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.