This REU in mathematics was hosted by UNICAMP (Universidade de Campinas), a state University in Sao Paulo. Campinas is located in a large industrial center in Sao Paulo state and is surrounded by technological and computational centers.

American Participants

Students and Faculty together

On the first day of the program participants were given the option to select the problem to work on. The problems had been presented to them a month before through the REU Website. On the last day of the program, students gave presentations of their work. They also wrote papers describing their results. Click on the title of the papers for a pdf version.

Symmetry breaking for optimal periodic composite membranes.
Welington Vieira Assuncao, Anne Caroline Bronzi, and Lyubov Chumakova.

It is shown that configurations with minimal first eigenvalue for the 1-periodic composite membrane in \R^2 (the membrane is a strip \Omega=R x [0,1]) are not necessarily the ones invariant under x-translations, for given basic data, if the period is taken sufficiently large.
A new computation of the codimension sequence of the Grassmann algebra.
Joel Louwsma, Adilson Eduardo Presoto, and Alan Tarr.

Krakowski and Regev compute the codimension se- quence of the T-ideal of polynomial identities of the Grassmann algebra using polynomial relations. We give an elementary proof of their main result and obtain some corollaries.

On the Application of Calculus toward Both Continuous and Discrete Optimization.
Chris Willmore and Anderson Tiago da Silva.

The search for the maximum or minimum of a function is often an essential problem in both pure and applied mathematics; it can be found anywhere from finding the best route to the supermarket to finding the best mix of chemicals for a needed reaction, from maximizing onešs crop yield for the year to minimizing the stress on an iron beam used to construct a building. The canonical way of finding the maximum of a real function has been to find its derivative and then locate its roots, but sometimes that derivative is not readily available, and sometimes mitigating circumstances can get in the way of a simple derivative reading as that. This is especially true for more complex systems which cannot be represented as just an equation from R to R. We will explore two problems and explore their solutions, investigating ways to find maxima and minima of complex systems: one maximizing a complex function along a line, and one minimizing a function from Rn to R with very large n.
Closed and Exact Differential Forms.
Patricia R. Cirillo, Jose Regis A. Varao Filho, and Sharon M. Lutz.

We show in this paper that if every closed 1-form defined on a domain of the plane is exact then such a domain is simply connected. We also show that this result does not hold in dimensions 3 and 4.

The prospective teachers also worked on research problems and compared the curriculum of teacher preparation of their universities. Thier papers as well as the compariosn essay is below.
Parallels in Hyperbolic Geometry.
Racheal Allen, Leonardo Barichello, Roberta C. Carrocine, and Maria G. Uribe.

Consider two parallel lines in Euclidean plane and a transversal intersectin the lines at points A and B. If fix A and let B go to infinity, the family of rotations given by the composition of reflections in the transversal and in the line containing B converges to a translation. We study this problem in the Hyperbolic plane and show that this result generalizes only when the two given lines are critically parallel.

An open historical Calculus problem.
Racheal Allen, Leonardo Barichello, Roberta C. Carrocine, and Maria G. Uribe.

This problem was first explored by Japanese Mathematicians during the period in which Japan was isolated from the western culture (1603-1867). It is important to observe that during this era, the Japanese didnt have access to the progress that was being made in the calculus eld. However, they were still able to work on geometry problems using their own techniques of Calculus. In this problem we found that even by using advanced calculus techniques, it was still difficult to solve by hand. Probably because of this difficulty we never found a satisfactory answer in the literature.

A comparison between the American and Brazilian teaching preparation curricula.
Racheal Allen and Maria G. Uribe.