My 2009 group will work on problems in topological graph theory, combining knot theory and graph theory. A good place to learn some background is "The Knot Book," by Colin Adams. Check out chapter 8 in particular, dealing with intrinsically linked and intrinsically knotted graphs. We might also discuss issues related to planarity of graphs (see Kuratowski's Theorem in most graph theory books.) You might also want to check out:
Some participants in our 2003 program in the Adirondacks:
Earlier that same day at nearby Owl's Head Mountain: