Mathematics Colloquium

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Numerical Algebraic Intersection with Applications

Speaker: Charles Wampler, General Motors and University of Notre Dame

Location: Warren Weaver Hall 1302

Date: Monday, April 28, 2014, 3:45 p.m.

Synopsis:

In numerical algebraic geometry, algebraic sets are represented by witness sets, which can be computed with numerical homotopy methods (a.k.a. continuation). After discussing this basic construct, we will describe an algorithm, based on the regeneration technique, that solves the following problem: given a witness set for a pure-dimensional algebraic set, say Z, along with a system of polynomial equations defined on Z, compute a numerical irreducible decomposition of the zero set of the polynomials on Z. Also treated is the case where Z is the cross product of two or more pure-dimensional sets, each given in terms of a witness set. Two existing algorithms, diagonal intersection and the homotopy membership test, can be seen as special cases of the new algorithm. In addition to this unification, the method also extends the range of problems that can be solved using numerical algebraic geometry. We will discuss application of these techniques to robot kinematics and sphere packing problems.