*Grand Valley
State University Department of Mathematics*

*Research
Experiences for Undergraduates Program*

For many 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. 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, now working as a software developer). 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, now an
Assistant Professor at the University of the Pacific) and Audrey
Malagon (Powers) (then at Agnes Scott College, now an Associate
Professor at Virginia Wesleyan University). We continued the investigation of
the geometry of the Hausdorff metric Dominic and I began in summer 2000. We
classified circles and disks in H(**R*** ^{n}*) (the space of all non-empty, compact subsets of

In our REU 2003, Christopher Bay (then at Truman
State University, now Director of Education -
Curriculum at LaunchCode in St. Louis, Missouri), Amber
Swift (Lembcke) (then at Concordia College, now an actuary for Ameriprise Financial) and I had a great
time extending the previous REU results about the Hausdorff metric geometry.
Chris and Amber worked on separate but connected problems. Amber investigated
the geometry of two-point sets and Chris was able to determine some general
properties of Hausdorff lines. These results appear in the paper “When
Lines Go Bad In Hyperspace” in *Demonstratio
Mathematica*. Chris presented his work at Mathfest in 2003 and at
the Undergraduate Poster Session at the annual Joint Mathematics Meetings (JMM)
of the American Mathematical Society and the Mathematical Association of
America in 2003. Chris won a best presentation award at Mathfest and a best
poster presentation award at JMM.

The 2004 REU, with Kris Lund (then
at GVSU, now working at Foremost Insurance) and Patrick Sigmon
(then at Wake Forest University, went to Harvard School of Law, now an
associate in Davis Polk’s Tax Department) 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. Results of some of our work appear in the paper “Fibonacci sequences in the
space of compact sets” in *Involve*.

In 2005, my
students Chantel Blackburn (then at Andrews University, now an
Associate Professor at Pacific Union College) 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 began 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 teaching mathematics) and Dan Schultheis
(then at the University of Washington, now at Smith College) 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* and several of these sequences are in the On-Line Encyclopedia of Integer Sequences.
Lisa and Dan presented their work at the Undergraduate Poster Session at the
annual Joint Meetings of the American Mathematical Society and the Mathematical
Association of America in 2006 and won a best poster presentation award.

I worked with Katrina Honigs (then at Grinnell
College, now at the University of Utah) and Vincent Martinez
(then at The College of New Jersey, now at Tulane 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.
Her work appears in “Missing edge
coverings of bipartite graphs and the geometry of the Hausdorff metric”,

My 2010 REU group consisted of
Michael Sanchez (then at New Mexico State University and now working in the
University of New Mexico system) and Jon VerWys (then at Grand Valley State
University, now working at Eastfield College). We worked on orthogonality in H(**R*** ^{n}*).

In 2014 my students were Pallavi
Aggarwal (at the California Institute of Technology) and Ryan Swartzentruber
(then at Eastern Mennonite University). We investigated the notion of
orthogonality in the space of compact sets, and some of the results of our work
appear in “Pythagorean
Orthogonality of Compact Sets”, *Involve*.

*Other Research Experiences with Undergraduates*

My first experience doing
research with an undergraduate student occurred before I came to Grand Valley,
when I was on the faculty at Luther College in Decorah, Iowa. The student, Kevin
Dennis, was interested in fractal geometry and we started working on
generalizing the construction of the Sierpinski triangle to other polygons. The
result was a paper titled “Sierpinski
Polygons” which appeared in the spring 1995 issue of the *Pi Mu Epsilon Journal*.
This paper won Kevin a Richard V. Andree Award - sharing second place -in the *Pi Mu Epsilon Journal's* National Student
Paper Competition. Kevin has since received his Ph.D. in mathematics with an
emphasis in college teaching from Central Michigan University. He is now a
faculty member at Saint Mary’s University of Minnesota.

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.

Geoff Patterson did his senior thesis (MTH 496 in fall 2008, Project Title “A Sieve for Betweenness of Compact Sets”) under my supervision. Geoff earned his Ph.D. in mathematics from the University of Hawaii.

In the summer of 2015 I supervised the senior thesis of Burritt Watrous. Burritt began an investigation into applying the Hausdorff metric to the word metric on groups to define a geometry of subgroups of a group. Burritt was able to classify all four-point geometries that arise in this way. Burritt has recently founded High Strata Marketing.

In summer 2017 Casey Koch-LaRue expanded the work from Burritt’s 2015 senior thesis in a research project sponsored by the Student Summer Scholars Program at GVSU. Casey applied the Hausdorff metric to the word metric on groups to define a geometry of subgroups of a group. Using this method, Casey was able to create new finite geometries from the subgroup structure of a group. Casey has a lot of really good results and is in the process of preparing them for publication. He is now a graduate student at the University of Washington.

The winter semester of 2018 found Liah Renaud and Shannon
Napier conducting a research project for their honor’s thesis under my
supervision. Liah and Shannon investigated what they called “Cross
Configurations” in H(**R*** ^{n}*)
and discovered an infinite family of previously undocumented sequences. Liah is
now a Risk Analyst at Ally

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This page was last updated on 06/29/2018