'Let X x A^1 = A^n. Please solve for X', and related questions -- Brent Doran

To what extent can highly refined algebro-geometric structure be captured by appropriately formalized topological intuition? Where do analogies with classical geometric topology hold true, and where do they break down? Using the question of the title as a touchstone, we compare some structural results in the category of smooth manifolds and analogs in algebraic geometry. By beginning with the study of contractible manifolds we touch upon: exotic smooth structures, algebraic spaces arising in surprising places, the limits of a theory of classifying spaces for bundles in algebraic geometry, and tantalizing hints at geometric interpretations for higher (rational) connectivity in algebraic geometry using A^1-homotopy theory as a foundation. Applications to toric varieties and quadric (and some higher degree) hypersurfaces will be discussed if time allows. In particular we will see that quite a lot of structure in algebraic geometry is missed by traditional ``cohomological" invariants (e.g., the motive is far weaker than motivic homotopy type), suggesting an approach more in line with the classification theory of manifolds may be useful.