I’m offering a couple of possible projects for third-year students at Bristol (i.e., for MATH32001 or MATH32200 depending on amount of time) in the upcoming academic year (2012/13). Other topics in areas such as spectral graph theory or computational number theory could also be possibilities; students are welcome to get in touch (via my magdt@bristol account) to arrange a more in-depth discussion.
1) The Mathematics of Social Networks
This project would start with an introduction to the key concepts of graph theory, before turning to applications to ‘social’ graphs. For instance, how might we identify the most influential scientists from the graph of their collaborations? If two countries come into conflict, who are their neighbours more likely to ally with? And how can it be that, on average, your friends have more friends than you do- yet the same is true for them?
2) Discrete and Computational Geometry
Although the study of geometry is almost as old as mathematics itself, discrete and computational geometry – at the intersection of computer science and mathematics – has only attracted attention in the last few decades. Nonetheless, it has found many applications, such as terrain reconstruction, robot motion planning, and remarkable advances in the discipline of ‘mathematical origami’.
A project in this area might suit a student with a strong interest/ability in programming (particularly related to computer graphics), but the material can also be approached in a purely theoretical way.