Spring 2017

Cox, Little, O'Shea

An Introduction to Computational Algebraic Geometry and Commutative Algebra

4th Ed., Springer-Verlag, 2015.

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Day | Topics | Reading |
---|---|---|

Th. 1/19 | Short intro to course. |
IVA 1.1,2 |

Videos | The polynomial ring in one variable: Main Properties | IVA 1.5, H. Ch. 1-5 |

Tu. 1/24 | Ideals in polynomial rings. Varieties: Solutions of an ideal. |
IVA 1. 2,4-5 |

Videos | Rings, ideals, homomorphisms. The Chinese Remainder Theorem. | H. Ch. 3, 6 |

Th. 1/26 | Parametrization and implicitization. Computing using SageMathCloud |
IVA 1.3,4,IVA 2.1 |

Tu. 1/31 | Monomial Orderings Monomial ideals and Dickson's Lemma |
IVA 2.2 IVA 2.4 |

Th. 2/2 | Groebner basis and division in multivariate polynomial rings. |
IVA 2. 3,5 |

Tu. 2/7 | The ascending chain condition Hilbert basis theorem. |
IVA 2.4 |

Tu. 2/9 | Problems solved with Groebner bases. Computations in Sage. S-polynomials. |
IVA 2.4 |

Tu. 2/14 | Syzygy polynomials Buchberger's algorithm. |
IVA 2.5, 6,7 |

Th. 2/16 | Buchberger's algorithm, proof. | IVA 2.6, 7, 9 |

Tu. 2/21 | TEST: Division, Groebner basis. Systems of representatives for a quotient ring. |
IVA Ch 1, 2 |

Th. 2/23 | Elimination and Extension theorems. Geometrical interpretation. |
IVA 3.1,2 |

Tu. 2/28 | Extension theorems, a proof. Computational examples of elimination and extension. |
IVA 3.1,2 |

Th. 3/2 | Parametrized varieties and implicitization. | IVA 1.3, 2.8(end), 3.3 |

Tu. 3/7 | Radical ideals and Hilbert's Nullstellensatz.
Zariski closure. |
IVA 4.1,2 |

Th. 3/9 | Hilbert's Nullstellensatz.and the algebra geometry dictionary
Radical ideals: computation questions. |
IVA 4.1,2,4 (1st pages), 5 (1st pages) |

Tu. 3/14 | Operations on ideals and geometric analogues. | IVA 4.3 |

Th. 3/16 | Computation of the intersection.
Ideal quotients. |
IVA 4.3,4 |

Tu. 3/21 | Ideal quotients and saturations. Primes and irreducibility. |
IVA 4.4,5 |

Th. 3/23 | Primes and irreducibility The closure theorem. |
IVA 4.4,5 |

Tu. 4/4 | Review problems. | IVA Ch 3, 4. |

Tu. 4/6 | TEST. | IVA Ch 3, 4. |

Tu. 4/11 | Localization/rings of fractions. | Notes. |

Tu. 4/13 | Localization/rings of fractions: Problem solving. | Notes. |

Tu. 4/18 | Cone varieties, homogeneous ideals, the cone over a variety. | VIII.1-4x. |

Th. 4/20 | Projective geometry. | VIII.1-4. |

Tu. 4/25 | Cone varieties, homogeneous ideals, Projective varieties: problem solving. |
Handout. |

Tu. 4/27 | No class: discuss projects with me. | . |

Tu. 5/2 | ?. | . |

Th. 5/4 | No class: discuss projects with me. | . |

Tu. 5/9 (Final 10:30-12:30) | Presentations. | . |