# Multivariate Splines and Algebraic Geometry

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- | The focus of the proposed minisymposium is on problems in approximation theory | + | A multivariate spline is a function defined on a domain in $R^d$ with $d>1$ |

- | that may be studied using techniques from commutative algebra and algebraic | + | that is piecewise a polynomial. These objects from approximation theory |

+ | may be studied using techniques from commutative algebra and algebraic | ||

geometry. Research interests of the participants relevant to the minisymposium | geometry. Research interests of the participants relevant to the minisymposium | ||

fall broadly under multivariate spline theory, interpolation, and geometric | fall broadly under multivariate spline theory, interpolation, and geometric |

## Revision as of 03:06, 23 December 2016

A multivariate spline is a function defined on a domain in $R^d$ with $d>1$ that is piecewise a polynomial. These objects from approximation theory 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 Hilbert function and algebraic structure of the spline module; 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!