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\noindent{}The proposed project is for a group of 
{\bf 3 undergraduate students} at the University
of Washington to do research and development
of algorithms in {\bf SAGE}, which is
an open source mathematics software project started
by the PI for research in algebra, geometry, number 
theory, cryptography, numerical analysis, statistics,
and other areas ({\tt http://sage.math.washington.edu/sage}).  
Student work would be done in conjunction
with Jim Morrow's highly successful REU.

By joining the SAGE project,
the PI hopes that participants will become involved
in mathematical research, gain
extensive knowledge about mathematical software, 
make long-term connections with a vibrant research and development community, and
contribute tools that will be used by teachers and expert 
researchers in numerous fields.
They will also learn
presentation and writing skills, and 
really understand some part of
mathematics deeply, which will 
prepare them to apply computational mathematics
in graduate school and industry. 


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\noindent{\bf The PI's involvement in Undergraduate Research:}

\mypar{}The PI, William Stein, has supervised over 20
undergraduate student research projects, ranging
from deep theoretical work on the Birch and Swinnerton-Dyer
conjecture and ranks of elliptic curves
to difficult software engineering projects involving
mathematical computation.  One of his students, Yi 
Qiang, recently received the prestigious Mary Gates
Research Grant, and gave an invited talk at an international worshop in Berkeley on his work with Stein on parallel computation.


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\noindent{\bf Intellectual Merit:}

\mypar{}Students will
research and implement algorithms, 
write proposals,
and give presentations to other REU
students.
They will design and implement  algorithms
for computing with mathematical structures such
as groups, rings, fields, graphs, matrix spaces, varieties,
manifolds, etc. 
Their work will be reviewed, commented on, and used
by the growing worldwide community of SAGE users.


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\noindent{\bf Broader Impact:}
\mypar{}In addition to designing new algorithms, students will create freely available implementations
of algorithms that have been developed over many years by
work of dozens of Jim Morrow's REU students.
  

Another impact is that instead of students (e.g.,
in future REU's) having no choice but to pay 
to buy mathematics software, they
will have the option to use SAGE for free.
At many institutions and US high schools, purchasing
computer software is a significant burden.
SAGE could help address this problem.

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