Los Angeles Math Circle

5/19 -- High School Circle: How many ways are there to get from A to B? Part I (Prof. Christian Haesemeyer)

We will investigate paths in geometric objects ("spaces") to answer the question: how many (essentially different) ways are there to get from point A to point B in the given object? It turns out that the best answer is not given by a number, but by algebraic structures called groups and groupoids that take into account how a given path can be broken down into pieces - and what happens when we go in circles. Our main examples of spaces will be graphs, and surfaces.