KTH/SU Mathematics Colloquium
08-06-11
Andrew J. Sommese, Director of the Center for Applied Mathematics and
V. Duncan Chair of Mathematics at Notre Dame
A Brief Introduction to Numerical Algebraic Geometry
Numerical Algebraic Geometry is a new discipline aimed at
nonlinear problems, including those that previously were approached
using linear algebra methods. The first half of the talk will be an
introduction to Numerical
Algebraic Geometry and the numerical methods that underly it.
We will briefly discuss some recent work of J. Hauenstein,
B. Hu, and A. Sommese applying Numerical Algebraic Geometry to the
solution of diffusion equations. The polynomial systems that arise,
which easily have over 50 equations, are beyond the capability of
any other polynomial solver besides the new program Bertini
developed by D. Bates, J. Hauenstein, A. Sommese, and C. Wampler.
As time permits we will also discuss some recent advances:
-
regeneration, a new equation-by-equation method, developed
by J. Hauenstein, A. Sommese, and C. Wampler;
-
a local dimension algorithm, developed by D. Bates,
J. Hauenstein, C. Peterson, and A. Sommese; and
-
an algorithm, developed by D. Bates, J. Hauenstein,
C. Peterson, and A. Sommese, to decompose the algebraic set
where a matrix of polynomials has rank less than or equal to k,
where k is any nonnegative integer.