Detailed Schedule -- Seminar in Computational Commutative Algebra

Fall 2006, Prof. Little

As always, topics may be added, deleted, or rearranged during the course of the semester. Any changes will be announced in class and here.

Date	Class Topic	Reading (``IVA'')

8/30	Course Intro, polynomials and affine space	1.1
9/1	Affine varieties	1.2
9/4	Parametrizations of affine varieties	1.3
9/6	Ideals	1.4
9/8	Division in k[x], Euclidean algorithm	1.5
9/11	Lab Day 1 -- The generalized Euclidean algorithm	1.5
9/13	Introduction to Gr"obner bases	2.1
9/15	Monomial orders	2.2
9/18	Division in k[x₁,...,x_n]	2.3
9/20	Monomial ideals and Dickson's Lemma	2.4
9/22	Hilbert Basis Theorem; Gr"obner bases	2.5
9/25	Buchberger's Criterion and Algorithm	2.6, 2.7
9/27	Lab Day 2 -- Computing Gr"obner bases	2.7
9/29	First applications	2.8
10/2	The Elimination Theorem	3.1
10/4	The Extension Theorem	3.1
10/6	Geometry of Elimination	3.2
10/9	No Class -- Columbus Day Break
10/11	Implicitization	3.3
10/13	Lab Day 3 -- Singularities and envelopes (Midterm Problem Set out)	3.4
10/16	The resultant	3.5
10/18	Proof of the Extension Theorem	3.6
10/20	Spare day (Midterm Problem Set due)
10/23	The Nullstellensatz	4.1
10/25	The ideal-variety correspondence	4.2
10/27	More on the ideal-variety correspondence	4.2
10/30	Sums, products, intersections of ideals	4.3
11/1	Zariski closure, irreducibility	4.4, 4.5
11/3	Irreducible decomposition of varieties	4.6
11/6	Coordinate ring of a variety	5.1
11/8	Quotient rings and Gr"obner bases	5.2, 5.3
11/10	Lab Day 4 -- more on quotient rings	5.3
11/13	Applications of Gr"obner bases	Chapter 6
11/15	Applications	Chapter 6
11/17	Applications	Chapter 6
11/20	Applications	Chapter 6
11/22,24	No Class -- Thanksgiving Break	4.1
11/27	Applications	Chapter 6
11/29	Final project presentations
12/1	Final project presentations
12/4	Course wrap-up

The final exam for this course will be given as a take-home problem set.

Last modified: August 30, 2006