Great problems of geometry and space

To do mathematics is to raise great mathematical problems, and try to solve them. Eventually to solve them. This time, we shall treat problems of geometry and space, and we shall classify geometric objects in dimensions 2 and 3. Dimension 2 is classical: it’s the classification of surfaces, which are obtained by attaching handles on spheres. One can also describe surfaces by using the Poincaré −Lobatchevsky upper half plane. What happens in higher dimensions? In dimension ≧5, Smale obtained decisive results in 1960. Last year, Thurston published great results in dimension 3. He conjectured the way such objects can be constructed starting with simple models, and also how one could obtain them from the analogue of the upper half plane in 3 dimensions. He proved a good part of his conjectures. We shall describe Thurston’s vision