Unit COMBINATORICS
- Course
- Mathematics
- Study-unit Code
- 55A00045
- Curriculum
- Matematica per la crittografia
- Teacher
- Rita Vincenti
- Teachers
-
- Rita Vincenti
- Hours
- 42 ore - Rita Vincenti
- CFU
- 6
- Course Regulation
- Coorte 2019
- Offered
- 2020/21
- Learning activities
- Affine/integrativa
- Area
- Attività formative affini o integrative
- Academic discipline
- MAT/03
- Type of study-unit
- Opzionale (Optional)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- English
- Contents
- Galois Geometries. Projective varieties. Linear codes and projective systems.
- Reference texts
- R.Vincenti, Finite fields, projective geometries and related topics, Molracchi Ed. 2021
- Educational objectives
- The aim is to address students to research on combinatorial problems and topics.
- Prerequisites
- Algebra and Geometry for the first two years of the bachelor.
- Teaching methods
- The lectures will be accompanied by exercises, examples and open research problems.
- Other information
- In office hours and by arrangement students will be followed in a personalized way.
- Learning verification modality
- The exam may be substituted by a seminar on a shared topics, chosen by the student or may consist of an oral proof on the whole programm. Material for the preparation of the seminars will be found in all the texts that will be recommended and available in our library.
- Extended program
- Galois fields, basis, algebraic extensions, effective constructions, norms and traces, equations.The squares and the non-squares. The finite geometries PG(r,q), r =1, projective incidence properties, duality. The plane PG(2,q), the conics for q odd. The axiomatic projective plane: the ternary ring, increasing algebraic properties in the ternary ring while adding central collineations, translation planes, semifields, quasifields. Partitions, spreads, and translation planes. The linear groups: GL(n,q), PGL(n,q), Sylow subgroups, trasvections, representation of GL(n,q). Projective varieties: quadrics in PG(r,q), r =2, rational normal curves, arcs. Grassmannians. The Veronese variety. Projective systems and linear codes. Applications.