Fall 2012, Thursdays, Milner 216, 4:00-4:50 PM

Date: October 18

"  Numerical Methods for Solving Schubert Problems "
  Nickolas Hein  


A Schubert problem is a geometric problem which may be expressed as a system of determinantal equations. Such systems are overdetermined (more equations than variables), which is good for solving with polynomial algebra (Gröbner bases), but bad for numerical methods. Using duality, we recast the problem as a square system (number of equation equals number of variables). Unfortunately this adds many variables to the system. However, we replace high-degree polynomials by bilinear equations. The advantage of low-degree equations may outweight the disadvantages with this approach.
Numerical methods apply to the square system. Regeneration (solving equation by equation), may solve otherwise infeasible Schubert problems. Parameter homotopy is even faster when it may be used. Both methods parallelize wonderfully.
In this talk I will give background to understand the geometric problem involved. Then we see how to trade determinantal equations for bilinear equations. I will detail the strengths of each approach.