Abstract: In 2001 Peterzil and Starchenko worked to expand the notions of complex analysis to the algebraic closures of real closed fields. The cornerstone of this process was their ability to define a reasonable notion of the winding number and use that to recover many of the standard theorems of complex analysis. In this talk, we will go over this construction of the winding number and, time permitting, we will look at some of its consequences and how it might be used to recover the standard theorems of complex analysis.
