This is the home page for Math 497C, a course in the MASS program at Penn State for fall 2013. It centers around the notion of “winding number”.

The *winding number* is one of the most basic invariants in topology: it is an integer that measures the number of times a moving point P goes around a fixed point Q, provided that P travels on a path that never goes through Q and that the final position of P is the same as its starting position. For example, the winding number of the tip of the minute hand of a clock (P), about the center of the clock (Q), between 11 a.m. and 4 p.m. the same day, is -5. This simple idea has far-reaching applications in almost every area of mathematics. For instance, in the course we’ll learn about how the winding number (and its generalizations)

- Help us show that every polynomial equation has a root (the fundamental theorem of algebra)
- Guarantee a fair division of three objects in space by a single planar cut (the ham sandwich theorem)
- Show why every simple closed curve has an inside and an outside (the Jordan curve theorem)
- Allow you to subtract infinity from infinity and get a finite answer (Toeplitz index theory)
- And other applications still to be revealed…

Other pages on this site link to course lecture notes, official syllabus, homework assignments, and additional resources.