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'') |
|---|---|---|
| 1/23 | Course Intro, polynomials and affine space | 1.1 |
| 1/25 | Affine varieties | 1.2 |
| 1/28 | Parametrizations of affine varieties | 1.3 |
| 1/30 | Ideals | 1.4 |
| 2/1 | Division in k[x], Euclidean algorithm -- Problem set 1 due | 1.5 |
| 2/4 | Lab Day 1 -- The generalized Euclidean algorithm | 1.5 |
| 2/6 | Introduction to Gr"obner bases | 2.1 |
| 2/8 | Monomial orders -- Problem set 2 due | 2.2 |
| 2/11 | Division in k[x1,...,xn] | 2.3 |
| 2/13 | Monomial ideals and Dickson's Lemma | 2.4 |
| 2/15 | Hilbert Basis Theorem; Gr"obner bases -- Problem set 3 due | 2.5 |
| 2/18 | Buchberger's Criterion and Algorithm | 2.6, 2.7 |
| 2/20 | Lab Day 2 -- Computing Gr"obner bases | 2.7 |
| 2/22 | First applications -- Problem set 4 due | 2.8 |
| 2/25 | The Elimination Theorem | 3.1 |
| 2/27 | The Extension Theorem | 3.1 |
| 3/1 | Geometry of Elimination -- Problem set 5 due | 3.2 |
| 3/4,6,8 | No Class -- Spring Break | |
| 3/11 | Implicitization | 3.3 |
| 3/13 | Lab Day 3 -- Singularities and envelopes | 3.4 |
| 3/15 | The resultant -- Problem set 6 due | 3.5 |
| 3/18 | Proof of the Extension Theorem | 3.6 |
| 3/20 | The Nullstellensatz | 4.1 |
| 3/22 | Midterm Exam | Chapters 1 - 3 |
| 3/25 | The ideal-variety correspondence | 4.2 |
| 3/27 | More on the ideal-variety correspondence | 4.2 |
| 3/29 | Sums, products, intersections of ideals -- Problem set 7 due | 4.3 |
| 4/1 | Zariski closure, irreducibility | 4.4, 4.5 |
| 4/3 | Irreducible decomposition of varieties | 4.6 |
| 4/5 | Coordinate ring of a variety -- Problem set 8 due | 5.1 |
| 4/8 | Quotient rings and Gr"obner bases | 5.2, 5.3 |
| 4/10 | Lab Day 4 -- more on quotient rings | 5.3 |
| 4/12 | Applications of Gr"obner bases -- Problem set 9 due | Chapter 6 |
| 4/15 | Applications | Chapter 6 |
| 4/17 | Applications | Chapter 6 |
| 4/19,22,24 | No Class -- Easter Break and Academic Conference | |
| 4/26 | Applications -- Problem set 10 due | Chapter 6 |
| 4/29 | Final project presentations | |
| 5/1 | Final project presentations | |
| 5/3 | Final project presentations | |
| 5/6 | Course wrap-up -- Final project papers due |
Last modified: January 9, 2019