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Infinity may be an immeasurable item, but Greg Francos ’05 (Haddonfield, N.J.) helped prove a “limitless” math problem.
Francos, a mathematics major and economics & business minor, devoted his time to finding the best possible place to locate a computer network server in a grid that has no boundaries.
“Specifically, what we did on this project was to look at an optimal network,” says Francos, who worked on the problem under the guidance of Gary Gordon, professor of mathematics. “We took any size grid and looked at where to place the root of the network, which is the server. The more computers you add, the more the problem becomes difficult very quickly.”
Francos was one of three Lafayette students conducting mathematics research with peers at other top undergraduate institutions from around the country in the National Science Foundation’s Research Experience for Undergraduates program on campus this summer. Their work will be submitted as a paper to a scholarly journal
About 100 students have participated in REU at Lafayette since 1992, according to Gordon, who coordinates the program. Most have published papers in professional journals and/or presented talks on their summer research at national mathematics conferences, and are in graduate school or have earned a Ph.D.
The National Science Foundation funding for the Lafayette students is supplemented by Lafayette’s distinctive EXCEL Scholars program, in which students conduct research with faculty while earning a stipend. The program has helped to make Lafayette a national leader in undergraduate research. Many of the more than 160 students who participate each year share their work through articles in academic journals and/or conference presentations.
“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 students with faculty and/or other research mentors and access to appropriate facilities and professional development opportunities.”
Computer engineers of the future might use Francos’ work to design a streamlined network that saves money and time by avoiding redundant connections, Gordon says.
The solution to the problem could also help engineers identify the most vulnerable areas of a network and compel those people to spend more time strengthening the areas of that system.
The team first had to devise a written formula proving that a server must be placed in a specific position to get the most effective result, Gordon says.
“Technically, we tried to do an infinite problem,” he explains. “We just tried to do it in a finite number of steps.”
Finding, developing, and writing down those steps in the form of a proof was the hard part.
The more components added to the network and the larger its boundaries become, the harder it is to devise the proof, Francos says.
“If you have a tiny grid, you might be able to do something,” he notes. “But the larger the problem — it gets almost impossible.”
Francos learned a tremendous amount about optimality and mathematics through the project.
“When you’re in the classroom, you’re typically getting problems that you expect you can solve,” he says. “You’re doing problems out of the math book that are solvable and you know they are solvable. Here, we didn’t know whether the problem was solvable, what kind of tools we would use to solve it or how we could look at this problem differently to solve it.”
For students, learning through research brings a higher level of frustration than does classroom learning, Gordon says.
“When you’re doing research, not only might you be stuck on a problem for which you don’t know whether it’s true or false, but you might not have any idea how to proceed, or you might have an idea and it doesn’t work and you don’t know what to do next,” he notes.
On the other hand, solving questions in research has the same thrill as solving a complicated puzzle, he adds.
“You have to figure out what combinations of things put together in the right way will solve the problem,” Gordon says. “In the classroom, everything is often predigested puzzle pieces you know will fit together. Here the puzzle pieces could be all over the floor but they might not even fit in the puzzle.”
The professor found Francos to be well suited to the work.
“He gets a lot of ideas and he’s really tenacious,” Gordon says. “Once he gets an idea, he really works on it. He may have to modify it, but he doesn’t give up very easily and that’s really important. He also has a lot of talent for abstract math reasoning skills.”
As Francos used those skills to slowly work his way to finding the solution of this complicated problem, he learned he has the abilities required to pursue his goals of becoming a professor and full-time researcher, he says.
“I definitely gained some confidence from it,” Francos says. “I learned what it takes to succeed and I saw — with other students around me who were so strong — what can be done and the work that goes into success. But also, maybe I wasn’t sure when I got here what I could do or if I’d be able to hang with the best and the brightest, but now I’m seeing that I can.”
In addition to his research, Francos has been involved in the mathematics department’s Problem Solving Group and competed in its Math Bowl and the Lehigh Valley Association of Independent Colleges Math Contest. He has played intramural frisbee and plays guitar and sings in The Catfish Blues Band.
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 the last annual conference in April.