Research Experiences with Undergraduates

Steve Schlicker

 

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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 2008 REU students can be found here. A summary discussion of the results of our REU research can be found at The Strange World of the Hausdorff Metric Geometry page.  Our current REU grant runs through the summer of 2009 and I plan to continue studying this geometry in the future. 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, now at the University of Virginia). 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 (than at the University of California, Fullerton, now at the University of Southern California) and Audrey Malagon (Powers) (then at Agnes Scott College, now at Emory 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 n) as well as determine quite a bit about lines in this space. A paper summarizing the work done to August 2002 on this geometry can be found here:

·         The Geometry of the Hausdorff Metric

Our results appear 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 at SUNY Stony Brook), Amber Swift (Lembcke) (Concordia College) 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 Demonstratio Mathematica (No. 3, Volume 38 (2005), p. 689-701).

·         Amber’s Final Report

·         Chris’ Final Report

The 2004 REU, with Kris Lund (then at GVSU, now at the University of Nebraska – Lincoln) and Patrick Sigmon (then at Wake Forest University, now at North Carolina State University) 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. We have submitted two papers for publication.

·         Kris and Patrick’s Final Report

In 2005, my students Chantel Blackburn (Andrews University, now at the University of Arizona) and Alex Zupan (Gustavus Adolphus College, soon to be in graduate school somewhere), 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. One paper has been submitted for publication (with the 2004 group) and one other is in preparation.

·         Chantel’s Final Report

·         Alex’s Final Report

The 2006 REU, with Lisa Morales (California State Polytechnic University Pomona) and Dan Schultheis (University of Washington, will attend the graduate program 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 (while will hopefully appear some time soon at The Strange World of the Hausdorff Metric Geometry. We also discovered an infinite family of previously undocumented (at the On-Line Encyclopedia of Integer Sequences, hopefully we will add them soon) sequences from Polygonal Chains. There is good potential to obtain a published paper from this work. 

·         Lisa and Dan’s Final Report

My 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. 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.

Preprints of both of these papers are available here:

·         Sierpinski Polygons

·         Sierpinski Polyhedra

                     

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This page was last updated on 12/22/2005