Laurea magistrale in Matematica

Knowledge and understanding

The course uses basic mathematics concepts as an essential tool to work on more advanced contents, in order to understand the historical development and the technical contents of a fundamental moment in the development of the discipline, in particular geometry, also connecting them to recent developments intertwined with technological development. The student then takes up previously acquired concepts, improving their mastery and ability to use them. The course expands the basic knowledge of the Bachelor's Degree, developing skills in abstraction and mastery of the scientific method, and provides a solid preparation in theoretical and applied mathematics.

Therefore students at the end of the lectures will achieve the following objectives: 1, 2, 3, 9. 

Ability to apply knowledge and understanding

The course aims to develop critical thinking in students, the ability to support mathematical reasoning, soliciting reports and short seminars during lessons.

During the course, exercises and didactic activities are proposed to get the student used to apply the studied theory for solving new problems and autonomously producing proofs of statements concerning the topics of the course, possibly making use of appropriate IT tools.

Therefore students at the end of the lectures will achieve the following objectives: 3, 4, 5. 

Making judgements

The students, basing on the learned knowledge, acquire specific skills and competences, in particular they are capable of:

1) starting research activities on specific topics;

2) framing what has been learned within the historical development of mathematics;

3) working in team and doing problem solving activities.

4) using the literature to investigate new problems in an independent way

Therefore students at the end of the lectures will achieve the following objectives: 1, 2, 4, 6. 

Communication skills

The suggested texts for individual learning are all in English, making the student accustomed to the use of English for scientific communications. The exam, both written and oral, forces the student to express himself in a mathematically rigorous way.

Therefore students at the end of the lectures will achieve the following objectives: 1, 2. 

Learning skills

The work required for following the lectures is a useful first step in the development of critical thinking in mathematics and a flexible and useful mentality for third level studies.

Therefore students at the end of the lectures will achieve the following objectives: 1, 2.

At the end of the lectures students know the technical content of the Erlangen program in a modern form. Specifically they know Projective Geometry as founding a family of related geometries : Affinities G.,  Similarities G., Euclidian G., Hyperbolic G.). The students will know and be able to apply the fundamental theorems of all the geometries above.

According to the topic choice they have done they know one of the following subjects:

- elliptic geometry as a subgeometry of projective geometry;

- elements of Computer Vision;

- how visualisation concur to geometric conceptualisation.

Teaching consists of 48 hours of frontal teaching and seminar activities. Lectures are carried out at the blackboard and/or showing slides. The frontal teaching consists of theoretical lessons and exercises. The last 12 hours are chosen by the students attending the lectures of one of the following three topics: Elliptic Geometry, Computer Vision, Visualisation in Geometry.   Attendance is optional but recommended

The assessment of students’ learning is achieved through three tools:

1. Weekly exercises to be carried out during the semester (weight: 20%)
2. A final written task consisting of a problem to be solved and of short reports on the studied topics (weight 40%)
3. An oral part consisting of two questions, randomly drawn from a numbered list, which catalogues all the topics that students have chosen (weight 40%).

NB. Since June the exams will be on line, always with one part for the written part and the other for the oral part. 

Summary of Projective Geometry (both from a synthetic and an analytic standpoint). Transformation groups of affinities, similarities, equiareal, euclidean, hyperbolic geometries, as subgroups of projective transformations.

The students can choos one of the following topics:

- elliptic geometry 

- computer vision: with 1, 2 points of view; computation of the fundamental matrix with the internal and external parameters of a camera.

- visualisation in the processing of geometrical concepts as a cognitive counterpart of the algebraic component.

The lecture notes of the course will be available at the Press Center of the Department and in the Moodle page.

Fishback,W.T., 1969, Projective and Euclidean Geometry, Wiley & Sons: New York

R. Hartley e A. Zisserman, 2003, Multiple View Geometry in Computer Vision, CAMBRIDGE UNIVERSITY PRESS: Cambridge (UK), Second Edition.