Combinatorial Geometry

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/03 Geometry.

Examination method

Oral exam.

Learning
objectives

The aims of the course are to:

Provide an introduction to geometry over fields.
Provide the tools for the study of Coding Theory, with particular regard to Linear Codes.
Discuss the most important results, and illustrate the main techniques of proof and problem solving.

Syllabus

Review of projective geometry, the axioms of projective space, examples of projective spaces, structure of a projective geometry, quotient geometries, finite projective spaces, affine geometries, an application to communication algorithms. Theorems of Pappus and Desargues in projective spaces P(V) coordinatised by a division ring; homogeneous coordinates; reguli and hyperbolic quadrics of three-dimensional projective space; rational normal curves; proof that every projective geometry of dimension greater than 2 is Desarguesian; Moulton planes. Central collineations; the translation group; proof that every Desarguesian projective space is isomorphic to a space coordinatised by a division ring; collineations and projective collineations. Quadratic sets; index of a quadratic set; the case of low-dimensional spaces; quadratic sets in finite projective spaces; elliptic, parabolic and hyperbolic quadratic sets; outline of quadrics. Basic notions of binary coding theory; Hamming distance; linear codes and their parity-check matrix; Hamming codes and perfect codes; MDS codes and their geometric construction; Reed-Muller codes.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

know and understand the language of combinatorial geometry and have acquired a basic knowledge of the topics presented in the course, with particular regard to the arithmetic of Galois fields
are able to apply the knowledge acquired to the study and solution of concrete examples, making correct use of proof techniques;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Knowledge and skills acquired on the course topics, the student’s presentation skills and command of language, the ability to apply the knowledge acquired to the solution of simple problems, the ability to support a discussion with examples and counterexamples, and command of the mathematical tools used in the course.

Combinatorial Geometry