Research Opportunities at Grinnell
Current MAPs (Summer 2013)
- Prof. Chris French
Title: Noncommutative, Imprimitive Association schemes of Rank 6
Description: Association schemes are one generalization of groups. It is known that the smallest noncommutative group has 6 elements. My collaborator, Paul-Hermann Zieschang, and I have been trying to show that the set of all noncommutative association schemes with rank 6 fall into three different classes. To do this completely, one needs to show that no other examples, beyond those that are known, can exist. One can make progress on such a problem by introducing extra assumptions that an association scheme might satisfy, and see if such assumptions lead to new examples. Zieschang and I have shown that one particular set of assumptions does not lead to new examples. I have another similar set of assumptions in mind that one could make, and I would like a student to investigate whether or not new association schemes can be found with these assumptions.
Prerequisites: This would be a good project for a student with interests in both Algebra and Combinatorics.
- Prof. Jeff Jonkman
Title: Randomization Inference: Beyond the First Statistics Course
Description: Randomization methods are now widely used to introduce the logic of statistical inference in introductory courses. However, such methods are not well-developed or widely accepted for many problems that may be encountered in a second course, such as two-way ANOVA, logistic regression, and meta-analysis. MAP participants will research the availability of randomization methods for these and perhaps other problems, and evaluate their statistical properties, ease of implementation, and feasibility for use in courses at Grinnell. If no suitable methods are identified, the MAP participants will attempt to apply basic principles of randomization inference to develop a logical framework for randomization methods in these more complex situations. This project has the potential to contribute to curriculum development at Grinnell, as well as the literature on statistics education and/or statistical methodology.
Prerequisites: MAT 209. MAT 335-336 preferred. Some programming experience, especially with R, is also desirable.
- Prof. Joe Mileti
Title: Computing Primes in Rings
Description: The prime numbers have fascinated mathematicians for millennia and now play an important role in cryptography. In order to use them, we must have fast computational procedures to determine whether a number is prime. Over the past few centuries, mathematicians have developed ingenious methods that work reasonably quickly even on very large numbers.
One typically thinks about the prime numbers in the usual integers, but mathematicians have been led to the study of generalizations of the integers, called rings, and the primes they contain. I recently showed the existence of a computable ring, in fact a computable UFD, where it is impossible (not merely difficult) to computably determine which elements are prime. This MAP will build on that work and will investigate the subtle differences between prime and irreducible elements in computable rings. In particular, is there a computable ring where one of these sets is computable while the other is not?
Prerequisites: Ideally, students will have taken Abstract Algebra, but a sufficiently motivated student who has completed Elementary Number Theory may be considered. Also, some programming experience (such as at the level of CSC 151) is important.
Prof. Jen Paulhus
Title: Jacobian Varieties of Hurwitz Curves
Description: We will study the factorization of mathematical objects called Jacobian varieties for a special family called the Hurwitz curves. A technique has been developed, using a subject called representation theory, to factor the Jacobian Variety of a curve. This technique has been used to explore factorizations of special curves, specifically those of low genus (those defined by a polynomial of small degree). The next natural question is to study the decompositions of Jacobian varieties of infinite families of curves. Hurwitz curves have very unique properties and are of great interest to mathematicians in arithmetic geometry. Questions about factorization of these varieties relate to problems in cryptography and questions of rank of elliptic curves, both of wide interest to the mathematics community. We will learn some basic representation theory and then work on decompositions of Jacobian Varieties of Hurwitz curves both from the computational and theoretical perspectives.
Prerequisites: Math 321. Some very basic programming skills or some prior knowledge about basic representation theory are a bonus
Prof. Michael VanValkenburgh
Title: Complex Geometrical Optics and Complex Fourier Integral Operators ("FIO").
Description: The project has two parts: one primarily analytic and one primarily algebraic. The analytic project is an extension of certain integrals, arising in optics, to the complex domain. The algebraic project is a study of complex symplectic linear algebra, the geometric/algebraic side of complex FIO.
- Archive: List of previous Mentored Advanced Projects (MAPs) in the Department of Mathematics and Statistics (PDF).