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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

**Participants:**

* Lyubov Chumakova*, University of Wisconsin

*Joel Louwsma *, University of Michigan

*Sharon Lutz *, University of Colorado, Boulder

* Alan Tarr*, Pomona College

*Chris Willmore*, Purdue University

*Racheal Allen*, California State University Northridge

*Maria G. Uribe*, California State University Northridge

* Adilson Eduardo Presoto *, Universidade Federal de Sao Carlos(UFSCar)

*Anderson Tiago da Silva *, Universidade Federal de Vicosa

*Anne Caroline Bronzi*, Universidade Estadual de Campinas (UNICAMP)

* Jose Regis Azevedo Varao Filho *, (UNICAMP)

*Leonardo Barichello*, UNICAMP

*Patricia Romano Cirilo*, Universidade Federal de Minas Gerais

*Roberta Camelucci Carrocine*, (UFSCar)

*Welington Vieira Assuncao*, Universidade Estadual do Estado de Sao Paulo, Rio Claro

Brazilian participants were supported by the University Foundation, **FAEP**.

**Organizers:** * M. Helena Noronha,* California State University Northridge, and *Marcelo Firer,* UNICAMP.

**Faculty Advisors: ** * Renato H. L. Pedrosa, Plamen Emilov Koshlukov, Ana Friedlander * (UNICAMP), * Yuriko Baldin *, (UFSCar), and * M. Helena Noronha,* California State University Northridge.

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.

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.

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.

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 one¹s 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.

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.

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.

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.