| Day | Topics | Sections in IVA |
|---|---|---|
| Mon. 1/24 | Introduction: some problems in algebraic geometry. | . |
| Wed. 1/26 | Polynomial rings and affine varieties. | I.1,2 |
| Mon. 1/31 | Parametrization and Implicitization. | I.2,3. |
| Wed. 2/2 | Ideals in polynomial rings. | I.4. |
| Mon. 2/7 | Univariate polynomial rings versus multivariate polynomial rings. | I.5. |
| Wed. 2/9 | Monomial Orderings | II.1,2 |
| Mon. 2/14 | Division in multivariate polynomial rings. | II.3 |
| Wed. 2/16 | Dickson's Lemma. | II.4 |
| Mon. 2/21 | Hilbert basis theorem, Grobner basis and applications. |
II.5-8 |
| Wed. 2/23 | S-polynomials and Grobner basis. | II.6,8 |
| Mon. 2/28 | Buchberger's algorithm. The syzygy module. | II.6,8,9 |
| Wed. 3/2 | Bases for the syzygy module. | II.9 II.1, 2 |
| Mon. 3/7 | Elimination theorem. Extension theorem Closure theorem |
III.1,2 |
| Wed. 3/9 | Parametrization | III.3 |
| Mon. 3/14 | Sylvester matrix, resultants. | II.5 |
| Wed. 3/16 | Proof of the Extension theorem. | III.6 |
| Mon. 3/21 | Hilbert's Nullstellensatz. The algebra-geometry dictionary. Zariski topology. |
IV.1,2 |
| Wed. 3/23 | Sums, products and intersections of ideals. Ideals and homomorphisms of rings. Computation of the intersection of two ideals. |
IV.3. |
| Mon. 4/4 | Ring homomorphisms and quotients. The coordinate ring of a variety. |
V.1,2. |
| Wed. 4/6 | Finite varieties. Radical, prime, maximal ideals. |
V.3. IV.5 |
| Mon. 4/11 | Irreducible varieties and minimal decomposition. | V.5,6 |
| Wed. 4/13 | Short project presentations. | . |
| Mon. 4/18 | Colon ideal and set difference of two varieties. Project presentations. |
V.4,6. . |
| Wed. 4/20 | p-adic numbers and formal power series. | . |
| Mon. 4/25 | Cone varieties,homogeneous ideals, the cone over a variety. | VIII.1-4. |
| Wed. 4/27 | Projective geometry. | VIII.1-4. |
| Mon. 5/2 | Polynomial maps on a variety. The coordinate ring. | V.1-4. |
| Wed. 5/4 | Rational maps and parametrization. | . |
| Mon. 5/9 | Shayne, Raymond, | V.1-4. |
| Wed. 5/11 | Val, Marcel, Jeremy. | . |
| Mon. 5/16 | Colleen, Dan, David, Alina. | . |