Seminar: Constructive Algebraic Geometry

Topics:

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

Discrete Algebraic Structures - Lecture 5

Themen:

• Überabzählbarkeit der Menge der reellen Zahlen
• Kontinuumshypothese
• Elementare Zählprinzipien
• Doppelte Abzählung, Inklusion-Exklusion
• k-Kombinationen von n, Binomialzahl.

Computational Biology - Lecture 5

Topics:

• General algebraic statistical models, DiaNA
• maximum likelihood estimation (MLE)
• fully observed toric Markov model
• fully observed Markov model, MLE.

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.

Discrete Algebraic Structures - Lecture 4

Themen:

• Permutationen vom Grad n, Zykeldarstellung, Signum
• vollständige Induktion
• Peano-Axiome, Aufbau der natürlichen Zahlen
• Abzählbarkeit
• 1. Diagonalisierungsmethode (Cantor).

Computational Biology - Lecture 4

Topics:

• Normal cones and normal fans
• pollytope algebra
• Newton polytopes
• polytope propagation 
• finding the optimal alignments for all scores - example

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.

Discrete Algebraic Structures - Lecture 3

Themen:

• Digraphen, Wege und Kreise,
• Äquivalenzrelationen, Äquivalenzklassen, Quotientenmengen, Partitionen
• Halbordnungen, halbgeordnete Mengen
• Abbildungen
• Gleichheit und Komposition von Abbildungen
• Spezielle Abbildungen

Computational Biology - Lecture 3

Topics - Combinatorial Geometry:

• convex sets and polytopes, affine sets, cones, dimension
• hyperplanes, supporting hyperplanes
• faces, normal cones, and normal fan of polytope.

Seminar: Constructive Algebraic Geometry

Topics:

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

Discrete Algebraic Structures - Lecture 2

Themen:

• Mengen, Mengenverknüpfungen, Rechenregeln, Potenzmenge, Mengensysteme, Antinomien
• Geordnete Paare, Paarmengen, n-Tupel, kartesisches Produkt
• Relationen, homogene Relationen, inverse Relation, Allrelation, Gleichheitsrelation
• Komposition von Relationen, Rechenregeln.

Computational Biology - Lecture 2

Topics:

• Sum-product decomposition of marginal probability
• Tropicalization of sum-product decomposition
• Needleman-Wunsch algorithm, example.