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REVIEW:
What is the intellectual merit of the proposed activity?
What is the intellectual merit of the proposed activity?
The proposal is in the area of algebraic geometry and computer-aided
geometric design. It mainly concerns rational surfaces, which are surfaces
in n-dimensional space parametrized by rational functions (or polynomials).
There are at least three descriptions of such a surface: parametric
(by rational functions), implicitly (set of equations cutting out the surface),
Grobner basis (basis for ideal of which surface is zero set). The PI's
wish to consider a fourth representation: mu-basis, which is a basis of the
module of syzygies between the rational functions in a parametrization.
Such a basis is not unique, so one imposes some minimality properties
on such a basis, minimal degree in certain parameters (with respect to a
suitable partial order). Mu-bases were proposed by Sederberg and some of
the PI's in the early 1990's as an alternative representation of such surfaces
which may be more economical under some circumstances.
From them such data a parametrization of the surface can
be easily recovered, while an implicit representation is harder to find.
The possible value of mu-bases is that they may provide a useful compact computer representation
of such surfaces from which basic questions about the surface can be quickly
answered computationally.
The proposal concerns establishing fundamental algorithmic results for
construction of mu-bases, and of conversion between mu-bases and other representations
of space curves and surfaces. For example at present no effectively computable
algorithm for finding a mu-basis of a general rational surface is known. The
PI's propose to construct one and analyze its complexity (based on factorization of
polynomial matrices (work of Lin [18]). They raise a number of algorithmic questions:
computing singular loci of rational surfaces, computing mu-bases for rational space
curves, determining singular points and implicit equations for such curves. They
also wish to study the moving curve ideal associated to a rationally parametrized
curve and to figure out its structure and what information is easy to extract from
it- there is a connection with the dual curve. They propose some specific conjectures
which if true would bound the complexity of algorithms to solve these questions.
This proposal seems strong. It raises a number of theoretical questions
about mu-bases which need to
be answered in the process of finding out if they yield a useful tool in computer-aided
geometric design. It definitely formulates some new directions. The co-PI's are well-known
in this research area and can be expected to make progress on these questions. They
have a working relationship established by previous publications.
The mathematics PI (Cox-PhD 1975) is a senior researcher
well-known for work in computational algebraic
geometry, and the computer science PI (Goldman-PhD 1973) is a senior researcher
well-known in computer-aided geometric design. Both PI's have strong research track
records, and have previous joint work.
The research will be carried on with a consultant (Falai Chen-PhD 1994;
now at Univ. of Science and Technology-China) with excellent implementation skills,
who has joint work with Cox.
What are the broader impacts of the proposed activity?
Positive solution of the problems raised in this grant would facilitate use of
mu-basis representations in computer-aided geometric design.
The grant will support one grad student (Goldman) and one undergraduate (Cox).
Summary Statement
What are the broader impacts of the proposed activity?
Summary Statement
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