Conference Agenda

The focus of the proposed minisymposium is on problems in approximation theory that may be studied using techniques from commutative algebra and algebraic geometry. Research interests of the participants relevant to the minisymposium fall broadly under multivariate spline theory, interpolation, and geometric modeling. For instance, a main problem of interest is to study the dimension of the vector space of splines of a bounded degree on a simplicial complex; recently there have been several advances on this front using notions from algebraic geometry. Nevertheless this problem remains elusive in low degree; the dimension of the space of piecewise cubics on a planar triangulation (especially relevant for applications) is still unknown in general.

We study the space of spline functions defined on polyhedral cells. These cells are the union of 3-dimensional polytopes sharing a common vertex, so that the intersection of any two of the polytopes is a face of both. In the talk, we will present new bounds on the dimension of this spline space. We provide a bound on the contribution of the homology term to the dimension count, and prove upper and lower bounds on the ideal of the interior vertex which depend only on combinatorial (or matroidal) information of the cell. We use inverse systems to convert the problem of finding the dimension of ideals generated by powers of linear forms to a computation of dimensions of so-called fat point ideals. The fat point schemes that comes from dualizing polyhedral cells is particularly well-suited and leads to the exact dimension in many cases of interest that will also be presented in the talk.

The following conjecture was proposed by myself and Tom McKinley:

Let p and f be homogeneous polynomials in n variables such that p(gradf)=0. Then p(grad)f=0.

This intriguing conjecture is closely related to the work of Gordan and Noether on polynomials with vanishing Hessians and with some density problems proposed by Pinkus and Wainryb. In my talk I will indicate some particular cases when the conjecture holds true. In particular when the number of variables is at most 5 and when the deg(p)=2.

Recently, a novel approach to approximation of functions on submanifolds has been made: It is based on extending the function constantly along the normals and approximating this extension by functions that exist in the ambient space with tensor product splines as a prominent example. For those, the concept is able to essentially reproduce the convergence orders on the submanifold one can expect in the ambient space.

In the talk, we will give an introduction into the basic concept along with some theoretical results and numerical experiments.

Trimmed NURBS are the standard for industrial surface modeling, and all common data exchange formats, like IGES or STEP, are based on them. Typically, trimming curves have so high degree and so complex knot structure that it seems to be impossible to match them properly to neighboring geometry. Thus, surfaces built from several trimmed NURBS patches are known to reveal gaps along inner boundaries, and it is a cumbersome and sometimes nontrivial task for designers to keep the magnitude of these gaps below an acceptable tolerance.

In this talk, we present a novel methodology to construct trimmed NURBS surfaces with prescribed low order boundary curves, facilitating the representation of watertight surface models within the functionality of standard CAD systems.