Mathematics

PAOLO PAPI
PAOLO PAPI

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.

At the end of the course the student will have learned to apply linear algebra techniques to deal with problems of affine, Euclidean and projective geometry. He will be familiar with linear submanifolds in arbitrary finite dimensional spaces and with conics and quadrigas in two-dimensional and three-dimensional spaces. He will have received the first notions of projective geometry and the first examples of the action of groups of matrices in a geometric context.

The Linear Algebra course is an essential prerequisite. No mandatory exam prior to this is required

1. Complements of Linear Algebra.
1.1 Scalar products, orthogonal 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 Isometry Group.
3.1 Analysis of certain matrix groups and their geometric interpretation
4. Projective geometry:
4.1 projective spaces, subspaces, relations between projective spaces and related spaces.
4.2 Projectivity
4.3 Projective invariants; birational invariants.
5. Projective, affine and Euclidean classification of conics and quadrics.

Sernesi, Geometria 1, Ed. Boringhieri, Seconda Edizione Riveduta e Ampliata.
Fortuna, Frigerio, Pardini, Geometria proiettiva, Springer

M. Artin, Algebra, Ed. Boringhieri

Lessons and exercise classes

In presence. Not mandatory but strongly recommended.

The exam aims to evaluate learning through a written test (consisting of solving exercises) and an oral test (consisting of the discussion of the most relevant topics illustrated in the course). The written test will last approximately three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place halfway through the course and the second immediately at the end of the course.
To pass the exam you must achieve a grade of no less than 18/30. The student must demonstrate that he has acquired sufficient knowledge of the topics of commutative algebra and number theory, 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 excellent knowledge of all the topics covered during the course and be able to connect them in a logical and coherent way.

It is not considered useful, as it is contrary to the spirit of the basic courses , to provide lists of questions or standard exercises. In any case, it is important that the student knows the exact formulation of definitions and theorems, accompanied by appropriate examples and counterexamples.