Week 9: We've spent the week attending lectures for Charles' University's Midsummer Combinatorial Workshop. I'll be honest in saying that most of the talks have been completely over my head, but it's been a lot of fun learning about how every branch of math is coming together to solve these tough problems. We've also been exploring the city and some of the countryside. I had no idea a place could have so many castles.

Week 8: After a cancelled flight and an hour-and-a-half delay in Paris, we finally landed in Prague. We've had special guest speakers come in and teach us about various topics in combinatorics and graph theory. We've also had some problem solving sessions (like the bridge sessions at DIMACS), and I'm starting to see that these problems are very, very difficult. Although they are tough, I always feel like I've learned something whenever I solve one.

Week 7: This is my final week at DIMACS before I head off to Prague. These last few days have been an attempt to construct some sort of geometry with the lemmata I have developed, but I haven't had much success. I'm thinking that I need some other criteria for what makes a point and what makes a line. Everything I have come up with so far has been related to the existence of cycles of a certain size within the graph, but nothing has lent itself to a geometry.

Week 6: I have continued to explore the Petersen Graph. The lemmas I came up with last week were useful, but they really didn't lead me anywhere. The space the Petersen Graph was too "small" to give me any sort of interesting geometry, so I expanded it. This week consisted of me looking at concenctric 5-cycles and other variations of the Petersen Graph in order to create a geometry.

Week 5: I've spent the weeke exploring the finte geometries embedded within the Petersen Graph. My original ideas about what consititues a "point" and a "line" weren't leading me anywhere, so Gene suggested another approach. By the end of the week we had proven three conjectures regarding the nature of perfect matchings within the Petersen Graph. I now want to use these properties to more fully define what properties the embedded geometry has.

Week 4: The tough conjecture has finally been proven. I was a making it much harder than it needed to be, but I guess that's better than assuming it's going to be easier than it acutally is. I tried writing a program for Maple that would leta user specify a graph and output its eigenstuff. However, it was unwieldy and not very efficient, so I'm investigating the open-source Sage project as an alternative. I've also began exploring the relationships between graphs and finite geometries.

Week 3: Of the 6 conjectures I originally came up with, I proved 3. One of the others I was trying to prove was very difficult, so Gene suggested I generalize the problem and work on that instead. So I did, but I hit a roadblock. That was until Gene said he came up with an idea to solve it.

Week 2: I spent the first half of the week doing some more reading on Linear Algeba and Finite Geometry. The last half I spent some time coming up with conjectures related to graphs and their adjacency matrices eigenvalues. I came up with an upper bound for the eigenvalues and a general form for the adjacency matrices of complete graphs raised to any power.

Week 1: I met with Dr. Fiorini a few times to narrow down my options for the research project. The options were graph pebbling, finite geometry, or extremal graph theory. All were very interesting, but finite geometry sounded really exotic and interesting. I was given some books to read, and I believe there is a connection between finite geometry, graph theory, and linear algebra. I also made my webpage and gave my first presentation.