Mathematical Sciences: Combinatorics of Splines and         Polytopes
{'Code': '03040100', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Michael Steuerwalt'}
The proposed research is on the combinatorial and algebraic theory          of smooth piecewise polynomial functions over polyhedral subdivisions           and its relationship to the theory of facial enumeration in convex              polyhedra.  A particular emphasis will be given to the problem of               determining a simple generating set for the algebra of all smooth               splines, as was already done in the case of continuous piecewise                polynomials, and to the problem of determining for which subdivisions           does this algebra have a free basis over the ring of global polynomial          functions.  The PI  has shown these problems to be related to certain           enumeration questions for convex polytopes, and he hopes to develop this        connection further here.                                                            Piecewise polynomial functions have long been used for solving              partial differential equations by the finite element method:  this work         was originally motivated by questions arising in this context.  More            recently, they have found extensive application in the area of surface          modelling for computerized design and control.  As a result of earlier          work on this project, they are now beginning to find new applications in        the theory of structural rigidity.
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1987&All=true', 'xml': '8703370.xml'}