of Cellular Auotmata and he is a member of the Advisory Board of the Computer Science Journal of Moldova. He is a member of WG 1.5 in the TC1 of IFIP.
He is very active in the field of cellular automata. He introduced an original method in order to implement these automata in hyperbolic spaces. This has very interesting connections with elementary theory of numbers and the theory of languages. It may also have surprising applications. He published many papers on this topic and a two-volumed book, "Cellular Automata in Hyperbolic Spaces", an important scientific event. He is also a contributor to Springer Encyclopedia of Complexity and Systems Science.
Maurice Margenstern edited several special issues in international journals of Computer Science: in Theoretical Computer Science, in Fundamenta Informaticae and a new issue is currently planned for the International Journal of Foundations of Computer Science.
Maurice Margenstern organised a cycle of conferences, called "Machines, Computations and Universality", MCU-conferences, which hold each third year starting from 1995. Each edition of the conference is followed by a special issue of a well know journal: TCS for the first three editions, FI for the fourth and fifth editions (Saint-Petersburg, 2004 and Orle'ans, 2007). The sixth edition is to be held at Pittsburgh, USA, in September, 21-25, 2010. The proceedings will be published in EPTCS and a special issue of IJFCS will follow the conference, devoted to its topics.