For my non-math audiences:

• Compute for me the following: 1+1-1+1-1-1. Did you get 0? If so, great!

• Now compute for me the following:

  • 1+1+1-1-1-1-1-1-1+1+1+1+1-1-1-1+1+1+1-1

• This computation is probably a little trickier because there are way more terms involved. So perhaps some of you, instead of just adding and subtracting 1's, decided to count the number of +1's and count the number of -1's and you saw that if the number of +1's and -1's are the same, you get 0!

• If you prefer to use some visual aids, say the +1's are blue blocks and the -1's are red blocks. The only way we get 0 is if the height of the blue blocks and the red blocks are the same.

• Through my research, I seek to answer a similar question. If someone gives me a random combination of elements (like +1's and -1's), can I tell you when that combination is 0? If I always can, I've given you a solution to "The Word Problem." I seek to provide a nice solution to the Word Problem for groups known as Euclidean Artin Groups.

For my math audience:

• In general, I enjoy the pleasant intersection of algebra, geometry and topology that I find in the field of Geometric Group Theory.

• If you read the above, you can see that I've been thinking about the Word Problem for Euclidean Artin groups. In general, the word problem for Artin groups remains unanswered though progress towards solutions for certain types of Artin groups exist.

• In particular, no "nice" solutions exist for Euclidean Artin groups. Recently, I've been looking at these solutions and attempting to create nicer solutions. Using topics and techniques like that of Garside structures and Stallings foldings, I am working to write a set of finite state automata that will serve as a solution to the word problem for a particular Euclidean Artin group.