Classes in the school will begin promptly at 9:00 on Monday morning (18th June 2007).

The School will end with Friday afternoon's tutorials. Following the very successful format we have used for previous schools, there will be two intensive courses (about 6 hours of instruction in each) plus afternoon tutorials Monday-Friday.

Course 1	Lorenzo Robbiano	Approximate Methods in Commutative Algebra	tutor: Martin Kreuzer
Course 2	Aldo Conca	Betti Numbers and Generic Initial Ideals	tutor: Anna Bigatti

	18	19	20	21	22
9-10:15	lecture 1	lecture 1	lecture 1	lecture 1	lecture 1
11-12:15	lecture 2	lecture 2	lecture 2	lecture 2	lecture 2
14:30-15	CoCoA	CoCoALib			14:45 ApCoCoA
15-16	tutorial 1	tutorial 2	tutorial 1	tutorial 2	tutorial 1
16-17	-	-	-	-	16:20 Certificates group photo
17-18	tutorial 2	tutorial 1	tutorial 2	tutorial 1	tutorial 2
19		dinner

Approximate Methods in Commutative Algebra

A course (with tutorials) by Lorenzo Robbiano and Martin Kreuzer

1. Polynomial System Solving (Lex-GB method, Eigenvalue method)
2. The Buchberger-Möller Algorithm (vanishing ideals of points, BM algorithm)
3. Border Bases (theory, numerical stability, BB algorithm)
4. Approximate Algorithms (approx. BM algorithm, approx. BB algorithm)

References

1. M.Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer, Heidelberg 2000 [especially Section 3.7]
2. M.Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer, Heidelberg 2000 [especially Sections 6.3 and 6.4]
3. D. Cox, Solving equations via algebras, in: A. Dickenstein and I. Emiris (eds.), Solving Polynomial Equations, Springer, Berlin 2005
4. H. Stetter, Numerical Polynomial Algebra, SIAM, Philadelphia 2004
5. D. Heldt, M. Kreuzer, S. Pokutta and H. Poulisse, Approximate computation of zero-dimensional polynomial ideals, Preprint 2006, available at M. Kreuzer's web pages

Betti Numbers and Generic Initial Ideals

A course by Aldo Conca, with tutorials by Anna Bigatti

1. Introduction to the basic invariants: Hilbert functions, Betti numbers, regularity. Initial ideals and deformations.
2. Monomial ideals, strongly stable, Borel fixed, lex-segments and their Betti numbers.
3. Generic initial ideals: existence and main properties.
4. Bounds on Betti numbers Macaulay Theorem, Bigatti-Hullett and Pardue Theorem. Rigidity: Herzog-Hibi-Aramova Theorem and extensions.

Notes

Course 1 Monday slides tutorial Tuesday slides tutorial a tutorial b Wednesday slides cocoa tutorial Thursday slides tutorial Friday slides tutorial

Course 2 Monday slides tutorial Tuesday slides tutorial Wednesday slides tutorial Thursday slides tutorial Friday slides tutorial

slides-cocoa.pdf slides-cocoalib.pdf