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*Grand Valley
State University Department of Mathematics*

*Research
Experiences for Undergraduates Program*

For the last several years, I have been involved with the Grand Valley State University Department of Mathematics Research Experiences for Undergraduates Program (GVSU-REU). The program is funded by the National Science Foundation. A list of REU sites in mathematics is available here. The area in which I have been working with my REU students is in the geometry of the Hausdorff metric. Reading material for the 2016 REU students can be found here. A summary discussion of some past results of our REU research can be found at The Strange World of the Hausdorff Metric Geometry page. If you are interested in working in this area, please apply to our REU.

In the summer of 2000, I worked with a student, Dominic
Braun, (then at the University of North Carolina at Asheville, graduate
school at the University of Virginia, but I don’t know where Dominic is now).
Together we studied the Hausdorff metric in the space of all non-empty compact
subsets of *n*-dimensional real space.

In 2002, I had the pleasure of working with two students, John
Mayberry (then at the University of California, Fullerton, graduate school
at the University of Southern California, now at a one of the California state
universities, but I forget which one) and Audrey Malagon (Powers) (then at Agnes Scott College, graduate
school at Emory University, now at Mercer university). We continued the
investigation of the geometry of the Hausdorff metric Dominic and I began in
summer 2000. We have been able to classify circles and disks in H(R * ^{n}*) (the space
of all non-empty, compact subsets of R

· The Geometry of the Hausdorff Metric

Our results appear in the paper “A Singular Introduction to the Hausdorff
Metric Geometry” in the *Pi Mu Epsilon Journal*
, Vol. 12, No. 3, p. 129-138, 2005.

In our REU 2003, Christopher Bay (then at *Demonstratio Mathematica, *No. 3, Volume 38 (2005),
p. 689-701.

The 2004 REU, with Kris Lund (then at GVSU, now
working at Foremost Insurance) and Patrick Sigmon (then at Wake Forest University, now at the
Harvard School of Law) focused on line segments in the Hausdorff metric
geometry. We found some interesting connections in this geometry to the
Fibonacci and Lucas numbers, an amazing result about the number 19, and other
fun stuff. You can read all about it in Kris and Patrick’s final report.
Results of some of our work appear in the paper “Fibonacci sequences in the
space of compact sets” in *Involve*,
Vol. 1 (2008), No. 2, 197-215.

· Kris and Patrick’s Final Report

In 2005, my students Chantel Blackburn (then at
Andrews University, now at the University of Arizona) and Alex
Zupan (then at Gustavus Adolphus College, now at
the University of Iowa), and I made connections between infinite and finite
configurations in H(R * ^{n}*) that
allowed us to complete the work bean in the 2004 REU related to the number 19.
We also discovered some interesting applications of the Hausdorff metric
geometry to graph theory. A paper on our work (connected to the work of the 2004 group) titled “A
Missing Prime Configuration in the Hausdorff Metric Geometry” appears in the

The 2006 REU, with Lisa Morales (then at California State
Polytechnic University Pomona, now at the University of California at
Riverside) and Dan Schultheis
(then at the University of Washington, now at the University of California, San
Diego) was again a productive one. We created an algorithm to compute the
number of sets at each location in a finite configuration and discovered an
infinite family of previously undocumented sequences from Polygonal Chains. The
results of the latter appear in “Polygonal chain sequences in the space of
compact sets” in the *Journal
of Integer Sequences*, Vol. 12 (2009), Article 09.1.7 and several of
these sequences are in the On-Line Encyclopedia of
Integer Sequences. (Another interesting site about integer sequences is LearnStuff – thanks to Erin Williams for this
link).

I worked with Katrina Honigs (then at Grinnell
College, now at the University of California at Berkeley) and Vincent Martinez (then at The College of New Jersey, now
at Indiana University) in the 2007 REU. Katrina made connections between finite
configurations in H(R * ^{n}*) and
bipartite graphs that allowed her to extend the work of the 2004 and 2005
groups to show that there are other numbers that “missing” in this geometry.
She has submitted a paper for publication. Vincent began a study of convexity
in this space and come up with two different types of convexity to study. This
is an interesting avenue of investigation and one that can be built upon in the
future by an interested student.

In 2008, I supervised David Montague (University of Michigan). David obtained
some fascinating results about when we can have finite sets between two sets.
David has a really nice paper to submit, and is still working to extend his
results.

In 2008, I also worked with two
other students, Aaron Shatzer
(then at Luther College, now at the University of Oregon) and Haggai Nuchi (then at Carleton College,
now at the University of Pennsylvania) on problems involving roots of
polynomials. I hope to provide some additional information about this project
at a later date.

My 2010 REU group consisted of
Michael Sanchez of New Mexico State University and Jon VerWys
at Grand Valley. Pictures and additional details will be posted at some
undetermined date in the future.

In 2014 my students were Pallavi Aggarwal of the California Institute of Technology and Ryan Swartzentruber of Eastern Mennonite University. We investigated the notion of orthogonality in the space of compact sets, and have submitted a paper on the topic. Additional details may appear on this page sometime in the future.

*My Other Research Experiences with Undergraduates*

My first experience doing research with an undergraduate student occurred
before I came to *Pi Mu Epsilon Journal*. Kevin
has since received his Ph.D. in mathematics with an emphasis in college
teaching from

Not long after I came to GVSU, the Division of Science and Mathematics
instituted the Summer Undergraduate Research Program (SURP), whose goal is to
support and promote collaborative research between undergraduate students and
faculty. A student, Aimee Kunnen, and I received SURP
support for a project titled "Sierpinski Polyhedra". This project generated the construction
ideas began in "Sierpinski Polygons" to regular
polyhedra. The resulting paper, "Regular Sierpinski Polyhedra"
appeared in the spring 1998 issue of the *Pi Mu Epsilon Journal*. Aimee is
currently at home enjoying her children.

Preprints of both of these papers are available here:

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This page was last updated (minimally) on 10/20/2015