Abstract: Imagine you are looking down on the floor of a factory, where you have a set of robots that run around and, for instance, move boxes. To save money, these robots all run along tracks embedded in the floor. Of course, you do not want your robots to run into each other. So how can these robots move around?
Almost 100 years ago, Emile Artin introduced a family of groups called braid groups. These important groups have implications in many areas of science, including knot theory, group theory, cryptography, and genetics. Braid groups can be defined topologically, in terms of configurations of points running around on a disk. What happens if you replace that disk with some other space, like a graph?
The answer to both of these questions lies in the subject of braid groups on graphs. These braid groups model motions of robots moving on a factory floor, and are natural generalizations of Artin's groups. In this talk, I will discuss braid groups on graphs, give a short survey of known results, and present my work on the subject. My results include theorems on presentations of the group, results on group cohomology, a solution to the isomorphism problem in some cases, and the relationship between graph braid groups and classical braid groups as well as right-angled Artin groups. Some of the key tools I have used include configuration spaces, Forman's discrete Morse theory, algebraic topology, and the language of differential forms.
