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\noindent{}The proposed project is for a group of 
{\bf 40 undergraduate students} at the University
of Washington (8 each academic
year) to do research and development on {\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.  

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
teaching 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 Intellectual Merit:}

\mypar{}Students will
research and implement algorithms, 
write proposals,
and give presentations at seminars 
and conferences.
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{}One potential impact of this project
would be to provide tools for advanced 
research mathematics
that are more general and modern than anything 
available today (both free or
commercial).  
In particular, SAGE provides a
uniform interface to many different aspects of computation, and
this is an attractive feature for its
instructional use---learn the interface once, and you 
have convenient access to all sorts of different software.

Another impact is that instead of students having to pay 
to buy mathematics software
for classroom instruction, they
will have the option to use SAGE for free.
Many teachers have expressed concern that
currently they teach their students to use tools that the
students (or their employers) might not be able to afford 
after they graduate.
At many institutions and US high schools, purchasing
computer software is a significant burden.
SAGE could help address this problem.

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