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Research Experiences for Undergraduates Program
Summer 2009
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 over the past 12 summers by 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: Solving Games on Graphs, Fast, directed by Sasha
Rubin
A parity game is played by two players who move a token along the edges
of a finite graph. The players are trying to control the vertices
that occur infinitely often along the resulting path. This project
involves looking for a deterministic polynomial-time (i.e., fast) algorithm
that, given a game graph as input, decides which of the two players has
a winning strategy. This is a longstanding open problem in theoretical
computer science and appears naturally in a number of settings: infinite
duration two player games, formal verification, and automata theory. We
will first look at known algorithms that solve simpler games; for instance,
where player I (II) tries to reach (avoid) a target set of vertices. However,
new ideas are needed to deal with the more general case of parity games. Ideal
candidates should have a solid background in at least one of the following:
logic, design and analysis of algorithms, programming; however, candidates
with other backgrounds in discrete mathematics (combinatorics, graph
theory, probability theory, number theory, etc) are also encouraged. More
on this project
PROJECT 3: Groups via Actions, directed by Collin Bleak
Students in this project will investigate properties of infinite groups
through close analysis of their actions on various topological spaces. The
groups we will be investigating are the higher-dimensional piecewise
integral projective groups of W. Thurston, the R. Thompson groups F < T < V and
their generalizations, and various groups of homeomorphisms of the unit
circle and of the cantor set. All are quite accessible, their
definitions are concrete, and participating students will be exposed
to real research questions very quickly. On the other hand, the groups
are also quite mysterious, and real creativity will be required if we
are to discover some of these groups’ hidden properties! Knowledge
of group theory, point set topology, and dynamics will help the applicant,
but it is not expected or required. Also, individual students may choose
to work more theoretically or more computationally as their interests
and abilities allow.
WHEN: June 8 – July 31, 2009 (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:
- 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.)
- Send the following via email to mathreu@cornell.edu (preferred)
or mail to REU Program, Mathematics Department,
Malott Hall, Cornell University, Ithaca, NY 14853-4201:
- A copy of your college transcript; unofficial copies are acceptable.
- Two letters of recommendation.
DEADLINE: February 26, 2009. ALL materials
must be received by this date. Late applications will not be accepted.
You will receive notification sometime in March.
If you have comments, questions, or concerns, please send
e-mail to the REU coordinators at mathreu@cornell.edu.
Last modified:February 11, 2009
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