REU Information and Links

In 2012, we will investigate spatially embedded graphs. 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). As part of our investigations, we might try to classify all maximally flat graphs, but this is probably a difficult problem (see Thomas Bohme, On spatial representations of graphs. Contemporary methods in graph theory, 151-167, Bibliographisches Inst., Mannheim, 1990).

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. We also might use some group theory, so experience with a modern/abstract algebra course could be helpful.

Students in my group will be attending the UnKnot Conference at Denison University, July 15-18, 2011.

Joel Foisy's REU group history:

2011: hiatus

2010: Flexibly planar and flexibly flat graphs. Paper is still in the editing process.

2009: Intrinsically linked signed graphs in projective space. (Paper submitted.)

2008: Intrinsically 3-linked graphs in projective space. (Paper submitted.)

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

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).

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:

SUNY Potsdam/Clarkson 2012 REU web page, with application forms.

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Some participants in our 2003 program in the Adirondacks:

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