In Mathematical Careers and Ideas, the Department of Mathematics is offering a series of talks for undergraduates.   This is the next event in the series.

Abstract:   Start with a piece of paper whose boundary forms a polygon, choose some pairs of edges and glue them together. This yields what mathematicians call a "surface".

Now, take a closed curve C in that surface. Imagine that you can shrink, stretch and slide C on the surface with the only restriction that the curve cannot leave the surface. All the curves C that you can make this way are "equivalent" to C and the whole set of curves you can obtain this way is called the free homotopy class of C.

The number of times a curve C crosses itself is the self-intersection number of C. The self-intersection of a free homotopy class of curves is the smallest number of crossings of curves in the class.

In this talk I will discuss algorithms to compute the self-intersection number of a class of curves and some statistics related to the self-intersection number.