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CSUMS: Undergraduate Computational Research in Arithmetic Geometry

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\noindent{}The proposed project is for a group of 6 undergraduate
students each year to do research with a strong computational emphasis
in number theory and arithmetic geometry, where each project will be
relevant to research on the Birch and Swinnerton-Dyer conjecture (BSD
conjecture) or modular functions.  Participants will become well
versed in the practical use of computation in advanced mathematical
research, gain knowledge about mathematical software, make long-term
connections with a vibrant research and development community, and
contribute tools that will be used by expert researchers and students.
Number theory is a venerable research area that draws strongly from
many areas of mathematics. The BSD conjecture is one of the deepest
problems in number theory.  Modular curves and the functions which
uniformize them lie at a crossroads of number theory, geometry,
complex analysis, and differential equations.  Student research will
thus make connections with a wide range of mathematics. The program is
structured so that students will learn teaching and writing skills,
which will prepare them to apply computational mathematics techniques
in graduate school and industry.


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\noindent{\bf Intellectual Merit:}

\mypar{}Students will do research on a central problem in number
theory and implement algorithms and give presentations at seminars and
conferences.  Their work will enhance our understanding of and ability
to work with important mathematical objects.

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\noindent{\bf Broader Impact:}
\mypar{}This proposal has the potential to transform
research and education in several ways. One broader impact of
this project would be to provide tools for research and education in
number theory and related areas that are better than anything
available today.  Another impact is that instead of students and
researchers having to pay to buy mathematics software for number
theory courses and research, they will have the option to use Sage for
free.  The PI will also develop course materials for use by the
students that will be of broader interest.

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