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Mathematics majors Kevin Penderghest ’05 (Lindenwold, N.J.) and Prince Chidyagwai’05 (Marondera, Zimbabwe) traveled to Atlanta last month to present separate research projects at the joint national meeting of the American Mathematical Society and Mathematics Association of America.

Both presentations went well, says Chidyagwai, who spoke before an audience of about 40 mathematicians. Each student spoke at individual sessions, which allow 20 minutes for presentations and 10 minutes to answer questions. Chidyagwai, who also is a computer science major, says he handled two of the three questions thrown his way and received help from an audience member on a third.

Lafayette students also typically present their research at the annual Student Mathematics Conference hosted by the Pi Mu Epsilon (math honor society) chapter of Moravian College, where Penderghest shared prior work in 2003. This year’s conference will take place Saturday, Feb. 19. (Chidyagwai’s experience includes presentations at conferences in Philadelphia and Baltimore.)

Along with fellow math majors Greg Francos ’05(Haddonfield, N.J.) and Jacob Carson ’06 (New Richmond, Ohio), Penderghest and Chidyagwai partnered with peers from other undergraduate institutions to conduct mathematics research as part of the National Science Foundation’s Research Experience for Undergraduates program on campus.

Mentored by Lafayette mathematics professors, only ten of the more than 200 applicants participated in the program. Other institutions with student representatives included M.I.T, the Universities of Rochester, Akron, Nebraska, and Chicago, and Washington University of St. Louis.

According to Gary Gordon, professor of mathematics and program coordinator, about 100 students have participated in Research Experience for Undergraduates at Lafayette since 1992. Most have published papers in professional journals and/or presented talks on their summer research at national mathematics conferences, and are currently in graduate school or have earned their Ph.D’s.

Penderghest, who worked with John Meier, professor of mathematics, focused on knot theory. Meier likened the problem to a computer network in which wires from all of the components become tangled between the back of the machines and the wall outlet.

All prior theories assumed that nothing could be done to untangle those wires unless they become unplugged from the backs of the machines, Meier says.

The research group developed an easy test to determine if that three-dimensional knot can be undone without becoming unplugged. They worked with alternating graphs and the fundamentals groups of those graphs, and their results were encouraging.

“We showed that when a graph is alternating, the algebra of the group is very complicated, but not completely unsolvable,” Penderghest says.

The group has written a paper on its findings and submitted an article to The Journal of Knot Theory.

Chidyagwai worked with Cliff Reiter, professor of mathematics, and studied a structure that he compared to a snowflake. He grew quasicrystalline structures and analyzed their cellular automa growth.

He and Reiter also studied the structures to learn more about parallel algorithms, which they created in the laboratory. Chidyagwai used conanical projections, or projecting one dimension of the structure into two dimensions, to make the algorithms. He designed experiments that changed the parameters of the quasicrystalline structures. This allowed him to study their effect on the structures’ growth.

“We experimented with different patterns of growth,” Chidyagwai says. “Our research was very demanding on computing power, and one of the things we were able to do was to develop algorithms that can work effectively.”

Francos worked with Gordon, concentrating on graph theory and probability while attempting to find a rigorous proof for the best possible place to locate a network server in a grid of unknown dimensions.

“Specifically, what we did on this project is looked at an optimal network,” Francos says. “We took any size grid and looked where to place the root of the network, which is the server. The more computers you added, the more the problem became difficult very quickly.”

Finding the best place to locate a server has applications for a host of network problems. This could help computer engineers design a streamlined network that saves money and time by avoiding redundant connections, Gordon says.

“The solution to this problem could also help engineers identify the most vulnerable areas of a network and compel those people to spend more time strengthening those areas of that system,” he adds.

Carson’s research also focused on graph theory, but was applied to a popular 1960s game called Instant Insanity.

“The idea was that you have cubes where each cube is a different color and you stack them in a tower. You have a solution if on each side of the tower there is no one repeated color,” says Carson, who worked with Ethan Berkove, assistant professor of mathematics.

“The project was to look at a particular variation of Instant Insanity. A die has a square as its top and bottom faces. Instead, one could use a solid with a pentagon, hexagon, or any polygon with more sides for the top and bottom faces. This would generate a corresponding puzzle with more sides. We wanted to know how this generalization affects our ability to find a solution to the puzzle.”

Carson looked for conditions on the colors of the puzzle components, which guarantee a solution. He also tried to determine an efficient way to find a solution when it exists.

“Jacob has some interesting partial results,” says Berkove. “We intend to continue our exploration of this project in the spring under the EXCEL program.”

Carson says the problems stretched his mental capabilities by pushing him to focus on scrutinizing the mathematical puzzles.

The National Science Foundation funding for the Lafayette students is supplemented by the College’s EXCEL Scholars program, which funds student-faculty collaborative research.

“Active research experience is considered one of the most effective ways to attract talented undergraduates to and retain them in careers in science and engineering, including careers in teaching,” says the National Science Foundation. “REU projects feature high-quality interaction of the students with faculty and/or other research mentors and access to appropriate facilities and professional development opportunities.”

As a national leader in undergraduate research, Lafayette sends one of the largest contingents to the National Conference on Undergraduate Research each year. Forty-two students were accepted to present their work at last year’s annual conference.

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