For 2016, I am still considering various options involving spatially embedded graphs. As one project, we might study a certain topological property of graphs embedded in the projective plane.

There is no specific prerequisite background for working in my group, aside from strong mathematical maturity. I will be looking for students who are ready to follow up on their work beyond the summer program.

Joel Foisy's REU group history:

2015: The group started with a classic paper, "On the dimension of a graph," by Erdos, Harary and Tutte and then worked on some related problems. Perhaps a general classification for these problems would be "geometric graph theory." They wrote a paper, currently under revision, studying the dimension of a graph, with a definition of dimension similar to, but different from, the definition of Erdos, Harary and Tutte.

2014: The group studied three topics and wrote a whopping 3 papers. One has been submitted already (good job Madeleine!) The other two are in progress. Here are links to them: Intrinsically spherical linked graphs and Linking of n-spheres.

2013: We ended up studying a purely graph theoretic idea: weakly diameter m critical graphs. The main paper is going to appear in Graph Theory Notes of New York: 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,

and intrinsically linked permutation graphs.

2010: Flexibly planar and flexibly flat graphs.

2009: Intrinsically linked signed graphs in projective space. (Paper published, Discrete Mathematics, 312 (2012), no. 12-13, 2009-2022.)

2008: Intrinsically 3-linked graphs in projective space. (Recently accepted for publication.)

2007: Intrinsically linked graphs in projective space. (Paper published: Algebr. Geom. Topol. 9 (2009), no. 3, 1255--1274.)

2005: Lower dimensional versions of intrinsically linked graphs. Some of their results are here.

2004: Linkable and knottable graphs. (Paper published: On graphs for which every planar immersion lifts to a knotted spatial embedding, with REU students Amy DeCelles, Chad Versace, and Alice Wilson, Involve 1 (2008), no. 2, 145--158.)

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.

2000: Disjoint Linking property of spatial graphs. (Paper published: Graphs with disjoint links in every spatial embedding, with REU '99 and '00 students Anton Dochtermann, Stephen Chan, Jennifer Hespen, Trent Lalonde, Quincy Looney, Eman Kunz, Katherine Sharrow and Nathan Thomas. J. Knot Theory Ramif., (13), no. 6 (2004); 737-748.)

1999: Disjoint Linking property of spatial graphs. Their results were combined with the 2000 group's results in the paper listed above.

1998: hiatus

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, before my hair turned grey:

Earlier that same day at nearby Owl's Head Mountain: