###### Student Summer Research Projects

## Research Opportunities at Grinnell

## Current MAPs (Summer 2013)

Science Summer Research Application Form

Prof. Chris FrenchTitle: Noncommutative, Imprimitive Association schemes of Rank 6Description: 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 JonkmanTitle: Randomization Inference: Beyond the First Statistics CourseDescription: 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 RingsDescription: 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 CurvesDescription: 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.

###### Faculty Research Areas

Jeffrey Blanchard (Assistant Professor), Applied Mathematics, Applied and Computational Harmonic Analysis, Wavelets, Compressed Sensing.

Marc Chamberland (Myra Steele Professor of Natural Sciences), Experimental Mathematics, Classical Analysis, Number Theory, Dynamical Systems, Differential Equations.

Christopher French (Associate Professor), Algebra.

Jeff Jonkman (Associate Professor), Meta-analysis, inference for nonlinear models, analysis of observational studies, composite sampling.

Shonda Kuiper (Associate Professor) experimental design, statistics education, statistical consulting.

Joseph Mileti (Assistant Professor), Computability theory, reverse mathematics, set theory, Ramsey theory, and the interaction of mathematical logic with algebra and combinatorics.

Emily Moore (Senior Faculty Status) Combinatorics, especially difference sets and graph coloring

Tom Moore (Senior Faculty Status) statistics education and applications of statistics

Jennifer Paulhus (Assistant Professor), algebraic number theory, arithmetic geometry, elliptic and hyperelliptic curves, representation theory, Jacobian varieties.

Karen Shuman (Associate Professor), Analysis, Harmonic Analysis, Iterated Function Systems.

Nick Teff (Assistant Professor) Representation theory and algebraic combinatorics: symmetric group, Coxeter groups, integer partitions, and symmetric functions

Michael VanValkenburg (Assistant Professor) Microlocal Analysis, Harmonic Analysis

Royce Wolf (Associate Professor), Algebraic Topology, Combinatorial Group Theory, Knot Theory, Spherical Virtual Knot Theory, Quandles.

###### Faculty Presentations

Our faculty give talks all over the world. Here are some of the talks from the last year.

August, 2103Jeffrey Blanchard

“Greedy Algorithms for Joint Sparse Recovery”

CIMPA New Trends in Applied Harmonic Analysis

Mar del Plata, Argentina

July, 2013Marc Chamberland

"Borromean Five"

juried exhibition of mathematical art Bridges conference

Enschede, Netherlands

June, 2013Marc Chamberland

“The 3x+1 Problem”

Max Planck Institute for Mathematics

Bonn, Germany

May, 2013Shonda Kuiper

“Igniting a Passion for Change in Teaching Statistics”

Invited Opening Session Presentation

and

“Nurturing a Passion for Statistics by Finding Stories in Data”,(30% acceptance rate), USCOTS,

and

“Flipping the Classroom: Changing the Classroom Model to Enhance Student-centered Learning and Optimize Use of Time”

and

“Using Fun in the Statistics Classroom: An Exploratory Study of Hesitations and Motivations of USCOTS 2011 Attendees”

all at

United States Conference On Teaching Statistics

April, 2013Marc Chamberland

The Computer's Role in Mathematical Discovery

Westchester University

and

"A Feast of Experimental Mathematics"

Rutgers University

"Experimental Mathematical Seminar"

and

"The 3x+1 Problem"

Rutgers University Mathematics ColloquiumChristopher French

"Functors from Association Schemes."

American Mathematical Society Spring Sectional MeetingJoe Mileti

"Primes in Computable UFDs"

Buenos Aires Semester in Computability, Complexity, and Randomness

and

"Uniform Computable Reductions Between Mathematical Theorems"

American Mathematical Society Spring Sectional MeetingJen Paulhus

“Computing branching data and decomposing Jacobian varieties”

American Mathematical Society Spring Sectional Meeting

March, 2013Jen Paulhus

“Using Algebra to Study Curves”

Mathematics Colloquium

Universidad de Chile

Santiago, Chile

February, 2013Joe Mileti

"Computable Reductions Between Mathematical Theorems"

Mathematics Colloquium

Western Illinois University

January, 2013Marc Chamberland

"Complex Numbers and Geometry"

and

"Inner Square", Juried Exhibition of Mathematical Art

Joint Mathematics Meetings, San Diego

November, 2012Jeffrey Blanchard

“Greedy Algorithms in Compressed Sensing: Theory and Software”

Iowa/Nebraska Functional Analysis Seminar, Des MoinesMarc Chamberland

“The Computer's Role in Mathematical Discovery”

Colorado CollegeJoe Mileti

"Computable Combinatorics"

Logic Colloquium

University of California, BerkeleyJen Paulhus

“The abc-conjecture: A Snapshot of the World of Pure Mathematics”

Natural Science Colloquium

Illinois Wesleyan University

October, 2012Marc Chamberland

"A Beautiful Cantor-Like Function"

Iowa MAA Section MeetingJoe Mileti

"Uniform Reductions Between Combinatorial Problems"

Recursion Theory Seminar

University of California at BerkeleyJen Paulhus

“Decomposing Jacobian Varieties of Curves”

Berkeley Number Theory Seminar

September, 2012"Ramsey's Theorem and Computability Theory"

Reverse Mathematics Seminar, Berkeley

August, 2012Shonda Kuiper

“Playing Games with a Purpose: A New Approach to Teaching and Learning Statistics”

Topic Contributed Presentations

Joint Statistics Meetings. San Diego California

July, 2012Jeffrey Blanchard

"GPU Accelerated Greedy Algorithms for Compressed Sensing”

SIAM Annual Meeting, MinneapolisShonda Kupier

“2012 MERLOT Classics Award Invited Presentation – Stat2Labs”

SLOAN Fifth Annual International Symposium on Emerging Technologies for Online Learning

Las Vegas, Nevada

###### Faculty Awards and Grants

Shonda Kuiper

The 2012 MERLOT Classics Award in Statistics.

This award is given once a year for the best peer reviewed online resources designed to enhance teaching and learning.

Jeffrey Blanchard

“Large-scale Algorithm Analysis and GPU Implementations for Compressed Sensing and Matrix Completion”

National Science Foundation, Research in Undergraduate Institutions

August 2011- July 2014.

Shonda Kuiper

“Playing Games with a Purpose: A New Approach to Teaching and Learning Statistics.”

National Science Foundation Transforming Undergraduate Education in Statistics

May 2011-May 2014.