Research Experiences for Undergraduates

Summer 2000

ANALYSIS ON FRACTALS, MATH PROBLEMS FROM BIOLOGY AND ALGEBRAIC COMBINATORICS

WHAT: Undergraduates will work on research projects under the direction of Cornell University Mathematics Department members and visitors. The specific projects and faculty members directing them follow.

PROJECT 1: ANALYSIS ON FRACTALS (ROBERT STRICHARTZ) Students in this project will study properties of functions defined on fractals. For a certain class of self-similar fractals, including the familiar Sierpinski gasket (also called the Sierpinski triangle,) there is now a theory of "differential equations." 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 over the past 3 summers by REU students. (See the web site http://mathlab.cit.cornell.edu/~gibbons for a sample of this work.)  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. Experience from the past indicates that it is possible to learn a lot by doing carefully chosen computations. 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).

BACKGROUND: Students should have experience in programming (C, FORTRAN, or PASCAL) and/or the use of symbolic calculation packages (MAPLE, MATHEMATICA, MATHLAB). Students should have some courses in mathematical analysis and/or differential equations.

PROJECT 2: MATH PROBLEMS FROM BIOLOGY (RICK DURRETT) Ecology and genetics are a source of many interesting problems in mathematics. Among the problems we will consider are the following:

• Evolving predator prey systems. In the 60's Paine demonstrated that a predator can cause the coexistence of species that would not coexist in its absence. How many species can coexist in this way? What happens when new species are introduced by mutation or immigration.
• Transposable elements are like computer viruses in our DNA. The availability of complete genomic sequence for yeast and for fruit flies (to be released in Feb 2000) makes it possible to ask questions about the number of distinct families of TE's and when they entered the genome.
• Genome rearrangement. Our DNA evolves not only by small local changes but also by large inversions within a chromosome or exchange between two chromosomes. The 15 October 1999 issue of Science magazine gives a fairly complete picture for mammals but many questions remain about the relationship between plant genomes.

BACKGROUND: Students should have some experience in programming. No knowledge of biology is required but an undergraduate course in probability would be useful.

PROJECT 3: ALGEBRAIC COMBINATORICS (RICHARD EHRENBORG AND MARGARET READDY) Algebraic combinatorics links many different areas of mathematics including polytopes, algebra, discrete geometry and partially ordered sets, to name a few. In this REU project we will explore different interactions between these areas. The planned projects include:

• Exploring flag vectors of polytopes starting with zonotopes and hyperplane arrangements.
• Extending classical permutation statistics, such as descent set, excedance set and inversions, to set-valued ones.
• Finding appropriate enumerative properties of classes of matrices, such as alternating signed matrices and their analogues.

BACKGROUND:  Some programming experience is desirable, and a course in abstract algebra or group theory would be helpful, but neither is required.

WHEN: June 19 - August 11, 2000 (8 weeks)

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

STIPEND: $2750 plus free housing. Housing for participants is provided at the Cayuga Lodge Cooperative. The Lodge has spacious living room, kitchen and laundry facilities. Students will have private bedrooms. Since this is a cooperative, students will be expected to help with light housekeeping chores.

ELIGIBILITY: Funding for this program comes from the National Science Foundation, which has set the following requirements:

1. participants must be US citizens or permanent residents
2. enrolled in an undergraduate program.

High school students and graduating seniors are not eligible. These requirements cannot be waived.

HOW TO APPLY: There is no official application form. You should receive acknowledgment when your application is complete.

1. Send a letter about your background and interests, indicating your first choice of projects. (Be sure to include information about your computer experience.)
2. Send a copy of your college transcript (unofficial copies are acceptable).
3. Arrange to have two letters of recommendation sent.

WHERE TO SEND APPLICATION MATERIALS: We encourage e-mail submissions to <reu@math.cornell.edu >. You may also send regular mail to: REU Program, Math Department, Malott Hall, Cornell University, Ithaca, NY 14853.  You may also view the REU Web page at: <http://www.math.cornell.edu/~math/Educate/REU/REU.html>.

DEADLINE: February 24, 2000.  ALL materials must be received by this date.  Late applications will not be accepted. You will receive notification sometime in March. Please make sure your application letter includes an e-mail or regular address where you can be reached if you are going to be away from your campus address during spring break.

If you have comments, questions or concerns, please send e-mail to the REU Coordinators at <reu@math.cornell.edu>.