Complements of Geometry

Credits

Prerequisites

Geometry 2.

Scientific-disciplinary sector (SSD)

MAT/03 Geometry.

Examination method

Passing an oral exam.

Learning
objectives

The course aims to introduce the basic notions of various branches of geometry: algebraic geometry, differential geometry, algebraic topology and combinatorial geometry.

Syllabus

Review of general topology and commutative algebra. Introduction to algebraic geometry. Algebraic sets in affine space. Algebraic varieties and their characterisations. Introduction to differential geometry. Differentiable manifolds. Examples of differentiable maps. Differential forms. Tensors and tensor calculus. Riemannian manifolds. Homotopy of functions and of spaces. Paths. Fundamental group of a pointed space. Geometric structure of projective geometry, synthetic and defined from a vector space. Finite projective spaces. Affine geometries. Basic notions of coding theory. Linear codes. Hamming codes. MDS codes.

Expected learning
outcomes

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

know and understand the topics covered in lectures, with particular regard to algebraic varieties and differentiable manifolds, the fundamental group and coding theory;
are able to apply the knowledge acquired in the study and solution of simple problems related to the course topics;
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 related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Complements of Geometry