Complements of Geometry

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Complements of Geometry

Credits

6

Prerequisites

Geometry 2.

Scientific-disciplinary sector (SSD)

MAT/03 Geometry.

Examination method

Passing an oral exam.

Learning
objectives

1. to develop and explore in a critical way some topics of general topology;

2. to acquire the fundamental preliminary tools for the study of topological manifolds;

3. to discuss the main proof techniques in the areas described in the previous points;

4. to acquire the ability to place the notions learnt and the most important results in a more application-oriented context.

Syllabus

Plane algebraic curves, affine and projective. Singularities of a plane algebraic curve, intersection of curves and Bezout’s theorems, rational curves, inflection points and Plücker formulas, classification theorems for cubics.

Differentiable and regular curves. Length of a curve, differentiable curves in the Euclidean plane, curvature of a regular curve, evolutes, involutes and osculating circle of a plane curve, skew curves, curvature and torsion of a regular curve, Frenet-Serret equations, osculating, normal and rectifying planes. Canonical representation of a curve, the fundamental theorem for curves in space. Regular surfaces in 3-dimensional space, tangent plane, first and second fundamental forms. Surfaces of revolution and ruled surfaces, orientability of a surface, non-orientable surfaces, Gauss’s Theorema Egregium.

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.