Research Experiences for Undergraduates Program
Summer 2013

Visit the REU web site for a sample of students’ work on previous projects, including the ongoing Analysis on Fractals project.

PROJECT 1: Analysis on Fractals, directed by Robert Strichartz

Students in this project will study properties of functions defined on fractals. For certain fractals, including the Sierpinski gasket, the Sierpinski carpet, and some of the classical Julia sets, there is now a theory of “differential equations.” (See my book, Differential Equations on Fractals, a tutorial, Princeton University Press, 2006.) One of the goals of this project is to obtain more information about solutions of these fractal differential equations, following up on work that has been done by past REU students. Most of the work on this project will involve both computer experimentation and theoretical study, but individual students may put more emphasis on one or the other. We expect that students will be involved in all stages of the process: planning what examples to study, doing the programming for the computations, and interpreting the results (and attempting to prove the conjectures that come out of the process).

PROJECT 2: Generating Sets for Finite Groups, directed by R. Keith Dennis

This project will study the “linear algebra” of finite groups. For example, one can define the analogue of a “basis” for any finite group. However, the behavior of these sets can be quite different from bases for vector spaces (e.g., they need not all have the same size). Results from standard linear algebra and the theory of modules are used to suggest questions that should be investigated in the general case. Many such questions have not been previously studied and sometimes offer a new framework to interpret previously isolated results in group theory. One such is the study of certain manipulations of bases which arose in recent years in computational group theory (the product replacement algorithm). Students for this program should have a firm understanding of undergraduate linear algebra and abstract algebra. New topics, which although elementary, are not usually developed in standard undergraduate algebra, will arise naturally here. Students will develop computational tools to study examples using the computer algebra systems GAP and Magma. [More information]

PROJECT 3: High Dimensional Data Analysis, directed by Matthew Hirn (Yale)

Data is everywhere in this day and age, and much of it is very complicated, or, in other words, high dimensional. Many times though, there is a much smaller number of intrinsic parameters, hidden within the complicated high dimensional observations, that drive the underlying structure within these types of data sets. Examples include various types of imagery data, dynamical systems, medical data, social networks, financial data, and many others. Attempts to find these intrinsic parameters have driven a large class of research lately, leading to subfields such as nonlinear dimensionality reduction and manifold learning. This project will focus on extending these ideas in new ways and using them to enhance our understanding of the structure of certain high dimensional data sets. There will also be an opportunity to develop algorithms for interpolating, or filling in, missing data in an optimal way. Students will do at least some programming during the project, and most likely it will be a central part of the experimentation process. The mathematical ideas that we will use and develop are drawn from analysis, probability, statistics, geometry, and theoretical computer science. Students of any background with an interest in applied mathematics and at least a limited knowledge of programming in Matlab (or a willingness to learn) are encouraged to apply.

WHEN: June 17 – August 9, 2013 (8 weeks)

WHERE: Mathematics Department, Malott Hall, Cornell University, Ithaca, NY 14853-4201.

STIPEND: $5000, includes stipend and living expenses. Participants will arrange for their own housing; we will assist with local contact information.

ELIGIBILITY: Funding for this program comes from the National Science Foundation, which has set the following requirements: (1) Participants must be U.S. citizens or permanent residents; (2) Participants must be enrolled in an undergraduate program. High school students and graduating seniors are not eligible. These requirements cannot be waived.

HOW TO APPLY: Starting December 17th you can submit an application (through the REU web site) that includes a statement about your background, educational goals, and your scientific interests. Include whatever further information you consider relevant. (Be sure to include information about your computer experience.)  Also on this date you will be able to upload letters of recommendation and transcripts (unofficial transcripts accepted) via the REU website.  Two letters of recommendation are required.  Uploading your letters and transcripts is preferred and is the most efficient means of processing.

If necessary you can send letters and transcript (unofficial transcripts accepted) via email to mathreu@cornell.edu or mail them to the REU Program, Mathematics Department, Malott Hall, Cornell University, Ithaca, NY 14853-4201. Please avoid sending duplicate copies of recommendation letters and transcripts.

DEADLINE: February 27, 2013. All materials must be received by this date. Late applications will not be accepted. You will receive notification some time in March.

If you have comments, questions, or concerns, please send e-mail to the REU coordinators at mathreu@cornell.edu.

Last modified:December 13, 2012