Undergrads pursue Summer Research Projects
Although summer is typically a time for students to kick back and take a break from their studies, that's not the case for several mathematics students who are in residence in the Mathematics Department. With support provided by the National Science Foundation and Purdue alumni Andy Zoltners and Joel Spira, eleven math undergrads have been engaged in research projects under the guidance of mathematics faculty.
Working with Prof. Jim McClure, sophomore Alec McGail has been reading in David Bressoud's book A Radical Introduction to Real Analysis. Alec has also been working on a project to reconstruct all of the standard properties of sin and cos by using only facts about their power series. Since Alec has a special interest in philosophy, Prof. McClure asked him to write a philosophical reflection on whether the sin and cos he creates this way are "the same" as the ones he knew in high school.
In independent discussions with Prof. Uli Walther, senior Kathryn Marsh has investigated various topics in mathematics, touching on subjects in group theory, topology, knot theory, and prime numbers. Weekly meetings supplement and guide individual reading. Between students, communication is in the form of editing a class Rhea page on topics they are individually researching.
Seniors Brett Ernst, Sean Fancher and junior Abi Komanduru, who are working with Golomb Assistant Prof. Erik Lundberg, are using complex variables in order to model the lensing effect imposed by a spiral galaxy on background sources. Since mass bends light, any collection of mass acts as a lens, magnifying, distorting, or even creating multiple images of a single background source. By comparing models of gravitational lenses to observed lensing events, astronomers can make new discoveries about the universe and the structure of galaxies. There is no widely accepted theory for the formation of arms in a spiral galaxy, and the model Abi, Brett, and Sean are developing could lead to evidence favoring one of the competing theories.
In a project with Prof. Jon Peterson, junior Lirong Yuan and senior Jingdan Liu focused on studying the probabilistic properties of eigenvalues of random matrices and roots of random polynomials. While the statements of the problems they were interested in were very basic, the research brought up a variety of topics not often encountered by undergraduates: Rouche's theorem, Cardano's formula, weak convergence of probability distributions, and the change of variables formula for n-dimensional integrals. In addition to learning some more advanced mathematics, Lirong and Jingdan were able to experience how research in mathematics works. In particular, they were able to see that hard problems are attacked by first solving smaller simplifications of the problem, and that sometimes one needs to try several different approaches to a problem before finding one that works. Lirong likened mathematical research to climbing a mountain: when standing at the base of the mountain there are many possible routes to the top, but it might take several attempts to find the best route to the summit; moreover, the joy when reaching the summit makes all the previous failed attempts worthwhile.
Under the direction of Prof. Edray Goins and his doctoral student Jamie Weigandt, seniors Roland Archer, Benito Martinez, junior Han Liu, sophomore Steve Mussman, and Sergio Garcia Curras (a visiting student from University of Puerto Rico) are working on "Squares and Cubes in Arithmetic Progressions." The project has its origins in classical number theory and seeks explicit examples of four squares and three cubes in arithmetic progressions, and to recast many ideas by performing a complete 2-descent of quadratic twists of certain elliptic curves. The group plans to present their research at MAA MathFest, to be held at the University of Wisconsin at Madison August 2-4, 2012.