# Senior Seminar

The Senior Seminar is the culmination of the Mathematics and Computer Science majorâ€²s experience in the department. In the Seminar, students engage in independent research on a topic or question of their own choosing under the direction of a faculty member. Throughout the Fall and Spring semesters they present what they have found to other students and the faculty. The final presentation in the Spring is a public talk, open to the entire Rhodes College community. In both semesters, students write a paper summarizing their work.

Writing and presenting form an integral part of the Senior Seminar experience---one goal of the Spring semester is for the student to produce a work of publishable quality. Students are encouraged to study the suggested writing and presenting guidelines and to talk with their faculty advisors and Seminar coordinator.

## Senior Seminar Projects

The following are some possible Senior Seminar topics/projects. They are grouped by the proposer--see that person for more information and assistance. The lists are not intended to be complete so if you have other ideas or interests that you would like to explore, please feel free to talk with the faculty.

## Gottlieb

**Rigidity theory **Suppose you join a number of rigid rods of possibly different lengths at non-rigid joints. What configurations are rigid? In some cases, the answer depends on the dimension of the space in which the configuration lives. For example, consider two triangles that share an edge. This configuration is rigid in two dimensions but not in three dimensions; the triangles can fold towards each other. On the other hand, a triangle is rigid in all dimensions. If you choose this project, you will read a book that describes three ways of looking at rigidity. This subject is interesting in that it makes use of ideas from combinatorics, algebra, and analysis. It also is a powerful illustration of the power of moving from a nonlinear approach to a linear approximation thereof.

**Subpolytopes of the permutahedron**

Think of a permutation in word form as a vector in Rn with the ith letter occupying the ith coordinate. The convex hull of all of the permutations in Sn is a uniform n-1 dimensional polytope (i.e., higher dimensional polyhedron) called the permutahedron; we will denote it by Pn. A polytope is uniform if it has certain nice properties. For example, P3 is a regular hexagon; P4 is a truncated octahedron. If you choose this project, you will explore the properties of the permutahedron or of polytopes formed by taking the convex hull of some suitably defined subset of Sn. Be aware that this project is likely to be challenging.

**Mathematical programming **There are a number of problems on the campus that can be modeled using mathematical programming, such as the scheduling of classes, the assignment of rushees to greek organizations, and the admission of students to the school. If you choose this topic, you will choose a topic (possibly among the ones listed above, but not necessarily), learn a bit about mathematical programming, and use your knowledge along with some suitable solver (such as Excel or AMPL) to develop a model that will give recommendations as to how to proceed.

## Seaton

**The Mathematics of Soap Films **

A soap bubble is in the form of a sphere because this shape has the minimal surface area for a fixed volume. Similarly, a soap film bounded by a wire will stretch itself in such a way so that the surface area is minimal.

The book below introduces enough differential geometry and complex analysis to understand the rather complicated behavior of soap films. It is suitable for any student who has taken Calc III and Linear Algebra. The book is written for Maple (R), but it would not be difficult to apply these same ideas in Mathematica. I would love to go over this book with a student, either working with Maple or Mathematica.

*J.* *Oprea, The Mathematics of Soap Films: Explorations with Maple (R), American Mathematical Society, 2000.*