Abstract:

Algebraic geometry studies algebraic varieties, i.e. objects defined by polynomial equations and their invariants. Algebraic topology concerns with algebraic invariants of topological spaces. Can algebraic varieties and topological spaces have similar algebraic invariants? I'll try to show how algebraic invariants, which were first defined in an algebraic setting, were later successfully applied in topology and vice versa. For example, the functor K, defined first by Grothendieck in an algebraic setting, has led to the creation of topological K-theory. Like a boomerang, it returned back to algebra in the form of algebraic K-theory. I'll also try to sketch recent developments by Voevodsky who considerably enriched the interaction between algebra and topology.