Mathematics

SIMONE DIVERIO
SIMONE DIVERIO
PAOLO BRAVI
PAOLO BRAVI

General objectives: acquiring the techniques of diagonalization of quadratic forms and basic knowledge of affine, euclidean and projective geometry.

Specific objectives:

Knowledge and understanding: at the end of the course students will have acquired basic results on diagonalizability of quadratic forms and of symmetric operators, as well as elemetary notions of affine, euclidean and projective geometry, and of the natural transformations in each of these ambients.

Applying knowledge and understanding: at the end of the course students will be able to solve simple problems requiring the use of diagonalizability of quadratic forms, and to solve elementary problems in affine, euclidean and projective geometry.

Critical and judgmental skills: students will have acquired the necessary maturity to recongnize the close relationship between linear algebra and geometry, with specific reference to the notions acquired during the Linear Algebra course; they will have also acquired the tools to formulate and solve classical geometry problems in a modern language.

Communication skills: ability of exposition with clarity of notions, definitions, theorems and problem solutions during the written and oral part of the exam.

Learning skills: the acquired knowledge will allow the students to undertake with maestry the subsequent study of more technical and abstract geometry theories such as topology and differential geometry.

General objectives: acquiring the techniques of diagonalization of quadratic forms and basic knowledge of affine, euclidean and projective geometry.

Specific objectives:

Knowledge and understanding: at the end of the course students will have acquired basic results on diagonalizability of quadratic forms and of symmetric operators, as well as elementary notions of affine, euclidean and projective geometry, and of the natural transformations in each of these ambients.

Applying knowledge and understanding: at the end of the course students will be able to solve simple problems requiring the use of diagonalizability of quadratic forms, and to solve elementary problems in affine, euclidean and projective geometry.

Critical and judgmental skills: students will have acquired the necessary maturity to recongnize the close relationship between linear algebra and geometry, with specific reference to the notions acquired during the Linear Algebra course; they will have also acquired the tools to formulate and solve classical geometry problems in a modern language.

Communication skills: ability of exposition with clarity of notions, definitions, theorems and problem solutions during the written and oral part of the exam.

Learning skills: the acquired knowledge will allow the students to undertake the subsequent study of more technical and abstract geometry theories such as topology and differential geometry.

A basic course of Linear Algebra is necessary in order to attend the course Geometria I: fundamentals on vector spaces and linear maps until the notion of eigenvalue and eigenvector and the problem of diagonalization (included). No mandatory class prior to this one is required.

1. Complements of Linear Algebra:
1.1 Scalar products, orthoromal bases, Fourier coefficients, projections
1.1 symmetric bilinear forms and Sylvester's theorem
1.2 Hermitian forms
1.3 spectral theorems for self-adjoint operators, normal operators
1.4 unitary operators
2. Affine spaces and Euclidean spaces.
3. Affinity Group and Isometrie Group.
3.1 Analysis of groups of matrices of geometric interest
4. Projective geometry:
4.1 projective spaces, subspaces, relationships between projective spaces and affine spaces.
4.2 projectivity
4.3 projective invariants; cross ratio.
5. Projective, affine and Euclidean classification of conics and quadrics.

Teacher's lecture notes.
Sernesi, Geometria 1, Ed. Boringhieri, Seconda Edizione Riveduta e Ampliata.
Fortuna, Frigerio, Pardini, Geometria proiettiva, Springer
M. Artin, Algebra, Ed. Boringhieri

theoretical course and exercises

In presence. Not mandatory, but strongly recommended.

The exam aims to evaluate learning through a written test (consisting in solving problems) and an oral test (consisting in the discussion of the most relevant topics illustrated in the course). The written test will last about three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place in the middle of the course and the second immediately at the end of the course.
To pass the exam it is necessary to achieve a grade of not less than 18/30. The student must demonstrate that he has acquired sufficient knowledge of all the basic themes dealt with in the course, and that he is able to carry out at least the simplest of the assigned exercises.
To achieve a score of 30/30 cum laude, the student must demonstrate that he has acquired an excellent knowledge of all the topics covered during the course and be able to connect them in a logical and consistent way.

It is not considered useful to provide a list of typical questions: it will be necessary to understand all the basic definitions of the course as well as the main theorems.
The student may be asked to illustrate the main statements on specific examples and to provide examples and counterexamples.
The exercise classes will provide a useful guide for the written tests.