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Research Experiences for Undergraduates Program
Summer 2008
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.” (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 over the past 11 summers
by REU students. (See www.math.cornell.edu/~reu/ 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. 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). Note: Students in this project will begin by attending
a conference being held at Cornell on June 11–15. (An additional $100.00
will be added to the stipend to help with expenses.)
PROJECT 2: Games, Linear Orders, and Logic
(François Dorais)
Students in this project will play so-called Ehrenfeucht-Fraisse games
(aka EF-games) on linear orders. (A linear order is simply a set
of points together with an "is less than" relation.) The
main goal of the project is to count or estimate how many different linear
orders can be described by sentences of low complexity. We will
first explore some ideas of logic and model theory to learn more about
linear orders and the complexity of sentences, then we will see how these
relate to EF-games. After this, we will try to find good strategies
for playing EF-games on linear orders and try to extract pertinent information
from these strategies. We may also look at extensions of these
ideas to different structures such as partial orders and graphs. Students
should have completed undergraduate courses in linear and abstract algebra.
Prior knowledge of logic is a bonus, but it is not required. Participants
with programming skills may look at computer simulation of EF-games.
PROJECT 3: Topological Dynamics, Group Actions, and Cantor Set n-Cubes
(Collin Bleak)
Students in this project will investigate properties of the higher dimensional
analogues Bn of Richard Thompson’s
mysterious group V. The
groups Bn were discovered by Matthew Brin,
and can be thought of as particular infinite collections of one-to-one
and onto functions from a Cantor Set n-cube Cn to
itself, so that composition of functions in Bn results
in functions in Bn. In
general, the most successful studies of V have been through
analysis of the dynamics of element actions on the Cantor Set. We
will start with a similar viewpoint in our approach to the higher dimensional
analogues Bn, but with a certainty we will
need to create new and diverse techniques to really get to the heart
of these groups! The
groups Bn are relatively new and have many
open questions associated with them. We will be opportunistic in
choosing what questions we work on. Basic knowledge of group theory,
point set topology, and dynamics will help the applicant, but it is not
expected or required. Individual
students may choose to work more theoretically, or more computationally,
as their interests and abilities allow.
WHEN: June 16 – August 8, 2008 (8 weeks)
WHERE: Mathematics Department, Malott Hall, Cornell
University, Ithaca, NY 14853-4201.
STIPEND: $4000. Participants will arrange for their
own room and board; 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: Apply via email or U.S. mail. Please
include the following documents:
- Application Form. (A copy should
be printed out or cut & pasted into the body of your email.)
- A letter 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.)
- A copy of your college transcript; unofficial copies are acceptable.
- Arrange to have two letters of recommendation sent.
WHERE TO SEND APPLICATION MATERIALS: We encourage
email submissions to mathreu@cornell.edu.
You may also send regular mail to: REU Program, Mathematics Department,
Malott Hall, Cornell University, Ithaca, NY 14853-4201.
DEADLINE: February 28, 2008. 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 email 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 mathreu@cornell.edu.
You may see examples of past REU projects by contacting the REU Coordinators.
Also, www.math.cornell.edu/~reu/
has links to many web sites created by REU students over the past several
years.
Last modified:November 29, 2007
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