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 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 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 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 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 the paper “When Lines Go Bad In Hyperspace” 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 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 Journal of Geometry, (2009), 92, Numbers 1-2, 28-59.

·        Chantel’s Final Report

·        Alex’s Final Report

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). 

·        Lisa and Dan’s Final Report

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 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 (minimally) on 10/20/2015