MCS 595 Graduate Seminar, Fall 2004

1. Thu 2 Sep: "Parallel Algorithms in Algebraic Geometry". While homotopy continuation methods are embarassingly parallel, most modern homotopies are often used recursively, i.e.: paths leading to new solutions start at the end of other paths. Examples of this kind are polyhedral and Pieri homotopies. The goal of this seminar is to explore the parallellization of new homotopies recently developed in numerical algebraic geometry.
2. Thu 9 Sep: "The granularity of the monodromy breakup". The monodromy breakup is a new algorithm developed jointly with Andrew Sommese and Charles Wampler to factor positive dimensional solution sets into irreducible components. This algorithm scales quite well with increasing degrees. We show that the granularity can remain coarse, so the algorithm is relatively simple to parallellize.
3. Thu 16 Sep: "Implementation of a parallel monodromy breakup". We discussed the use of PHCpack as a state machine and listed the modifications to a parallel path tracker done so far to achieve a parallel factorization algorithm using monodromy.
4. Thu 23 Sep: "Running the Parallel Monodromy Breakup". The speedup of the preliminary version of the code is good, we discussed numerical issues regarding the reliability and recalled the concept of the linear trace.
5. Thu 30 Sep: "Efficiency Monodromy Breakup". We can run monodromy loops more efficiently using intrinsic coordinates and computing the linear span of a component in a preprocessing step.
6. Thu 7 Oct: "On parallel computation of Groebner bases" by Anton Leykin.
7. Thu 14 Oct: "Applying Pieri Homotopies to compute dynamic output feedback laws on a parallel computer" by Yusong Wang.
8. Thu 21 Oct: "Newton's Method with Deflation for Isolated Singularities of Polynomial Systems" by Ailing Zhao.
9. Thu 28 Oct: "Approximate Geometry of PDE: Determining Linearizable DE close to a given DE", by Greg Reid (University of Western Ontario, London, Canada).
10. Thu 11 Nov: Anton Leykin on "Deflation of polynomial systems at an isolated solution".

• MCS 595 Graduate Seminar, Spring 2004
• MCS 595 Graduate Seminar, Fall 2003