The topological properties of an algebraic variety (the set of solutions of some polynomial equations) will be studied in two ways, using structures which are special to them. First, there are notions of size and distance on the smooth part of an algebraic variety, inherited from its imbedding in space. Using this, one defines square-summable functions and, more generally, differential forms on the smooth part of the variety. The first problem to be studied is to what extent the periods of these functions or forms are determined by the topological properties of the variety, in particular, by a topological invariant called "the intersection homology group." The second problem is to find ways to relate quantitative measures of the intersection of cycles on the variety with its local topological properties. These geometrical investigations will not be confined to the non-singular case, rendering them more difficult, yet more suitable of application to spaces which arise in theoretical physics and in other analytic theories.
