Computations of cellular A^1-homology of smooth algebraic variety and applications

The cellular A^1-homology of smooth algebraic varieties over a field was introduced by "accident" by Anand Sawant and myself. It is an extremely simplified version of the notion of A^1-homology and as opposed to the latter is very computable. For instance the cellular A^1-homology of projective spaces is entirely computed, though the Suslin homology is not entirely known. I will give some introduction to these objects, give examples, and show how they behave very closely to singular homology for differentiable manifolds, for instance as far as Poincare' duality is concerned, the relation to orientability of the tangent bundle, the definition of the degree of morphisms between oriented varieties. If time allows, I will sketch how it may be used to (hope to) classify rational smooth projective varieties up to isomorphisms over a perfect field, in the lines of "geometric topology". This is based on joint work with Anand Sawant.