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(Rn) (the space of all non-empty, compact subsets of Rn) as well as determine quite a bit about lines in this space. Some of 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 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(Rn) 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 Journal of Geometry. Chantel and Alex both presented their work at Mathfest 2005: Chantel won the Council on Undergraduate Research award as best Pi Mu Epsilon research project and Alex won the Council on Undergraduate Research award as best MAA research project.
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(Rn) 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”, Journal of Geometry.
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. Katrina
and Vincent presented their work at Mathfest 2007 and won a best presentation
award in their session.
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(Rn).
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(Rn) and discovered an infinite family of previously undocumented sequences. Liah is now a Risk Analyst at Ally
Return to my home page or the department home page.
This page was last updated on 06/29/2018