Seminar: Constructive Algebraic Geometry

Topics:

• Divisors on curves
• geometric Goppa codes
• Ex.: Hermitian codes.

Seminar: Constructive Algebraic Geometry

Yesterday:

• Divisors on nonsingular projective curves
• Riemann's problem
• Riemann's theorem.

Computational Biology - Lecture 9

Topics:

• Gröbner bases: existence, minimal and reduced bases
• Hilbert's Basis Theorem.

Seminar: Constructive Algebraic Geometry

Topics:

• Nakayama's Lemma and its application to local rings
• Regular local rings and nonsingular varieties.

Seminar: Constructive Algebraic Geometry

Topics:

• Local orders, localization of polynomial rings
• ring of rational functions regular at a point of an affine variety
• standard bases in localized polynomial rings
• application to code ideals.

Computational Biology - Lecture 8

Topics:

• Monomial orders
• division algorithm in polynomial rings
• Dickson's Lemma
• Gröbner bases.

Seminar: Constructive Algebraic Geometry

Topics: Local rings and discrete valuation rings.

Seminar: Constructive Algebraic Geometry

Topics:

• Functions on affine varieties
• rings of regular functions
• valuation rings and basic properties.

Seminar. Constructive Algebraic Geometry

Topics:

• graded rings and homogeneous ideals
• graded modules and graded homomorphisms
• graded resolutions and Hilbert's syzygy theorem
• Hilbert function and Hilbert polynomial.

Seminar: Constructive Algebraic Geometry

Topics:

• Evaluation codes.
• Reed-Muller codes and Reed-Solomon codes as evaluation codes.
• Hermitian codes.
• A result of van Lint on evaluation codes.

Seminar: Constructive Algebraic Geometry

Topics:

• projective n-space
• homogeneous ideals
• projective algebraic sets
• Zariski topology
• irreducibility and dimension