MCS 595 Graduate Seminar, Spring 2004

This semester, we meet weekly in 712 SEO on Thursdays at 11:00. The theme of our seminar is "Introduction to Numerical Algebraic Geometry".
  1. Thu 15 Jan: "Introduction to Numerical Algebraic Geometry". While this can be seen as a continuation of the theme of last semester, we take a fresh look on solving polynomial systems. We defined what is involved in describing the positive dimensional solution sets. First, we want to decompose the solution set in pure dimensional sets. Second, we factor the pure dimensional sets into irreducible components.
  2. Thu 22 Jan: "Decomposing a Variety into Pure Dimensional Sets". In this lecture we defined a numerical representation of a solution component of dimension k as the subset of points which lie on the component and on k random hyperplanes. Using slack variables we defined an embedding for general systems. The application of Bertini's theorem led to a cascade of homotopies, and to a generalization of Bezout's theorem.
  3. Thu 29 Jan: "Bertini's theorems and the linear trace test". Bertini's 1st theorem (a generic member of a linear system is smooth outside its base locus) is applied in the cascade of homotopies to find witness sets. We apply Bertini's 2nd theorem (intersecting an irreducible algebraic set of dimension k with k-1 generic hyperplanes yields an irreducible curve) to the factorization of a pure dimensional solution set into irreducible factors. In the second half of the seminar we discussed the use of the linear trace to certify a factorization.
  4. Thu 5 Feb: no meeting. We had a makeup in the form of the Mathematics and Applications Seminar of Wed 28 Jan.
  5. Thu 12 Feb: Olga Kashcheyeva on "Primary Decomposition of Ideals".
  6. Thu 19 Feb: Yusong Wang on "a parallel path tracker in PHCpack".
  7. Thu 26 Feb: no meeting, as three of us attended the SIAM PP 04 meeting in San Francisco.
  8. Thu 4 Mar: Anton Leykin on "introduction to geometric resolution I".
  9. Thu 11 Mar: Anton Leykin on "introduction to geometric resolution II".
  10. Thu 18 Mar: Ailing Zhao on "Newton's method with deflation for isolated singular solutions of polynomial systems".
  11. Thu 25 Mar: no meeting because of Spring break.
  12. Thu 1 Apr: with intrinsic diagonal homotopies we can avoid the doubling of the number of variables when we intersect two irreducible algebraic sets.
  13. Thu 22 Apr: Anton Leykin on "Newton-Hensel Lifting".
  14. Thu 29 Apr: "An incremental solver for polynomial systems".
MCS 595 Graduate Seminar, Fall 2003