Colloquium Lectures

For a holomorphic function with an isolated singularity at the origin, there exists well-known Milnor fibration and lots of researches has been done. A mixed function is a complex valued analytic function in variables and their conjugates . We consider the hypersurface . This is a new approach to a real algebraic variety of real codimension one. We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show that there exists a canonical resolution for the hypersurface singularities defined by a mixed function . We discuss how much similarity this hypersurface has comparing with a holomorphic hypersurface. For example, we discuss Varchenko type formula for zeta functions.