In 2014, we will investigate links in spatially embedded graphs, and possibly in planar graphs as well. We will use methods from geometric topology as well as graph theory. Good introductory sources include Chapter 8 of Colin Adam's Knot Book, as well as any graph theory book that discusses Kuratowski's Theorem (texts by Rienhard Diestel and Douglas West are good).
There is no specific prerequisite background for working in my group, aside from strong mathematical maturity. It would, however, not hurt to have some experience with topology, graph theory, knot theory or LaTex.
Joel Foisy's REU group history:
2013: we ended up studying a purely graph theoretic idea: weakly diameter m critical graphs. The main paper is in the editing stages, but hopefully will be completed soon: On weakly diameter-m-critical graphs,
2012: Two topics. (Papers are still being edited, the first one has some deep issues that I hope will soon be resolved, the second is in the hands of one of the student authors and hopefully will be submitted very soon): the generalized conflict graph,
2010: Flexibly planar and flexibly flat graphs.
2008: Intrinsically 3-linked graphs in projective space.
2006: Lower dimensional versions of intrinsically linked graphs. (Paper published in Rose Hulman Journal: Intrinsically S^1 3-linked graphs an other aspects of S^1 embeddings).
2003: Knotted Hamiltionian cycles in spatial graphs. (Two papers published: Some results on intrinsically knotted graphs , with Tom Fleming and REU students Garry Bowlin, Paul Blain, Jacob Hendricks, and Jason LaCombe,J. Knot Theory Ramif., (16), no. 6 (2007); 749-760. Also: Knotted Hamiltonian cycles in spatial embeddings of complete graphs , with former REU '03 students Garry Bowlin, Paul Blain, Jacob Hendricks, and Jason LaCombe, New York J. Math. (13), (2007); 11-16.)
2002: Intrinsic Chirality. Their results were already known, as we learned later. They did complete a short paper, but I do not have it available electronically.
2001: Intrinsically linked graphs with an unused vertex.
1999: Disjoint Linking property of spatial graphs. Their results were combined with the 2000 group's results in the paper listed above.
1997: Perimeter minimizing enclosures, using a corner. (Paper published: J. Foisy, G. Christopher Hruska, Dmitriy Leykekhman, Daniel Pinzon, and Brian J. Shay; Rocky Mountain J. of Math., (31), no. 2 (2001); 437-482.)
You might want to check out:
Some participants in our 2003 program in the Adirondacks:
Earlier that same day at nearby Owl's Head Mountain: