Introduction to mixed singularities and their applications

Resumen:

Lecture 1:
First I will review the complex hypersurface singularity theory developped by Milnor and many others.
Then we recall weighted homogeneous polynomial and basic properties. We study Newton boundary of a function and the concept of  non-degeneracy of Newton boundary and the topology of the non-degenerate functions.

Lecture 2:
We start from a real algebraic knot defined by two real polynomials and then I will introduce the notion of mixed functions to describe the real knot, the Newton boundary and the concept of non-degeneracy. I will explain lots of properties such as Milnor fibrations,smoothness, resolution etc can be similarly treated similarly in this case. I will also expalin which properties can not be similar comparing with the complex singularities.

Lecture 3: 
If there will be still time, I will explain some applications such as 
· Mixed projective curves, Thom inequality, embedding of a Riemann surface with any genus as a degree 1 curve. 
· Non-degenerate mixed functions with strongly polar weighted homogeneous face type and the formula of Varchenko. 
· Contact structures on a certain mixed links and their symmplectic structure.

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