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REVIEW:
What is the intellectual merit of the proposed activity?
What are the broader impacts of the proposed activity?
Summary Statement
The proposal studies a novel formulation of syzygies called "u-bases" previously considered by the authors, which have a strong geometric foundation. As justification for this NSF program, they emphasize the applications of this method to geometric modeling and computer-aided design; their methods are for example promising for finding implicit equations for degenerate surfaces.
I am sure that the authors are sincerely interested in geometric modeling and computer-aided design, but this part of their proposal is least developed, bordering on genuflection for our benefit as reviewers. In fact, these ideas look promising as purely theoretical constructs; the study of syzygies in algebraic geometry could benefit from stronger geometric foundations, with many famous problems such as "Green's Conjecture" related to how syzygies interact in this interface between geometry and algebra. However, I can easily see how algebraic geometry panels could shove aside this work as too computational, and I want to see this research supported, so I recommend funding through this program.
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