Department of Mathematics

Second International Workshop on
Algebraic Geometry and Approximation Theory

Abstracts

The standard multivariate spline spaces are hard to use because their basic properties depend on the geometry of the underlying partition, and because changes at one point in general affect any interpolants everywhere. A standard remedy is to consider instead super or subspaces that do not have these effects. If the data are specified these are called macro element spaces. If the data are to be computed for example by approximating the solution of a differential equation the spaces are known as finite element spaces. I will describe the underlying ideas and several macro (or finite) elements in two and three variables.

---

In [On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1, Numerische Mathematik 57 (1990), 651-661] Alfeld and Schumaker give a formula for the dimension of the space of piece-wise polynomial functions (splines) of degree d and smoothness r on a generic triangulation of a planar simplicial complex Δ, for d > 3r + 1.

In [Cohomology vanishing and a problem in approximation theory, Manuscripta Math. 107 (2002), 43–58], Schenck and Stiller conjectured that the Alfeld-Schumaker formula actually holds for all d > 2r+1. In this note, we show that this is the best result possible; in particular, we present a simplicial complex Δ such that for any r, the dimension of the spline space in degree d = 2r is not given by the Alfeld-Schumaker formula. The proof relies on the explicit computation of the nonvanishing of the first local cohomology module described in [H. Schenck, M. Stillman, Local cohomology of bivariate splines, J. of Pure and Appl. Alg. 117 & 118 (1997), 535–548]. Note: the results in this presentation have been published under the same title in J. Approx. Theory 132 (2005), 72–76.

Address: Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, OH 45221-0025.
Email: stefan.tohaneanu@uc.edu

---

I will give an introduction to the use of algebraic methods in discrete geometry and combinatorics. We will begin with the f (face) vector of a convex polytope. Then I will introduce graded rings, specifically, polynomial rings and quotients. Associated to a simplicial polytope P (every face is "like" a triangle) is a graded ring called the Stanley-Reisner ring, which "remembers" everything about P, and gives us a beautiful algebra/combinatorics dictionary. I will sketch Stanley's solution to a famous conjecture (using this machinery), and also touch on connections between P and objects from algebraic geometry (toric varieties).

---

I will attempt to outline some of the general themes in mathematics of Approximation Theory in a very light and casual manner. These will include some classical principles of approximation and some that, as yet, do not exist.

---

For a d-dimensional polyhedral complex P, the dimension of the space of splines on P of smoothness r and degree k >> 0 is given by a polynomial f(P,r,k) of degree d. For P planar and simplicial, the Alfeld and Schumaker formula gives f and k >> 0. Using localization techniques and specialized dual graphs associated to codimension 2 linear spaces, we obtain the top three coefficients of f in the polyhedral case (which gives a complete answer for P planar). Joint work with Terry McDonald.

---

I will introduce scattered data fitting problems on the sphere and discuss their applications. I will define Bernstein-Bezier polynomials and describe how spherical splines can solve the problems.

---

In the first part of the talk I will give a description of ideal projectors in terms of their interpolation properties and associated zero-dimensional ideals, D-invariant spaces and sequences of commuting matrices.

In the second part I will introduce the Border scheme and discuss open problems in geometry, geography and geology of Hilbert schemes and their counterparts in Approximation Theory, Analysis, Linear Algebra and PDEs.

---

OPTIMAL RECOVERY OF CERTAIN CLASSES OF TWICE DIFFERENTIABLE MULTIVARIATE FUNCTIONS
S.V. Borodachov¹, V.F. Babenko², D.S. Skorokhodov²

We study the problem about optimal global recovery of the class of functions dened on a convex body in R^d, d ∈ N, whose second derivative in any direction is bounded by a given constant, and of the class of functions periodic with respect to a given full-rank lattice in R^d, with second derivative in any direction bounded by a given constant. The information about the function is given by its values and the values of its first order partial derivatives at n points (nodes) inside the domain. We consider the set of all algorithms, whose output function depends only on the information vector of the function from the class being recovered. For every given n, it is required to nd the positions of n nodes and the recovering algorithm, which minimize the supremum of the error over each class. The error is given by the uniform norm of the difference between the function being recovered and the approximating function.

For certain periodic two-variate classes, we show that optimal nodes are located at points of the hexagonal lattice. In the general case, we obtain asymptotic behavior, as n gets large, of the worst-case error of the optimal algorithm and asymptotically optimal sequence of configurations of nodes. For every fixed set of nodes in the domain, we describe the optimal algorithm with nodes at this set. The problem of finding the optimal set of nodes is related to the problem about the most economical covering of R^d by equal balls.

For any additional information, please, contact the workshop organizers: Tatyana Sorokina, Towson University and Luis Garcia, Sam Houston State University.